2.1 Topological modules

Recall that a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module $\mathsf {M}$ is separated if the intersection of the family of submodules ${\hbar }^n\mathsf {M}$ is trivial, and it is complete if the natural $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -linear map

$$ \begin{align*} \mathsf{M}\to \varprojlim_n( \mathsf{M}/{\hbar}^n \mathsf{M}) \end{align*} $$

is surjective, where the inverse limit is taken over the set $\mathbb {N}$ of nonnegative integers. In particular, $\mathsf {M}$ is both separated and complete if and only if the above map is an isomorphism. If $\mathsf {M}$ is separated, complete, and torsion-free as a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module, then it is said to be topologically free. This is equivalent to the existence of a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module isomorphism $\mathsf {M}\cong \mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ for a complex vector space $\mathsf {V}$ . Such an isomorphism is specified by a choice of complement $\mathsf {V}\subset \mathsf {M}$ to ${\hbar }\mathsf {M}$ :

More generally, if $\mathsf {M}$ is any $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module, then the space $\mathsf {V}=\mathsf {M}/{\hbar }\mathsf {M}$ is called the semiclassical limit of $\mathsf {M}$ . Similarly, the semiclassical limit of a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -linear map is the $\mathbb {C}$ -linear map uniquely determined by the commutativity of the diagram

As the following elementary result illustrates, the semiclassical limit of a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module homomorphism encodes important information about the original map.

The topological tensor product $\mathsf {M} \,{\widehat {\otimes }}\, \mathsf {N}$ of two $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules $\mathsf {M}$ and $\mathsf {N}$ is the ${\hbar }$ -adic completion of the algebraic tensor product $\mathsf {M}\otimes _{\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]}\mathsf {N}$ :

$$ \begin{align*} \mathsf{M} \,{\widehat{\otimes}}\, \mathsf{N}= \varprojlim_{n}(\mathsf{M}\otimes_{\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]}\mathsf{N}) /{\hbar}^n (\mathsf{M}\otimes_{\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]}\mathsf{N}). \end{align*} $$

If $\mathsf {M}$ and $\mathsf {N}$ are topologically free with $\mathsf {M}\cong \mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ and $\mathsf {N}\cong \mathsf {W}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ , then $\mathsf {M} \,{\widehat {\otimes }}\, \mathsf {N}$ is topologically free and isomorphic to $(\mathsf {V}\otimes _{\mathbb {C}} \mathsf {W})[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ .

In this article, we shall say that $\mathsf {M}$ is a topological module over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ if $\mathsf {M}$ is a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module which is both separated and complete. For any such module, we have

$$ \begin{align*} \mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]\,\widehat{\otimes}\, \mathsf{M} \cong \mathsf{M} \cong \mathsf{M}\,\widehat{\otimes}\, \mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]. \end{align*} $$

Similarly, by a topological algebra $\mathsf {A}$ over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ , we shall always mean that $\mathsf {A}$ is a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra which is both separated and complete as a module over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ . In particular, the multiplication m can be viewed as a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -linear map

$$ \begin{align*} m: \mathsf{A}\,{\widehat{\otimes}}\,\mathsf{A}\to \mathsf{A}. \end{align*} $$

A topological Hopf algebra $\mathsf {H}$ over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ is a topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra equipped with a counit $\varepsilon :\mathsf {H}\to \mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ , a coproduct $\Delta :\mathsf {H}\to \mathsf {H} \, \widehat {\otimes }\, \mathsf {H}$ , and an antipode $S:\mathsf {H}\to \mathsf {H}$ which collectively satisfy the axioms of a Hopf algebra with all tensor products given by the topological tensor product $\widehat {\otimes }$ . By modifying these definitions in the expected way, one obtains the notion of a topological coalgebra and bialgebra over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ .

If $\mathsf {M}$ and $\mathsf {N}$ are topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules and $\mathsf {N}\cong \mathsf {W}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ is topologically free, then the space of $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module homomorphisms $\mathrm {Hom}_{\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]}(\mathsf {M},\mathsf {N})$ is separated, complete, and torsion-free. If, in addition, $\mathsf {M}$ is topologically free with $\mathsf {M}\cong \mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ , then one has

$$ \begin{align*} \mathrm{Hom}_{\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]}(\mathsf{M},\mathsf{N})\cong \mathrm{Hom}_{\mathbb{C}}(\mathsf{V},\mathsf{W})[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]. \end{align*} $$

In particular, the $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -linear dual $\mathsf {M}^\ast :=\mathrm {Hom}_{\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]}(\mathsf {M},\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt])$ of a topologically free $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module $\mathsf {M}\cong \mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ satisfies $\mathsf {M}^\ast \cong \mathsf {V}^\ast [\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ .

2.2 Graded topological modules

Let us now turn toward the $\mathbb {Z}$ -graded analogues of the above definitions. Henceforth, we view $\mathbb {C}[{\hbar }]=\bigoplus _{k\in \mathbb {N}}\mathbb {C}{\hbar }^k$ as an $\mathbb {N}$ -graded ring. For brevity, we shall denote its $\mathbb {N}$ -graded quotient $\mathbb {C}[{\hbar }]/{\hbar }^n \mathbb {C}[{\hbar }]$ by $\mathsf {K}_n$ , for each $n\in \mathbb {N}$ .

Definition 2.2 We say that a topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module $\mathsf {M}$ is $\mathbb {Z}$ -graded if, for each $n\in \mathbb {N}$ , $\mathsf {M}/{\hbar }^n \mathsf {M}=\bigoplus _{k\in \mathbb {Z}}(\mathsf {M}/{\hbar }^n\mathsf {M})_k$ is a $\mathbb {Z}$ -graded $\mathsf {K}_n$ -module and the natural homomorphism

$$ \begin{align*} \mathsf{M}/{\hbar}^{n+1} \mathsf{M}\to \mathsf{M}/{\hbar}^{n} \mathsf{M} \end{align*} $$

is $\mathbb {Z}$ -graded. If $(\mathsf {M}/{\hbar }^n\mathsf {M})_k$ is trivial for $k<0$ , we say that $\mathsf {M}$ is $\mathbb {N}$ -graded.

A $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module homomorphism $\mathsf {M}\to \mathsf {N}$ between $\mathbb {Z}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules $\mathsf {M}$ and $\mathsf {N}$ is said to be $\mathbb {Z}$ -graded if the induced morphisms

$$ \begin{align*} \mathsf{M}/{\hbar}^n\mathsf{M} \to \mathsf{N}/{\hbar}^n\mathsf{N} \end{align*} $$

are all $\mathbb {Z}$ -graded. More generally, it is $\mathbb {Z}$ -graded of degree $a\in \mathbb {Z}$ if each of these induced morphisms is homogeneous of degree a.

The category of $\mathbb {Z}$ -graded topological modules is closed under the tensor product $\widehat {\otimes }$ . Indeed, this follows from the elementary observation that, given two $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules $\mathsf {M}$ and $\mathsf {N}$ , one has

$$ \begin{align*} (\mathsf{M}\, \widehat{\otimes}\, \mathsf{N})/ {\hbar}^n (\mathsf{M}\,\widehat{\otimes}\, \mathsf{N}) \cong (\mathsf{M} \otimes_{\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]} \mathsf{N})/ {\hbar}^n (\mathsf{M} \otimes_{\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]} \mathsf{N}) \cong \mathsf{M}/{\hbar}^n \mathsf{M} \otimes_{\mathsf{K}_n} \mathsf{N}/{\hbar}^n \mathsf{N}, \end{align*} $$

which can be equipped with the standard tensor product grading, provided $\mathsf {M}/{\hbar }^n\mathsf {M}$ and $\mathsf {N}/{\hbar }^n\mathsf {N}$ are both $\mathbb {Z}$ -graded.

Definition 2.3 A topological algebra $\mathsf {A}$ is said to be $\mathbb {Z}$ -graded if it is graded as a topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module and the multiplication map

$$ \begin{align*} m:\mathsf{A}\,\widehat{\otimes}\,\mathsf{A}\to \mathsf{A} \end{align*} $$

is a $\mathbb {Z}$ -graded homomorphism. Similarly, a topological Hopf algebra $\mathsf {H}$ is $\mathbb {Z}$ -graded if it is $\mathbb {Z}$ -graded as a topological algebra and the structure maps

$$ \begin{align*} \Delta:\mathsf{H}\to \mathsf{H}\,\widehat{\otimes}\,\mathsf{H},\quad \varepsilon:\mathsf{H}\to \mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt], \quad S:\mathsf{H}\to \mathsf{H} \end{align*} $$

are all $\mathbb {Z}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module homomorphisms. Equivalently, a topological algebra or Hopf algebra $\mathsf {H}$ is $\mathbb {Z}$ -graded if the conditions of Definition 2.2 hold and each $\mathsf {H}/{\hbar }^n\mathsf {H}$ is a $\mathbb {Z}$ -graded algebra or Hopf algebra over $\mathsf {K}_n$ , respectively.

Of course, one also has the notion of a $\mathbb {Z}$ -graded topological coalgebra and bialgebra, which are defined by making the obvious modifications to the above definition.

The prototypical example of a $\mathbb {Z}$ -graded topological module over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ is $\mathsf {M}=\mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ where $\mathsf {V}=\bigoplus _{k\in \mathbb {Z}}\mathsf {V}_k$ is a $\mathbb {Z}$ -graded complex vector space. In this case, one has

$$ \begin{align*} \mathsf{M}/{\hbar}^n\mathsf{M} \cong \mathsf{V}[{\hbar}]/{\hbar}^n \mathsf{V}[{\hbar}], \end{align*} $$

which is naturally graded, as $\mathsf {V}[{\hbar }]$ is graded with $\mathsf {V}[{\hbar }]_k= \bigoplus _{n\geq 0} {\hbar }^n \mathsf {V}_{k-n}$ , and ${\hbar }^n\mathsf {V}[{\hbar }]$ is a graded submodule. The assertion that $\mathsf {M}$ is $\mathbb {Z}$ -graded may be recaptured as follows. For each $k\in \mathbb {Z}$ , set

$$ \begin{align*} \mathsf{M}_k:=\varprojlim_n \mathsf{V}[{\hbar}]_k/{\hbar}^n \mathsf{V}[{\hbar}]_{k-n} \cong \prod_{n\in \mathbb{N}} {\hbar}^n \mathsf{V}_{k-n} \subset \mathsf{V}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]. \end{align*} $$

Then each $\mathsf {M}_k$ is a closed subspace of $\mathsf {M}$ satisfying $\mathsf {M}_k/{\hbar }^n \mathsf {M}_{k-n}\cong (\mathsf {M}/{\hbar }^n\mathsf {M})_k$ , and $\mathsf {M}$ contains the $\mathbb {Z}$ -graded $\mathbb {C}[{\hbar }]$ -module $\bigoplus _{k\in \mathbb {Z}}\mathsf {M}_k$ as a dense $\mathbb {C}[{\hbar }]$ -submodule. Moreover, the ${\hbar }$ -adic topology on this submodule coincides with the subspace topology, so $\mathsf {M}$ is the ${\hbar }$ -adic completion of $\bigoplus _{k\in \mathbb {Z}}\mathsf {M}_k$ . If, in addition, $\mathsf {V}$ is $\mathbb {N}$ -graded, then $\bigoplus _{k\in \mathbb {Z}}\mathsf {M}_k$ coincides with the polynomial space $\mathsf {V}[{\hbar }]\subset \mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ .

The below lemma provides an equivalent characterization of the definition of a $\mathbb {Z}$ -graded topological module and algebra which generalizes this picture.

Lemma 2.4 A topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module $\mathsf {M}$ is $\mathbb {Z}$ -graded if and only if it admits a dense, $\mathbb {Z}$ -graded $\mathbb {C}[{\hbar }]$ -submodule

$$ \begin{align*} \mathsf{M}_{\mathbb{Z}}=\bigoplus_{k\in \mathbb{Z}}\mathsf{M}_k \subset \mathsf{M} \end{align*} $$

with each $\mathsf {M}_k$ a closed subspace of $\mathsf {M}$ and ${\hbar }^n \mathsf {M}\cap \mathsf {M}_{\mathbb {Z}}={\hbar }^n\mathsf {M}_{\mathbb {Z}}$ for all $n\in \mathbb {N}$ .

If, in addition, $\mathsf {M}$ has the structure of a topological algebra, then it is $\mathbb {Z}$ -graded if and only if the above conditions hold and $\mathsf {M}_{\mathbb {Z}}$ is a $\mathbb {Z}$ -graded $\mathbb {C}[{\hbar }]$ -subalgebra of $\mathsf {M}$ .

If $\mathsf {M}$ is a $\mathbb {Z}$ -graded topological module, then the kth component $\mathsf {M}_k$ of $\mathsf {M}_{\mathbb {Z}}$ from Lemma 2.4 is uniquely determined and recovered as the inverse limit

(2.1) $$ \begin{align} \mathsf{M}_k:=\varprojlim_n (\mathsf{M}/{\hbar}^n \mathsf{M})_k \subset \varprojlim_{n} \mathsf{M}/{\hbar}^n \mathsf{M} =\mathsf{M}. \end{align} $$

Moreover, one has $(\mathsf {M}/{\hbar }^n \mathsf {M})_k\cong \mathsf {M}_k/{\hbar }^n \mathsf {M}_{k-n}$ for all $n\in \mathbb {N}$ and $k\in \mathbb {Z}$ .

We further observe that, for each $k\in \mathbb {Z}$ , the system of linear projections $\pi _{n,k}:\mathsf {M}/{\hbar }^n\mathsf {M}\to (\mathsf {M}/{\hbar }^n\mathsf {M})_k$ gives rise to a projection

$$ \begin{align*} \pi_k:=\varprojlim_n \pi_{n,k}: \mathsf{M}\to \mathsf{M}_k \end{align*} $$

which restricts to the projection of $\mathsf {M}_{\mathbb {Z}}$ onto its kth homogeneous component. In particular, a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -linear map between $\mathbb {Z}$ -graded topological modules is graded if and only if $\pi _k^{\mathsf {N}}\circ \unicode{x3c4} =\unicode{x3c4} \circ \pi _k^{\mathsf {M}}$ for each $k\in \mathbb {Z}$ .

We conclude this preliminary subsection with two corollaries of the above discussion. The first shows that any topologically free $\mathbb {Z}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module is of the form described above Lemma 2.4.

Corollary 2.5 Suppose $\mathsf {M}$ is a $\mathbb {Z}$ -graded topologically free module over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ , and let $\mathsf {V}$ denote the $\mathbb {Z}$ -graded complex vector space $\mathsf {M}/{\hbar }\mathsf {M}=\bigoplus _{k\in \mathbb {Z}} \mathsf {M}_k/{\hbar } \mathsf {M}_{k-1}$ . Then

$$ \begin{align*} \mathsf{M}\cong \mathsf{V}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt] \end{align*} $$

as a $\mathbb {Z}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module. In particular, one has

$$ \begin{align*} \mathsf{M}_{\mathbb{Z}}=\bigoplus_{k\in \mathbb{Z}} \mathsf{M}_k \cong \bigoplus_{k\in \mathbb{Z}} \mathsf{V}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]_k \subset \mathsf{V}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt], \quad \text{ where }\;\mathsf{V}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]_k=\prod_{n\in \mathbb{N}} {\hbar}^n \mathsf{V}_{k-n}. \end{align*} $$

Proof This is a refinement of the elementary result, alluded to at the beginning of the section, that $\mathsf {M}\cong \mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ as a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module (see [Reference Kassel26, Proposition XVI.2.4], for instance). In more detail, an isomorphism of $\mathbb {Z}$ -graded topological modules $\mathsf {M}\cong \mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ is specified by choosing, for each $k\in \mathbb {Z}$ , a complement $\mathsf {V}_k\subset \mathsf {M}_k$ to ${\hbar }\mathsf {M}_{k-1}$ in $\mathsf {M}_k$ :

Setting $\mathsf {V}:=\bigoplus _{k\in \mathbb {Z}}\mathsf {V}_k\subset \mathsf {M}_{\mathbb {Z}}$ , we then have

where the third line is an identification of $\mathbb {Z}$ -graded modules. Here, we note that the second line follows from the definition of $\mathsf {V}_k$ and that $\mathsf {M}$ is a torsion-free $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module. Taking inverse limits, one finds that $\mathsf {M}\cong \mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ as $\mathbb {Z}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules.

Let us now shift our attention to the case where $\dot {\mathsf {M}}=\bigoplus _{k\in \mathbb {N}} \dot {\mathsf {M}}_k$ is an $\mathbb {N}$ -graded $\mathbb {C}[{\hbar }]$ -module. Any such module is automatically separated, and so embeds into its ${\hbar }$ -adic completion

$$ \begin{align*} \mathsf{M}=\varprojlim_n (\dot{\mathsf{M}}/{\hbar}^n \dot{\mathsf{M}}), \end{align*} $$

which is an $\mathbb {N}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module. Moreover, if $\dot {\mathsf {M}}$ is a torsion-free $\mathbb {C}[{\hbar }]$ -module, then $\mathsf {M}$ is topologically free as a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module. Since ${\dot {\mathsf {M}}}_{k-n}$ is trivial for $n>k$ , the submodule $\mathsf {M}_k$ of $\mathsf {M}$ (see (2.1)) coincides with ${\dot {\mathsf {M}}}_k$ and so, in the notation of Lemma 2.4, one has $\dot {\mathsf {M}}=\mathsf {M}_{\mathbb {N}}$ . These observations, coupled with Corollary 2.5 and that $\mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]_k=\mathsf {V}[{\hbar }]_k$ when $\mathsf {V}$ is $\mathbb {N}$ -graded, yield the following.

Corollary 2.6 Let $\dot {\mathsf {M}}$ be an $\mathbb {N}$ -graded, torsion-free $\mathbb {C}[{\hbar }]$ -module. Then $\mathsf {M}$ is a topologically free $\mathbb {N}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module. Moreover, we have

$$ \begin{align*} {\dot{\mathsf{M}}}_k=\varprojlim_{n}({\dot{\mathsf{M}}}_k/{\hbar}^n {\dot{\mathsf{M}}}_{k-n})=\mathsf{M}_k \quad \text{ for all }\; k\in \mathbb{N}. \end{align*} $$

Consequently, $\dot {\mathsf {M}}$ coincides with $\mathsf {M}_{\mathbb {N}}$ and there is an isomorphism of $\mathbb {N}$ -graded $\mathbb {C}[{\hbar }]$ -modules

$$ \begin{align*} \dot{\mathsf{M}}\cong \mathsf{V}[{\hbar}]=\bigoplus_{k\in \mathbb{N}}\mathsf{V}[{\hbar}]_k, \quad \text{ where }\; \mathsf{V}:=\dot{\mathsf{M}}/{\hbar}\dot{\mathsf{M}}. \end{align*} $$

Note that if $\dot {\mathsf {M}}$ is an $\mathbb {N}$ -graded $\mathbb {C}[{\hbar }]$ -algebra or Hopf algebra, then $\mathsf {M}$ is automatically an $\mathbb {N}$ -graded topological algebra or Hopf algebra, respectively.

2.3 The restricted dual

For a given $\mathbb {Z}$ -graded complex vector space $\mathsf {V}=\bigoplus _n \mathsf {V}_n$ , we let $\mathsf {V}^\star =\bigoplus _n (\mathsf {V}^\star )_n\subset \mathsf {V}^\ast $ denote the restricted, or graded, dual of $\mathsf {V}$ , where

$$ \begin{align*} (\mathsf{V}^\star)_n=\{f\in \mathsf{V}^\ast: f(\mathsf{V}_{m})\subset \mathbb{C}_{m+n}\}\cong (\mathsf{V}_{-n})^\ast \end{align*} $$

and $\mathbb {C}$ is given the trivial grading with $\mathbb {C}_0=\mathbb {C}$ and $\mathbb {C}_m=\{0\}$ for $m\neq 0$ . One can similarly define the restricted dual $\mathsf {M}^\star \subset \mathsf {M}^\ast $ in the category of $\mathbb {Z}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules. In this subsection, we will recall some properties of this duality operation in the $\mathbb {N}$ -graded setting which will be applied to construct the dual Yangian in Section 6.

Suppose that $\mathsf {M}$ is an $\mathbb {N}$ -graded, topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module with $\mathbb {N}$ -graded $\mathbb {C}[{\hbar }]$ -submodule $\mathsf {M}_{\mathbb {N}}=\bigoplus _{k\in \mathbb {N}}\mathsf {M}_k$ as in Lemma 2.4. For each $n\in \mathbb {N}$ , let $\mathsf {J}_k$ denote the ${\hbar }$ -adic completion of the ideal $\bigoplus _{k\geq n}\mathsf {M}_k$ of $\mathsf {M}_{\mathbb {N}}$ . The gradation topology on $\mathsf {M}$ is the topology associated with the descending filtration

$$ \begin{align*} \mathsf{M}=\mathsf{J}_0\supset \mathsf{J}_1 \supset \cdots \supset \mathsf{J}_n \supset \cdots. \end{align*} $$

Equipped with this terminology, we may make the following definition.

Definition 2.7 The restricted dual $\mathsf {M}^\star $ is defined to be the $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -submodule of $\mathsf {M}^\ast $ consisting of those f which are continuous with respect to the gradation topology:

$$ \begin{align*} \mathsf{M}^\star:=\{f\in \mathsf{M}^\ast: f(\mathsf{J}_k)\subset {\hbar}^n\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt] \quad \forall\; n\in \mathbb{N} \;\text{ and }\; k\gg0\}. \end{align*} $$

The restricted dual of any $\mathbb {N}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module $\mathsf {M}$ is easily seen to be separated, complete, and torsion-free. Let us now see that it admits a $\mathbb {Z}$ -graded structure.

For each $a\in \mathbb {Z}$ , let $\mathrm {Hom}_{\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]}^a(\mathsf {M},\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt])\subset \mathsf {M}^\star $ denote the (closed) subspace consisting of $\mathbb {Z}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module homomorphisms $f:\mathsf {M}\to \mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ of degree a. Equivalently,

$$ \begin{align*} \mathrm{Hom}_{\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]}^a(\mathsf{M},\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt])=\{f\in \mathsf{M}^\ast: f(\mathsf{M}_k)\subset \mathbb{C}[{\hbar}]_{k+a} \quad \forall\; k\in \mathbb{N}\}. \end{align*} $$

Then the sum $\sum _{a\in \mathbb {Z}}\mathrm {Hom}_{\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]}^a(\mathsf {M},\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt])$ is direct and the space

$$ \begin{align*} (\mathsf{M}^\star)_{\mathbb{Z}}:=\bigoplus_{a\in \mathbb{Z}}\mathrm{Hom}_{\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]}^a(\mathsf{M},\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]) \subset \mathsf{M}^\star \end{align*} $$

is a $\mathbb {Z}$ -graded $\mathbb {C}[{\hbar }]$ -submodule of $\mathsf {M}^\star $ .

It is not difficult to prove that, for each $n\in \mathbb {N}$ , one has

$$ \begin{align*} {\hbar}^n\mathsf{M}^\star \cap (\mathsf{M}^\star)_{\mathbb{Z}}={\hbar}^n(\mathsf{M}^\star)_{\mathbb{Z}} \quad \text{ and }\quad (\mathsf{M}^\star)_{\mathbb{Z}}/{\hbar}^n (\mathsf{M}^\star)_{\mathbb{Z}} \cong \mathsf{M}^\star/{\hbar}^n \mathsf{M}^\star. \end{align*} $$

Consequently, $\mathsf {M}^\star $ coincides with the ${\hbar }$ -adic completion of $(\mathsf {M}^\star )_{\mathbb {Z}}$ and, by Lemma 2.4, is a $\mathbb {Z}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module. We note that, although Definition 2.7 is strictly for an $\mathbb {N}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module $\mathsf {M}$ , one can define the restricted dual in the $\mathbb {Z}$ -graded setting precisely as the ${\hbar }$ -adic completion of the space $(\mathsf {M}^\star )_{\mathbb {Z}}$ .

If $\mathsf {M}$ is itself topologically free with $\mathsf {M}\cong \mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ for a graded vector space ${\mathsf {V}=\bigoplus _{k\in \mathbb {N}}\mathsf {V}_n}$ , then the natural homomorphism $\mathsf {M}^\star /{\hbar } \mathsf {M}^\star \to \mathsf {V}^\star $ is an isomorphism of graded vector spaces. As $\mathsf {M}^\star $ is topologically free, Corollary 2.5 yields the following.

We shall say that a topologically free $\mathbb {N}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module is of finite type if the graded components $\mathsf {V}_k$ of $\mathsf {V}\cong \mathsf {M}/{\hbar }\mathsf {M}$ from the above corollary are all finite-dimensional complex vector spaces.

Now suppose that $\mathsf {H}$ is an $\mathbb {N}$ -graded topological Hopf algebra with coproduct $\Delta $ , counit $\varepsilon $ , antipode S, product m, and unit $\iota $ . Since these are all $\mathbb {N}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module homomorphisms and $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]^\star \cong \mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ , taking transposes yields $\mathbb {Z}$ -graded maps

$$ \begin{gather*} \Delta^t: (\mathsf{H}\, \widehat{\otimes}\,\mathsf{H})^\star\to \mathsf{H}^\star,\quad \varepsilon^t: \mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]\to \mathsf{H}^\star, \quad S^t:\mathsf{H}^\star\to \mathsf{H}^\star \\ m^t:\mathsf{H}^\star\to(\mathsf{H}\, \widehat{\otimes}\,\mathsf{H})^\star, \quad \iota^t: \mathsf{H}^\star \to \mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt] \end{gather*} $$

which formally satisfy the axioms of a Hopf algebra. In particular, $\mathsf {H}^\star $ is a topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra with unit $\varepsilon ^t$ and product given by restricting $\Delta ^t$ . It is not, in general, a topological coalgebra (or Hopf algebra) as $m^t$ does not necessarily have image in $\mathsf {H}^\star \, \widehat {\otimes }\,\mathsf {H}^\star $ . However, this is the case when $\mathsf {H}$ is of finite type, as we now explain.

In general, for any two $\mathbb {N}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules $\mathsf {M}$ and $\mathsf {N}$ , there is a canonical injective homomorphism of $\mathbb {Z}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules

$$ \begin{align*} \unicode{x3b3}:\mathsf{M}^\star\, \widehat{\otimes}\,\mathsf{N}^\star \hookrightarrow (\mathsf{M}\, \widehat{\otimes}\,\mathsf{N})^\star. \end{align*} $$

If $\mathsf {M}$ and $\mathsf {N}$ are topologically free with $\mathsf {M}\cong \mathsf {V}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ and $\mathsf {N}\cong \mathsf {W}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ , then the semiclassical limit of $\unicode{x3b3} $ is the natural inclusion $\mathsf {V}^\star \otimes _{\mathbb {C}} \mathsf {W}^\star \hookrightarrow (\mathsf {V}\otimes _{\mathbb {C}} \mathsf {W})^\star $ which is an isomorphism provided the graded components of $\mathsf {V}$ or $\mathsf {W}$ are all finite-dimensional. This observation, together with Lemma 2.1, implies the following proposition.

Proposition 2.10 Let $\mathsf {M}$ and $\mathsf {N}$ be topologically free, $\mathbb {N}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules and suppose that either $\mathsf {M}$ or $\mathsf {N}$ is of finite type. Then $\unicode{x3b3} $ is an isomorphism of $\mathbb {Z}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules

Consequently, if $\mathsf {H}$ is a topologically free $\mathbb {N}$ -graded Hopf algebra of finite type, then $\mathsf {H}^\star $ is a $\mathbb {Z}$ -graded topological Hopf algebra over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ .

2.4 Homogeneous quantizations

Let us now recall some basic constructions from the theory of quantum groups, adapted to the graded setting.

A topological Hopf algebra $\mathsf {H}$ over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ is called a quantized enveloping algebra if it is a flat deformation of the universal enveloping algebra $U(\mathfrak {b})$ of a complex Lie algebra $\mathfrak {b}$ as a Hopf algebra. Equivalently:

• • The semiclassical limit $\mathsf {H}/{\hbar }\mathsf {H}$ of $\mathsf {H}$ is isomorphic to $U(\mathfrak {b})$ as a Hopf algebra.

• • $\mathsf {H}$ is topologically free, and thus isomorphic to $U(\mathfrak {b})[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ as a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module.

If $\mathsf {H}=U_{\hbar }\mathfrak {b}$ is a quantized enveloping algebra with semiclassical limit $U(\mathfrak {b})$ , then $\mathfrak {b}$ inherits from $U_{\hbar }\mathfrak {b}$ the structure of a Lie bialgebra with cocommutator $\delta _{\mathfrak {b}}:\mathfrak {b}\to \mathfrak {b}\wedge \mathfrak {b}\subset U(\mathfrak {b})^{\otimes 2}$ given by the formula

(2.2)

where $\dot {x}\in U_{\hbar }\mathfrak {b}$ is any lift of x. We refer the reader to Propositions 6.2.3 and 6.2.7 of [Reference Chari and Pressley4] for a detailed discussion of this point.

Conversely, if $(\mathfrak {b},\delta _{\mathfrak {b}})$ is a Lie bialgebra, then a quantization of $(\mathfrak {b},\delta _{\mathfrak {b}})$ is a quantized enveloping algebra $U_{\hbar }\mathfrak {b}$ with semiclassical limit $U(\mathfrak {b})$ , such that $\delta _{\mathfrak {b}}$ coincides with the cocommutator (2.2).

Now let us shift our attention to the graded setting. In what follows, we will say that a Lie bialgebra $(\mathfrak {b},\delta _{\mathfrak {b}})$ is $\mathbb {Z}$ -graded if $\mathfrak {b}=\bigoplus _{k\in \mathbb {Z}}\mathfrak {b}_k$ is $\mathbb {Z}$ -graded as a Lie algebra, and the cocommutator $\delta _{\mathfrak {b}}$ is a graded linear map of degree d, for some $d\in \mathbb {Z}$ . That is, $\delta _{\mathfrak {b}}\in \mathrm {Hom}_{\mathbb {C}}^d(\mathfrak {b},{\mathfrak {b}}^{\otimes 2})$ .

Note that the last condition guarantees that the grading on $U(\mathfrak {b})$ inherited from $U_{\hbar }\mathfrak {b}$ coincides with that induced by the Lie algebra grading on $\mathfrak {b}$ . In addition, since the coproduct $\Delta $ is homogeneous of degree zero, (2.2) implies that the cocommutator $\delta _{\mathfrak {b}}$ must be of degree $d=-1$ .

We shall employ similar terminology to the above in the $\mathbb {N}$ -graded setting over $\mathbb {C}[{\hbar }]$ . Namely, if $(\mathfrak {b},\delta _{\mathfrak {b}})$ is an $\mathbb {N}$ -graded Lie bialgebra, then a homogeneous quantization of $(\mathfrak {b},\delta _{\mathfrak {b}})$ over $\mathbb {C}[{\hbar }]$ is a space $ {U}_{\hbar }\mathfrak {b}$ such that:

1. (1) $ {U}_{\hbar }\mathfrak {b}$ is an $\mathbb {N}$ -graded torsion-free Hopf algebra over $\mathbb {C}[{\hbar }]$ .

2. (2) The semiclassical limit $ {U}_{\hbar }\mathfrak {b}/{\hbar } {U}_{\hbar }\mathfrak {b}$ is isomorphic to $U(\mathfrak {b})$ as a graded Hopf algebra, with the cocommutator $\delta _{\mathfrak {b}}$ given by (2.2).

Note that, by Corollary 2.6, such a quantization $ {U}_{\hbar }\mathfrak {b}$ is isomorphic to $U(\mathfrak {b})[{\hbar }]$ as an $\mathbb {N}$ -graded $\mathbb {C}[{\hbar }]$ -module, and its ${\hbar }$ -adic completion is a homogeneous quantization of $\mathfrak {b}$ over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ .

2.5 The Yangian Manin triple

The most well-known, nontrivial, example of a homogeneous quantization is the Yangian $Y_{\hbar }\mathfrak {g}$ associated with an arbitrary simple Lie algebra $\mathfrak {g}$ over the complex numbers. In this article, we shall encounter two other, closely related, examples: the dual Yangian ${\dot {\mathsf {Y}}}_{\hbar }\mathfrak {g}{\vphantom {)}}^\star $ and the Yangian double $\mathrm {D}Y_{\hbar }\mathfrak {g}$ . Collectively, these three quantum groups arise as a quantization of a restricted Manin triple structure on $(\mathfrak {t},{\mathfrak {t}}_{\scriptscriptstyle {+}},{\mathfrak {t}}_{\scriptscriptstyle {-}})$ , where

$$ \begin{align*} \mathfrak{t}:=\mathfrak{g}[t^{\pm 1}],\quad {\mathfrak{t}}_{\scriptscriptstyle{+}}:=\mathfrak{g}[t],\quad \text{ and }\quad {\mathfrak{t}}_{\scriptscriptstyle{-}}:=t^{-1}\mathfrak{g}[t^{-1}]. \end{align*} $$

In this section, we briefly recall how this structure is defined.

The Lie algebra $\mathfrak {t}=\mathfrak {g}[t^{\pm 1}]$ comes equipped with a nondegenerate, invariant bilinear form $\langle \,,\,\rangle :\mathfrak {t}\otimes \mathfrak {t}\to \mathbb {C}$ given by

(2.3) $$ \begin{align} \langle f(t), g(s)\rangle:= -\mathrm{Res}_t(f(t),g(t)), \end{align} $$

where $(\,,\,)$ is a fixed symmetric, invariant, and nondegenerate bilinear form on $\mathfrak {g}$ , which has been extended to a $\mathbb {C}[t^{\pm 1}]$ -valued bilinear form on $\mathfrak {t}$ by $\mathbb {C}[t^{\pm 1}]$ -linearity. The above form is a degree $1$ element of the restricted dual $(\mathfrak {t}\otimes \mathfrak {t})^\star $ , as defined in the beginning of Section 2.3. Namely, it vanishes on

$$ \begin{align*} (\mathfrak{t}\otimes \mathfrak{t})_k=\bigoplus_{a+b=k}{\mathfrak{t}}_a\otimes {\mathfrak{t}}_b \end{align*} $$

for any $k\neq -1$ , and restricts to a nondegenerate pairing ${\mathfrak {t}}_a\otimes {\mathfrak {t}}_{-a-1}\to \mathbb {C}$ for any $a\in \mathbb {Z}$ . Moreover, one has the polarization

with ${\mathfrak {t}}_{\scriptscriptstyle {+}}$ and ${\mathfrak {t}}_{\scriptscriptstyle {-}}$ isotropic, graded Lie subalgebras of $\mathfrak {t}$ , with gradings concentrated in nonnegative and nonpositive degrees, respectively. Said in fewer words, this collection of data gives rise to a restricted Manin triple $(\mathfrak {t},{\mathfrak {t}}_{\scriptscriptstyle {+}},{\mathfrak {t}}_{\scriptscriptstyle {-}})$ (see Sections 5.2 and 5.3 of [Reference Appel and Toledano Laredo2]).

Since each homogeneous component ${\mathfrak {t}}_k$ of $\mathfrak {t}$ is finite-dimensional, there are dual Lie bialgebra structures on ${\mathfrak {t}}_{\scriptscriptstyle {+}}$ and ${\mathfrak {t}}_{\scriptscriptstyle {-}}$ , obtained as follows. The residue form (2.3) yields isomorphisms of graded vector spaces

which are homogeneous of degree $1$ : $\mathsf {Res}_{\pm }({\mathfrak {t}}_{{\scriptscriptstyle \pm },n})=(\mathfrak {t}^\star _{\scriptscriptstyle \mp })_{n+1}$ for all $n\in \mathbb {Z}$ . Dualizing Lie brackets then gives rise to honest, degree $-1$ , Lie bialgebra cobrackets

$$ \begin{align*} \delta_{\scriptscriptstyle\pm}=[,]_{{\mathfrak{t}}_{\mp}}^t:{\mathfrak{t}}_{\scriptscriptstyle\pm}\to {\mathfrak{t}}_{\scriptscriptstyle\pm}\wedge {\mathfrak{t}}_{\scriptscriptstyle\pm}. \end{align*} $$

Observe that the Casimir tensor $\Omega _{\mathfrak {g}}\in (\mathfrak {g}\otimes \mathfrak {g})^{\mathfrak {g}}$ satisfies

$$ \begin{align*} ([x\otimes 1,\Omega_{\mathfrak{g}}],y\otimes z)_{\mathfrak{g}\otimes \mathfrak{g}}=-(x,[y,z]) \quad \forall\; x,y,z\in \mathfrak{g}, \end{align*} $$

where $(,)_{\mathfrak {g}\otimes \mathfrak {g}}=(,)\otimes (,)\circ (2\,3):\mathfrak {g}\otimes \mathfrak {g}\otimes \mathfrak {g}\otimes \mathfrak {g}\to \mathbb {C}$ . It follows readily from this observation and the definition of $\mathsf {Res}_\pm $ that $\delta _{\scriptscriptstyle +}$ and $\delta _{\scriptscriptstyle -}$ are given explicitly on each graded component by

$$ \begin{gather*} \delta_{\scriptscriptstyle +}(xt^k)=\sum_{a+b=k-1}\![x\otimes 1,\Omega_{\mathfrak{g}}]t^as^{b}\in \mathfrak{g}[t]\otimes \mathfrak{g}[s]={\mathfrak{t}}_{\scriptscriptstyle{+}}\otimes {\mathfrak{t}}_{\scriptscriptstyle{+}},\\ \delta_{\scriptscriptstyle -}(xt^{-k-1})=\sum_{a+b=k}[x\otimes 1,\Omega_{\mathfrak{g}}]t^{-a-1}s^{-b-1}\in t^{-1}\mathfrak{g}[t^{-1}]\otimes s^{-1}\mathfrak{g}[s^{-1}]={\mathfrak{t}}_{\scriptscriptstyle{-}}\otimes {\mathfrak{t}}_{\scriptscriptstyle{-}}, \end{gather*} $$

where we have used the natural identification of $(\mathfrak {g}\otimes \mathfrak {g})[t^{\pm 1},s^{\pm 1}]$ with $\mathfrak {g}[t^{\pm 1}]\otimes \mathfrak {g}[s^{\pm 1}]$ , and $a,b$ take values in $\mathbb {N}$ . Since

$$ \begin{align*} (z-w)\sum_{a=0}^kz^aw^{k-a}=z^{k+1}-w^{k+1}, \end{align*} $$

the linear map is given by the formula

$$ \begin{align*} \delta(f)(t,s)=\left[f(t)\otimes 1+ 1\otimes f(s), \frac{\Omega_{\mathfrak{g}}}{t-s}\right]\in \mathfrak{g}[t^{\pm 1}]\otimes \mathfrak{g}[s^{\pm 1}] \quad \forall\; f(t)\in \mathfrak{g}[t^{\pm 1}] \end{align*} $$

and defines a Lie bialgebra structure on the Lie algebra $\mathfrak {t}$ such that $({\mathfrak {t}}_{\scriptscriptstyle {+}},\delta _{\scriptscriptstyle +})$ and $({\mathfrak {t}}_{\scriptscriptstyle {-}}, -\delta _{\scriptscriptstyle -})$ are Lie sub-bialgebras. This construction identifies $\mathfrak {t}$ with the restricted Drinfeld double $D({\mathfrak {t}}_{\scriptscriptstyle {+}})$ of the $\mathbb {N}$ -graded Lie bialgebra ${\mathfrak {t}}_{\scriptscriptstyle {+}}$ , as defined in [Reference Appel and Toledano Laredo2, Section 5.4], for instance.

3.1 The Lie algebra $\mathfrak {g}$

We henceforth fix $\mathfrak {g}$ to be a finite-dimensional simple Lie algebra over the complex numbers, with invariant form $(\,,\,)$ as in Section 2.5. Let $\mathfrak {h}\subset \mathfrak {g}$ be a Cartan subalgebra, $\{\alpha _i\}_{i\in \mathbf {I}}\subset \mathfrak {h}^\ast $ a basis of simple roots, and $\{\alpha _i^\vee \}_{i\in \mathbf {I}}$ the set of simple coroots, so that $\alpha _j(\alpha _i^\vee )=a_{ij}=2(\alpha _i,\alpha _j)/(\alpha _i,\alpha _i)$ are the entries of the Cartan matrix $\mathbf {A}=(a_{ij})_{i,j\in \mathbf {I}}$ of $\mathfrak {g}$ . Let $\Delta ^+\subset \mathfrak {h}^\ast $ be the associated set of positive roots, and let $Q=\bigoplus _{i \in \mathbf {I}}\mathbb {Z}\alpha _i$ and $Q_+=\bigoplus _{i\in \mathbf {I}} \mathbb {N}\alpha _i$ denote the root lattice and its positive cone, respectively, where we recall that $\mathbb {N}$ denotes the set of nonnegative integers. Set

$$ \begin{align*} d_{ij}=\frac{(\alpha_i,\alpha_j)}{2} \; \text { and }\; d_i=d_{ii} \quad \forall\; i,j\in \mathbf{I}. \end{align*} $$

We normalize $(\,,\,)$ , if necessary, so that the square length of a short root is $2$ . In particular, we then have $\{d_i\}_{i\in \mathbf {I}}\subset \{1,2,3\}$ . Let $\{e_i,f_i\}_{i\in \mathbf {I}}$ denote the Chevalley generators of $\mathfrak {g}$ , as in [Reference Kac24, Section 1.3], and set

$$ \begin{align*} h_i=d_i\alpha_i^\vee, \quad x_i^+=\sqrt{d_i}e_i, \quad x_i^-=\sqrt{d_i}f_i \quad \forall \; i\in \mathbf{I}. \end{align*} $$

These normalized generators satisfy $(x_i^+,x_i^-)=1$ and $h_i=[x_i^+,x_i^-]$ for all $i\in \mathbf {I}$ .

3.2 The Yangian

We now recall the definition of the Yangian $Y_{\hbar }(\mathfrak {g})$ . Let $S_m$ denote the symmetric group on $\{1,\ldots ,m\}$ .

Definition 3.1 The Yangian $Y_{\hbar }(\mathfrak {g})$ is the unital associative $\mathbb {C}[{\hbar }]$ -algebra generated by $\{x_{ir}^\pm , h_{ir}\}_{i\in \mathbf {I},r\in \mathbb {N}}$ , subject to the following relations for $i,j\in \mathbf {I}$ and $r,s\in \mathbb {N}$ :

(3.1) $$ \begin{align} [h_{ir},h_{js}]=0, \end{align} $$

(3.2) $$ \begin{align} [h_{i0},x_{js}^\pm]=\pm 2d_{ij}x_{js}^\pm, \end{align} $$

(3.3) $$ \begin{align} [x_{ir}^+,x_{js}^-]=\delta_{ij} h_{i,r+s}, \end{align} $$

(3.4) $$ \begin{align} [h_{i,r+1},x_{js}^\pm]-[h_{ir},x_{j,s+1}^\pm]=\pm {\hbar} d_{ij}(h_{ir}x_{js}^\pm+x_{js}^\pm h_{ir}), \end{align} $$

(3.5) $$ \begin{align} [x_{i,r+1}^\pm,x_{js}^\pm]-[x_{ir}^\pm,x_{j,s+1}^\pm]=\pm {\hbar} d_{ij}(x_{ir}^\pm x_{js}^\pm+x_{js}^\pm x_{ir}^\pm ), \end{align} $$

(3.6) $$ \begin{align} \sum_{\pi \in S_{m}} \left[x_{i,r_{\pi(1)}}^{\pm}, \left[x_{i,r_{\pi(2)}}^{\pm}, \cdots, \left[x_{i,r_{\pi(m)}}^{\pm},x_{js}^{\pm}\right] \cdots\right]\right] = 0, \end{align} $$

where in the last relation $i\neq j$ , $m=1-a_{ij}$ , and $r_1,\ldots ,r_m\in \mathbb {N}$ .

The Yangian $Y_{\hbar }(\mathfrak {g})$ is an $\mathbb {N}$ -graded $\mathbb {C}[{\hbar }]$ -algebra, with grading

$$ \begin{align*} Y_{\hbar}(\mathfrak{g})=\bigoplus_{k\in \mathbb{N}}Y_{\hbar}(\mathfrak{g})_k \end{align*} $$

determined by $\deg x_{ir}^\pm =\deg h_{ir}=r$ for all $i\in \mathbf {I}$ and $r\in \mathbb {N}$ . Moreover, Definition 6.1 is such that $Y_{\hbar }(\mathfrak {g})$ provides an $\mathbb {N}$ -graded $\mathbb {C}[{\hbar }]$ -algebra deformation of the enveloping algebra $U({\mathfrak {t}}_{\scriptscriptstyle {+}})$ , where we recall that ${\mathfrak {t}}_{\scriptscriptstyle {+}}=\mathfrak {g}[t]$ . Indeed, the identification $Y_{\hbar }(\mathfrak {g})/{\hbar } Y_{\hbar }(\mathfrak {g}) \cong U({\mathfrak {t}}_{\scriptscriptstyle {+}})$ is induced by the graded algebra epimorphism $q:Y_{\hbar }(\mathfrak {g})\twoheadrightarrow U({\mathfrak {t}}_{\scriptscriptstyle {+}})$ given on generators by

$$ \begin{align*} q:\;x_{ir}^\pm \mapsto x_i^\pm t^r, \quad h_{ir}\mapsto h_it^r \quad \forall\; i\in \mathbf{I}\;\text{ and }\; r\in \mathbb{N}. \end{align*} $$

In addition, the relations (3.1)–(3.6) imply that the assignment

$$ \begin{align*} x_i^\pm \mapsto x_{i0}^\pm, \quad h_i\mapsto h_{i0} \quad \forall \; i\in \mathbf{I} \end{align*} $$

determines a $\mathbb {C}$ -algebra homomorphism $U(\mathfrak {g})\to Y_{\hbar }(\mathfrak {g})$ , which is injective as its composition with q is the identity map $\mathbf {1}_{U(\mathfrak {g})}$ on $U(\mathfrak {g})$ . Henceforth, we shall identify $\mathfrak {g}$ with its image in $Y_{\hbar }(\mathfrak {g})$ without further comment.

To specify the standard Hopf algebra structure on $Y_{\hbar }(\mathfrak {g})$ , we first note that $Y_{\hbar }(\mathfrak {g})$ is generated as a $\mathbb {C}[{\hbar }]$ -algebra by the set $\mathfrak {g}\cup \{t_{i1}\}_{i\in \mathbf {I}}\subset Y_{\hbar }(\mathfrak {g})$ , where

$$ \begin{align*} t_{i1}:=h_{i1}-\frac{{\hbar}}{2}h_{i0}^2 \quad \forall\; i\in \mathbf{I}. \end{align*} $$

More precisely, for each $s>0$ , $x_{is}^\pm $ and $h_{i,s+1}$ are determined by the recursive formulas

$$ \begin{align*} x_{is}^\pm= \pm\frac{1}{2d_i}\left[t_{i1},x_{i,s-1}^\pm\right]\quad \text{ and }\quad h_{i,s+1}=[x_{is}^+,x_{i1}^-]. \end{align*} $$

Now let $\mathsf {r}\in \mathfrak {n}_-\otimes \mathfrak {n}_+$ denote the canonical tensor associated with the pairing $(\,,\,)|_{\mathfrak {n}_-\times \mathfrak {n}_+}$ , where $\mathfrak {n}_\pm =\bigoplus _{\alpha \in \Delta ^+}{\mathfrak {g}}_{\pm \alpha }$ is the Lie subalgebra of $\mathfrak {g}$ generated by $\{x_i^\pm \}_{i\in \mathbf {I}}$ . Equivalently, $\mathsf {r}$ is the unique preimage of the identity map under the natural isomorphism , determined by $(\,,\,)|_{\mathfrak {n}_-\times \mathfrak {n}_+}$ . In addition, we set

$$ \begin{align*} \mathsf{r}_i:=[h_i\otimes 1,\mathsf{r}]\quad \forall\quad i\in \mathbf{I}. \end{align*} $$

If $x_\alpha ^\pm \in {\mathfrak {g}}_{\pm \alpha }$ are root vectors satisfying $(x_\alpha ^+,x_\alpha ^-)=1$ , then one has the formulae

$$ \begin{align*} \mathsf{r}=\sum_{\alpha\in \Delta^+} x_\alpha^- \otimes x_\alpha^+ \quad \text{ and }\quad \mathsf{r}_i= - \sum_{\alpha\in \Delta^+}\alpha(h_i)x_\alpha^- \otimes x_\alpha^+. \end{align*} $$

The following proposition describes the Hopf algebra structure on $Y_{\hbar }(\mathfrak {g})$ , where $m:Y_{\hbar }(\mathfrak {g})^{\otimes 2}\to Y_{\hbar }(\mathfrak {g})$ denote the multiplication map.

Proposition 3.2 The Yangian $Y_{\hbar }(\mathfrak {g})$ is an $\mathbb {N}$ -graded Hopf algebra with counit $\varepsilon $ , coproduct $\Delta $ , and antipode S uniquely determined by the requirement that $\mathfrak {g}$ is primitive and that, for each $i\in \mathbf {I}$ , one has

$$ \begin{gather*} \varepsilon(t_{i1})=0,\quad \Delta(t_{i1})=t_{i1}\otimes 1+1\otimes t_{i1}+{\hbar} \mathsf{r}_i,\quad S(t_{i1})=-t_{i1}+m({\hbar} \mathsf{r}_i). \end{gather*} $$

In particular, $Y_{\hbar }(\mathfrak {g})$ is an $\mathbb {N}$ -graded Hopf algebra deformation of $U({\mathfrak {t}}_{\scriptscriptstyle {+}})$ over $\mathbb {C}[{\hbar }]$ .

The crux of the proof of this proposition lies in showing that $\Delta $ is an algebra homomorphism. Though this is a consequence of [Reference Drinfel’d6, Theorem 2] and [Reference Drinfel’d8, Theorem 1] (see also [Reference Guay, Regelskis and Wendlandt22, Theorem 2.6]), a complete proof has only recently appeared in [Reference Guay, Nakajima and Wendlandt21] (see Theorem 4.9 therein).

The Yangian $Y_{\hbar }(\mathfrak {g})$ also admits a Q-grading compatible with the above $\mathbb {N}$ -grading; that is to say, it is $\mathbb {N}\times Q$ -graded as a Hopf algebra. This Q-grading arises from the adjoint action of the Cartan subalgebra $\mathfrak {h}\subset \mathfrak {g}$ on $Y_{\hbar }(\mathfrak {g})$ . Namely, one has $Y_{\hbar }(\mathfrak {g})=\bigoplus _{\beta \in Q}Y_{\hbar }(\mathfrak {g})_\beta $ , where $Y_{\hbar }(\mathfrak {g})_\beta $ is just the $\beta $ -weight space

$$ \begin{align*} Y_{\hbar}(\mathfrak{g})_\beta=\{x\in Y_{\hbar}(\mathfrak{g}): \; [h,x]=\beta(h)x \quad \forall\quad h\in \mathfrak{h}\} \quad \forall\; \beta\in Q. \end{align*} $$

3.3 Automorphisms

There are two families of (anti)automorphisms of $Y_{\hbar }(\mathfrak {g})$ which will play an especially pronounced role in this article: the shift automorphisms and the Chevalley involution. The former are a family $\{\tau _c\}_{c\in \mathbb {C}}\subset \mathrm {Aut}(Y_{\hbar }(\mathfrak {g}))$ which give rise to an action of the additive group $\mathbb {C}$ on $Y_{\hbar }(\mathfrak {g})$ by Hopf algebra automorphisms. In more detail, $\tau _c$ is defined explicitly by

(3.7) $$ \begin{align} \tau_c(x_i^\pm(u))=x_i^\pm(u-c)\quad \text{ and } \quad \tau_c(h_i(u))=h_i(u-c) \quad \forall \; i\in \mathbf{I}, \end{align} $$

where we have introduced the generating series $x_i^\pm (u)$ and $h_i(u)$ in $Y_{\hbar }(\mathfrak {g})[\kern-1.2pt\![u^{-1}]\!\kern-1.2pt]$ by

$$ \begin{align*} x_i^\pm(u)=\sum_{r\in \mathbb{N}} x_{ir}^\pm u^{-r-1} \quad \text{ and }\quad h_i(u)=\sum_{r\in \mathbb{N}} h_{ir} u^{-r-1}. \end{align*} $$

Replacing c by a formal variable z in (3.7), one obtains an $\mathbb {N}$ -graded embedding

(3.8) $$ \begin{align} \tau_z:Y_{\hbar}(\mathfrak{g})\hookrightarrow Y_{\hbar}(\mathfrak{g})[z] \end{align} $$

called the formal shift homomorphism, where $\deg z=1$ . Let us now turn to defining the Chevalley involution, beginning with the following lemma.

Lemma 3.3 The assignments $\omega $ and $\varsigma $ defined by

$$ \begin{gather*} \omega(x_i^\pm(u))=x_{i}^\mp(u), \quad \omega(h_i(u))=h_i(u),\\ \varsigma(x_i^\pm(u))=x_i^\pm(-u), \quad \varsigma(h_i(u))=h_i(-u), \end{gather*} $$

extend to commuting anti-involutions $\omega $ and $\varsigma $ of $Y_{\hbar }(\mathfrak {g})$ . Moreover, $\omega $ and $\varsigma $ satisfy

This result, which has appeared in various forms in the literature (for instance, [Reference Chari and Pressley3, Proposition 2.9]), is readily established using Definition 6.1 and the relations of Proposition 3.2. By the lemma, $\omega $ is an involutive Hopf algebra anti-automorphism of $Y_{\hbar }(\mathfrak {g})$ , which we call the Chevalley involution of $Y_{\hbar }(\mathfrak {g})$ . On $\mathfrak {g}\subset Y_{\hbar }(\mathfrak {g})$ , this recovers the standard Chevalley involution, given by

$$ \begin{align*} \omega(x_i^\pm)=x_i^\mp \quad \text{ and }\quad \omega(h_i)=h_i \quad \forall\; i \in \mathbf{I}. \end{align*} $$

Similarly, under the identification $Y_{\hbar }(\mathfrak {g})/{\hbar } Y_{\hbar }(\mathfrak {g})\cong U({\mathfrak {t}}_{\scriptscriptstyle {+}})$ , the semiclassical limit $\bar \omega : U({\mathfrak {t}}_{\scriptscriptstyle {+}})\to U({\mathfrak {t}}_{\scriptscriptstyle {+}})$ of $\omega $ coincides with the anti-involution of $U({\mathfrak {t}}_{\scriptscriptstyle {+}})$ uniquely extending the Lie algebra anti-automorphism

(3.9) $$ \begin{align} \bar{\omega}(x t^r)=\omega(x)t^r \quad \forall\; x\in \mathfrak{g}\; \text{ and }\; r\in \mathbb{N}. \end{align} $$

In addition, we note that the composite $\unicode{x3ba} :=\omega \circ \varsigma $ is an involutive algebra automorphism of $Y_{\hbar }(\mathfrak {g})$ , given explicitly by

(3.10) $$ \begin{align} \unicode{x3ba}(x_i^\pm(u))=x_i^\mp(-u) \quad \text{ and }\quad \unicode{x3ba}(h_i(u))=h_i(-u) \quad \forall\; i\in \mathbf{I}. \end{align} $$

This automorphism is itself often called the Chevalley or Cartan involution of $Y_{\hbar }(\mathfrak {g})$ , though here we shall reserve the former terminology for $\omega $ .

3.4 Poincaré–Birkhoff–Witt theorem

An important foundational result in the theory of Yangians is the Poincaré–Birkhoff–Witt theorem, which asserts the flatness of $Y_{\hbar }(\mathfrak {g})$ as an $\mathbb {N}$ -graded Hopf algebra deformation of $U({\mathfrak {t}}_{\scriptscriptstyle {+}})$ . It can be stated concisely as follows.

As $Y_{\hbar }(\mathfrak {g})$ is an $\mathbb {N}$ -graded algebra deformation of $U({\mathfrak {t}}_{\scriptscriptstyle {+}})$ , an isomorphism $Y_{\hbar }(\mathfrak {g})\cong U({\mathfrak {t}}_{\scriptscriptstyle {+}})[{\hbar }]$ can be obtained by specifying an ordered, homogeneous, lift $ {G}\subset Y_{\hbar }(\mathfrak {g})$ of any fixed homogeneous basis of the Lie algebra ${\mathfrak {t}}_{\scriptscriptstyle {+}}$ . For our purposes, it will be useful to specify a class of isomorphisms of this type with a number of useful properties.

For each $\beta \in \Delta ^+$ , we may choose $i(\beta )\in \mathbf {I}$ and $\mathbf {X}^\beta \in U(\mathfrak {n}_{+})_{\beta -\alpha _{i(\beta )}}\subset U(\mathfrak {g})$ such that

(3.11) $$ \begin{align} x_\beta^+:=\mathbf{X}^\beta\cdot x_{i(\beta)}^+\in {\mathfrak{g}}_{\beta} \quad \text{ and }\quad x_\beta^-:=\omega(x_\beta^+)\in {\mathfrak{g}}_{-\beta} \end{align} $$

satisfy the duality condition $(x_\beta ^+,x_\beta ^-)=1$ , where $\mathbf {X}^\beta $ acts on $x_{i(\beta )}^+$ via the adjoint action of $\mathfrak {g}$ on $U(\mathfrak {g})$ . In particular, we can (and shall) take $\mathbf {X}^{\alpha _i}=1$ for all $i\in \mathbf {I}$ , so that $x_{\alpha _i}^\pm =x_i^\pm $ . For each $k\in \mathbb {N}$ , we then set

$$ \begin{align*} x_{\beta,k}^+:=\mathbf{X}^\beta\cdot x_{i(\beta),k}^+\in Y_{\hbar}(\mathfrak{g})_\beta\quad \text{ and }\quad x_{\beta,k}^-:=\omega(x_{\beta,k}^+)\in Y_{\hbar}(\mathfrak{g})_{-\beta}, \end{align*} $$

where $\mathfrak {g}$ now operates on $Y_{\hbar }(\mathfrak {g})$ via the adjoint action. This definition is such that $q(x_{\beta ,k}^\pm )=x_\beta ^\pm t^k$ for all $\beta \in \Delta ^+$ and $k\in \mathbb {N}$ , and hence the set of elements

$$ \begin{align*} {G}:=\bigcup_{k\in \mathbb{N}}\{h_{ik},x_{\beta,k}^\pm\}_{i\in \mathbf{I},\beta\in\Delta^+} \end{align*} $$

reduces modulo ${\hbar }$ to the basis of ${\mathfrak {t}}_{\scriptscriptstyle {+}}$ consisting of all Cartan elements $h_i t^k$ and root vector $x_\beta ^\pm t^k$ . For each choice of total order $\preceq $ on $ {G}$ , the corresponding set of ordered monomials

$$ \begin{align*} B({G})=\{x_{1}x_2\cdots x_n: n\in \mathbb{N},\, x_i\in {G} \;\text{ and }\; x_i\preceq x_j \; \forall \; i<j\} \end{align*} $$

is therefore a homogeneous basis of the $\mathbb {C}[{\hbar }]$ -module $Y_{\hbar }(\mathfrak {g})$ , and so defines an isomorphism $\mathbb {N}$ -graded modules

(3.12)

uniquely determined by the property that $\unicode{x3bd} _{ {G}}|_{B( {G})}$ coincides with the restriction of the quotient map q to $B( {G})$ . We note that $\unicode{x3bd} _{ {G}}$ is automatically an isomorphism of $\mathfrak {h}$ -modules, and is thus Q-graded.

We shall single out a subclass of isomorphisms of this type which are compatible with Chevalley involutions and satisfy a triangularity condition. To make this precise, we must first recall that $Y_{\hbar }(\mathfrak {g})$ admits a triangular decomposition, compatible with the decomposition

Let us define $Y_{\hbar }^0(\mathfrak {g})$ and $Y_{\hbar }^\pm (\mathfrak {g})$ to be the unital associative subalgebras of $Y_{\hbar }(\mathfrak {g})$ generated by $\{h_{ir}\}_{i\in \mathbf {I},r\in \mathbb {N}}$ and $\{x_{ir}^\pm \}_{i\in \mathbf {I},r\in \mathbb {N}}$ , respectively. These are $\mathbb {N}\times Q$ -graded subalgebras of $Y_{\hbar }(\mathfrak {g})$ . The triangular decomposition of $Y_{\hbar }(\mathfrak {g})$ is then encoded by the following proposition, which is a well-known consequence of Theorem 3.4.

As a consequence of Part (1) of Proposition 3.5 and Corollary 2.6, one has $Y_{\hbar }^\pm (\mathfrak {g})\cong U({\mathfrak {n}}_\pm [t])[{\hbar }]$ as $\mathbb {N}$ -graded $\mathbb {C}[{\hbar }]$ -modules. Following the procedure outlined at the beginning of the section, let us fix an arbitrary total order $\preceq _{\scriptscriptstyle +}$ on the union

$$ \begin{align*} {G}_+=\bigcup_{k\in \mathbb{N}}\{x_{\beta,k}^+\}_{\beta\in\Delta^+} = {G}\cap Y_{\hbar}^+(\mathfrak{g}). \end{align*} $$

The set of ordered monomials $B( {G}_+)$ in $ {G}_+$ is a basis of $Y_{\hbar }^+(\mathfrak {g})$ , and thus gives rise to an isomorphism of $\mathbb {N}\times Q$ -graded $\mathbb {C}[{\hbar }]$ -modules

sending each ordered monomial in $ {G}_+$ to its image in $Y_{\hbar }^+(\mathfrak {g})/{\hbar } Y_{\hbar }^+(\mathfrak {g})\cong U({\mathfrak {n}}_+[t])$ . Using the Chevalley involution $\omega $ and its semiclassical limit $\bar \omega $ (see (3.9)), we then obtain an isomorphism

Combining $\unicode{x3bd} _\pm $ with $\unicode{x3be} $ from Part (2) of Proposition 3.5 outputs an isomorphism of $\mathbb {N}\times Q$ -graded $\mathbb {C}[{\hbar }]$ -modules

(3.13)

where is the multiplication map, which we extend trivially by $\mathbb {C}[{\hbar }]$ -linearity. By construction, $\unicode{x3bd} $ is compatible with the underlying triangular decompositions on $Y_{\hbar }(\mathfrak {g})$ and $U({\mathfrak {t}}_{\scriptscriptstyle {+}})$ and satisfies

$$ \begin{align*} \unicode{x3bd}\circ \omega =\bar\omega\circ \unicode{x3bd}. \end{align*} $$

The definition (3.13) is such that $\unicode{x3bd} =\unicode{x3bd} _{ {G}}$ for any total order $\preceq $ on $ {G}$ which restricts to $\preceq _{\scriptscriptstyle {+}}$ , satisfies $x^+\preceq h \preceq x^-$ for all $x^{\pm }\in {G} \cap Y_{\hbar }^\pm (\mathfrak {g})$ and $h\in Y_{\hbar }^0(\mathfrak {g})$ , and for which $\omega $ is a decreasing function on $ {G}$ . We will denote the inverse of $\unicode{x3bd} $ by $\mu $ :

$$ \begin{align*} \mu:=\unicode{x3bd}^{-1}: U({\mathfrak{t}}_{\scriptscriptstyle{+}})[{\hbar}]\to Y_{\hbar}(\mathfrak{g}). \end{align*} $$

Note that, for any total order on $ {G}$ , one has $\mu (x)=\unicode{x3bd} _{ {G}}^{-1}(x)$ for all $x\in {G}$ .

3.5 Quantization

As a consequence of Proposition 3.2 and Theorem 3.4, the Yangian $Y_{\hbar }(\mathfrak {g})$ provides a homogeneous quantization of an $\mathbb {N}$ -graded Lie bialgebra structure on the Lie algebra ${\mathfrak {t}}_{\scriptscriptstyle {+}}$ over $\mathbb {C}[{\hbar }]$ , with cocommutator $\delta $ determined by the formula (2.2). By Proposition 3.2, $\delta $ is uniquely determined by $\delta (\mathfrak {g})=0$ and

$$ \begin{align*} \delta(h_it)=\mathsf{r}_i-\mathsf{r}_i^{21}=[h_i\otimes 1, \Omega_{\mathfrak{g}}]=\left[h_it\otimes 1 + 1\otimes h_i s,\frac{\Omega_{\mathfrak{g}}}{t-s}\right]=\delta_{\scriptscriptstyle +}(h_i t) \quad \forall\; i\in \mathbf{I}, \end{align*} $$

and thus coincides with $\delta _{\scriptscriptstyle +}$ from Section 2.5. This recovers the following well-known result, originally due to Drinfeld [Reference Drinfel’d6, Theorem 2]:

As explained in Section 2.4, it follows immediately that the ${\hbar }$ -adic completion

(3.14) $$ \begin{align} Y_{\hbar}\mathfrak{g}:=\varprojlim_{n} ( Y_{\hbar}(\mathfrak{g})/{\hbar}^n Y_{\hbar}(\mathfrak{g}) ) \end{align} $$

is a homogeneous quantization of $({\mathfrak {t}}_{\scriptscriptstyle {+}},\delta _{\scriptscriptstyle +})$ over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ . We refer the reader to Definition 2.12 and Corollary 2.6 for a detailed discussion of this point.

Remark 3.7 Let $Y_{\hbar }^{\pm \!}\mathfrak {g}$ and $Y_{\hbar }^0\mathfrak {g}$ denote the topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebras

$$ \begin{align*} Y_{\hbar}^{\pm\!}\mathfrak{g}:=\varprojlim_{n} Y_{\hbar}^\pm(\mathfrak{g})/{\hbar}^n Y_{\hbar}^\pm(\mathfrak{g}) \quad \text{ and }\quad Y_{\hbar}^0\mathfrak{g}:=\varprojlim_{n} Y_{\hbar}^0(\mathfrak{g})/{\hbar}^n Y_{\hbar}^0(\mathfrak{g}). \end{align*} $$

It follows from Corollary 2.6 and Proposition 3.5 that these are subalgebras of $Y_{\hbar }\mathfrak {g}$ , with $Y_{\hbar }^0\mathfrak {g}$ isomorphic to $U(\mathfrak {h}[t])[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]\cong \mathsf {S}(\mathfrak {h}[t])[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ as an $\mathbb {N}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra, and $Y_{\hbar }^{\pm \!}\mathfrak {g}$ a topologically free $\mathbb {N}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra with semiclassical limit equal to $U({\mathfrak {n}}_{\pm }[t])$ . By Part (3) of Proposition 3.5, the product m on $Y_{\hbar }\mathfrak {g}$ gives rise to an isomorphism of $\mathbb {N}$ -graded topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules

(3.15)

where, following Remark 2.11, $\otimes $ should now be understood to be the topological tensor product $\widehat {\otimes }$ of $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -modules. For later purposes, we note that the product on $Y_{\hbar }\mathfrak {g}$ also gives rise to an isomorphism

which can be realized as $\unicode{x3ba} \circ m\circ (\unicode{x3ba} _-\otimes \unicode{x3ba} _0\otimes \unicode{x3ba} _+)$ , where m is the isomorphism (3.15), $\unicode{x3ba} $ is the involutive automorphism of $Y_{\hbar }\mathfrak {g}$ defined in (3.10) (extended by continuity), and .

3.6 The universal R-matrix

We complete our survey of $Y_{\hbar }(\mathfrak {g})$ by reviewing the construction of the universal R-matrix $\mathcal {R}(z)$ of the Yangian, whose existence and uniqueness was first established by Drinfeld in [Reference Drinfel’d6, Theorem 3]. We shall, however, need a refined version of Drinfeld’s theorem only recently proven in [Reference Gautam, Toledano Laredo, Wendlandt, Alekseev, Frenkel, Rosso, Webster and Yakimov17, Section 7.4], which reconstructs $\mathcal {R}(z)$ from the factors in its Gauss decomposition

$$ \begin{align*} \mathcal{R}(z)=\mathcal{R}^+(z)\mathcal{R}^0(z)\mathcal{R}^-(z). \end{align*} $$

Let us begin with a few preliminaries. For each positive integer n, let

$$ \begin{align*} Y_{\hbar}(\mathfrak{g})^{\otimes n}[z;z^{-1}]\!\kern-1.2pt]=\bigcup_{k\in \mathbb{N}} z^k Y_{\hbar}(\mathfrak{g})^{\otimes n}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]\subset Y_{\hbar}(\mathfrak{g})^{\otimes n}[\kern-1.2pt\![z^{\pm 1}]\!\kern-1.2pt] \end{align*} $$

denote the algebra of formal Laurent series in $z^{-1}$ with coefficients in $Y_{\hbar }(\mathfrak {g})^{\otimes n}$ . Following [Reference Wendlandt41, Section 4.2], we then introduce the subspace

$$ \begin{align*} \widehat{Y_{\hbar}\mathfrak{g}}{\vphantom{Y_{\hbar}\mathfrak{g}}}^{\scriptscriptstyle{(n)}}_z:= \prod_{k\in \mathbb{N}} (Y_{\hbar}(\mathfrak{g})^{\otimes n})_k z^{-k}\subset Y_{\hbar}(\mathfrak{g})^{\otimes n}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt], \end{align*} $$

where $(Y_{\hbar }(\mathfrak {g})^{\otimes n})_k$ is the kth graded component of the $\mathbb {N}$ -graded algebra $Y_{\hbar }(\mathfrak {g})^{\otimes n}$ . This is a $\mathbb {C}$ -algebra isomorphic to the completion of $Y_{\hbar }(\mathfrak {g})^{\otimes n}$ with respect to its grading. The $\mathbb {C}[z,z^{-1}]$ -submodule of $Y_{\hbar }(\mathfrak {g})^{\otimes n}[z;z^{-1}]\!\kern-1.2pt]$ that it generates is a $\mathbb {Z}$ -graded $\mathbb {C}[{\hbar }]$ -algebra

(3.16) $$ \begin{align} {L}(\widehat{Y_{\hbar}\mathfrak{g}}{\vphantom{Y_{\hbar}\mathfrak{g}}}^{\scriptscriptstyle{(n)}}_z):=\bigoplus_{k\in \mathbb{Z}} z^k \widehat{Y_{\hbar}\mathfrak{g}}{\vphantom{Y_{\hbar}\mathfrak{g}}}^{\scriptscriptstyle{(n)}}_z \subset Y_{\hbar}(\mathfrak{g})^{\otimes n}[z;z^{-1}]\!\kern-1.2pt]. \end{align} $$

Though for the moment we shall only be interested in the case where $n=2$ , such formal series spaces will reappear in later sections. In addition to the above, we shall make use of two functions $Q_+\to \mathbb {N}$ . First, we have the standard additive height function $\mathrm {ht}$ given by

$$ \begin{align*} \mathrm{ht}(\beta)=\sum_{i\in \mathbf{I}}n_i \quad \text{ for each }\; \beta=\sum_{i\in \mathbf{I}}n_i\alpha_i \in Q_+. \end{align*} $$

Second, we have an auxiliary function $\nu :Q_+\to \mathbb {N}$ defined by

(3.17) $$ \begin{align} \nu(\beta)=\min\{k\in \mathbb{N}: \exists\, \beta_1,\ldots,\beta_k\in \Delta^+\; \text{ with }\; \beta=\beta_1+\cdots +\beta_k\}, \end{align} $$

where it is understood that $\nu (0)=0$ .

Let us now recall the construction of the factor $\mathcal {R}^-(z)$ . Fix a Cartan element

$$ \begin{align*} \zeta\in \mathfrak{h}\setminus \bigcup_{\beta\neq 0}\mathrm{Ker}(\beta), \end{align*} $$

where the union runs over all nonzero $\beta \in Q_+$ . We then introduce

$$ \begin{align*} \mathcal{R}^-_\beta(z)\in (Y_{\hbar}^-(\mathfrak{g})_{-\beta}\otimes Y_{\hbar}^+(\mathfrak{g})_\beta)[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt] \quad \forall \; \beta \in Q_+ \end{align*} $$

by setting $\mathcal {R}^-_0(z)=1$ and defining $\mathcal {R}^-_\beta (z)$ inductively in $\mathrm {ht}(\beta )$ using the formula

(3.18) $$ \begin{align} \mathcal{R}^-_\beta(z)= {\hbar}\sum_{p\geq 0} \frac{\mathbf{T}(\zeta)^p}{(z\beta(\zeta))^{p+1}} \sum_{\alpha\in \Delta^+}\alpha(\zeta)\mathcal{R}_{\beta-\alpha}(z) (x_{\alpha}^- \otimes x_{\alpha}^+ ), \end{align} $$

where $\mathcal {R}_{\gamma }(z)=0$ whenever $\gamma \notin Q_+$ and $ \mathbf {T}(\zeta )=\mathrm {ad}(\mathrm {T}(\zeta )\otimes 1 + 1\otimes \mathrm {T}(\zeta )), $ with $\mathrm {T}:\mathfrak {h}\to Y_{\hbar }^0(\mathfrak {g})$ the embedding determined by

$$ \begin{align*} \mathrm{T}(h_i)=t_{i1} \quad \forall \; i\in \mathbf{I}. \end{align*} $$

Using the fact that, for each $p\in \mathbb {N}$ , $\mathbf {T}(\zeta )^p z^{-p-1}$ is a homogeneous operator on $ {L}(\widehat {Y_{\hbar }\mathfrak {g}}{\vphantom {Y_{\hbar }\mathfrak {g}}}^{\scriptscriptstyle {(2)}}_z)$ of degree $-1$ , one deduces from the recursive formula (3.18) that

$$ \begin{align*} \mathcal{R}^-_\beta(z)\in {\hbar}^{\nu(\beta)}{L}(\widehat{Y_{\hbar}\mathfrak{g}}{\vphantom{Y_{\hbar}\mathfrak{g}}}^{\scriptscriptstyle{(2)}}_z)_{-\nu(\beta)} \subset z^{-\nu(\beta)}Y_{\hbar}(\mathfrak{g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt] \quad \forall \; \beta \in Q_+. \end{align*} $$

As the set $\{\beta \in Q_+: \nu (\beta )\leq k\}$ is finite for any $k\in \mathbb {N}$ , we obtain a well-defined formal series

$$ \begin{align*} \mathcal{R}^-(z)=\sum_{\beta\in Q_+}\mathcal{R}^-_\beta(z) \in(Y_{\hbar}^-(\mathfrak{g})\otimes Y_{\hbar}^+(\mathfrak{g}))[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt] \end{align*} $$

which by construction satisfies $ \mathcal {R}^-(z)\in 1+{\hbar } {L}(\widehat {Y_{\hbar }\mathfrak {g}}{\vphantom {Y_{\hbar }\mathfrak {g}}}^{\scriptscriptstyle {(2)}}_z)_{-1}\subset \widehat {Y_{\hbar }\mathfrak {g}}{\vphantom {Y_{\hbar }\mathfrak {g}}}^{\scriptscriptstyle {(2)}}_z. $

By Theorem 4.1 of [Reference Gautam, Toledano Laredo, Wendlandt, Alekseev, Frenkel, Rosso, Webster and Yakimov17], $\mathcal {R}^-(z)$ is independent of the choice of $\zeta \in \mathfrak {h}$ made above and satisfies a number of remarkable properties. Notably, it intertwines $\tau _z\otimes \mathbf {1} \circ \Delta $ and the formal, deformed Drinfeld coproduct on $Y_{\hbar }(\mathfrak {g})$ , as defined in [Reference Gautam, Toledano Laredo, Wendlandt, Alekseev, Frenkel, Rosso, Webster and Yakimov17, Section 3.4]. We will not make direct use of these properties here, and refer the reader to [Reference Gautam, Toledano Laredo, Wendlandt, Alekseev, Frenkel, Rosso, Webster and Yakimov17] for a detailed treatment of $\mathcal {R}^-(z)$ .

Let us now recall the definition of the abelian R-matrix $\mathcal {R}^0(z)\in Y_{\hbar }^0(\mathfrak {g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]$ from [Reference Gautam, Toledano Laredo, Wendlandt, Alekseev, Frenkel, Rosso, Webster and Yakimov17, Section 6]. Let $\mathbf {B}=(d_i a_{ij})$ denote the symmetrization of the Cartan matrix $\mathbf {A}$ . Given an indeterminate v, we let $\mathbf {B}(v)=([d_i a_{ij}]_v)\subset {\mathrm {GL}}_{\mathbf {I}}(\mathbb {Q}(v))$ be the associated matrix of v-numbers, where

$$ \begin{align*} [m]_v=\frac{v^m-v^{-m}}{v-v^{-1}}. \end{align*} $$

Then it is known [Reference Gautam and Toledano Laredo16, Theorem A.1] that the auxiliary matrix

$$ \begin{align*} \mathbf{C}(v)=(c_{ij}(v))=[2\kappa]_v \mathbf{B}(v)^{-1} \end{align*} $$

has entries $c_{ij}(v)$ in $\mathbb {N}[v,v^{-1}]$ , where $4\kappa $ is the eigenvalue of the Casimir element of $\mathfrak {g}$ in the adjoint representation.

Next, for each index $i\in \mathbf {I}$ , we introduce the series $t_i(u)\in {\hbar } Y_{\hbar }^0(\mathfrak {g})[\kern-1.2pt\![u^{-1}]\!\kern-1.2pt]$ and its inverse Borel transform $B_i(u)\in {\hbar } Y_{\hbar }^0(\mathfrak {g})[\kern-1.2pt\![u]\!\kern-1.2pt]$ by

$$ \begin{align*} t_i(u)={\hbar} \sum_{r\geq 0}t_{ir} u^{-r-1}=\log(1+{\hbar} h_i(u)) \quad \;\text{ and }\; \quad B_i(u)={\hbar}\sum_{r\geq 0}\frac{t_{ir}}{r!}u^r. \end{align*} $$

Note that $t_{i1}$ coincides with the element of the same name introduced in Section 3.2. From these data, we obtain an element $\mathcal {L}(z)\in ({\hbar }/z)^2 Y_{\hbar }^0(\mathfrak {g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]$ defined by

$$ \begin{align*} \mathcal{L}(z)=T^{2\kappa}\sum_{i,j\in \mathbf{I}}c_{ij}(T)B_i(\partial_z)\otimes B_j(-\partial_z) (-z^{-2}), \end{align*} $$

where T is the shift operator $T(f(z))=f(z+\frac {{\hbar }}{2})$ on $Y_{\hbar }(\mathfrak {g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]$ . Equivalently,

$$ \begin{align*} T=\exp\!\left(\frac{{\hbar}}{2}\partial_z\right):Y_{\hbar}(\mathfrak{g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]\to Y_{\hbar}(\mathfrak{g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]. \end{align*} $$

As $\mathcal {L}(z)\in {\hbar } z^{-2}Y_{\hbar }^0(\mathfrak {g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]$ and $Y_{\hbar }^0(\mathfrak {g})^{\otimes 2}$ is torsion-free, there is a unique solution $\mathcal {S}(z)$ to the formal difference equation

$$ \begin{align*} \mathcal{L}(z)=\mathcal{S}(z+2\kappa{\hbar})-\mathcal{S}(z) \quad \text{ with }\; \mathcal{S}(z)\in z^{-1}Y_{\hbar}^0(\mathfrak{g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]. \end{align*} $$

If $g(z)\in z^{-1}\mathbb {C}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]$ is the unique solution of $-z^{-2}=g(z+1)-g(z)$ , then by Proposition 6.6 of [Reference Gautam, Toledano Laredo, Wendlandt, Alekseev, Frenkel, Rosso, Webster and Yakimov17], we have

(3.19) $$ \begin{align} \mathcal{S}(z)=\frac{T^{2\kappa}}{(2\kappa{\hbar})^2}\sum_{i,j\in \mathbf{I}}c_{ij}(T)B_i(\partial_z)\otimes B_j(-\partial_z) \left(g\!\left(\frac{z}{2\kappa{\hbar}}\right)\right), \end{align} $$

where $g(z/2\kappa {\hbar })$ is viewed as an element of $ {L}(\widehat {Y_{\hbar }\mathfrak {g}}{\vphantom {Y_{\hbar }\mathfrak {g}}}^{\scriptscriptstyle {(2)}}_z)$ . The abelian R-matrix $\mathcal {R}^0(z)$ is defined to be the formal series exponential of this solution:

$$ \begin{align*} \mathcal{R}^0(z)=\exp(\mathcal{S}(z))\in 1+z^{-1}Y_{\hbar}^0(\mathfrak{g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]. \end{align*} $$

Equivalently, it is the unique formal solution in $1+z^{-1}Y_{\hbar }^0(\mathfrak {g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]$ of the equation

$$ \begin{align*} \mathcal{R}^0(z+2\kappa{\hbar})=\mathcal{A}(z)\mathcal{R}^0(z), \end{align*} $$

where $\mathcal {A}(z)=\exp (\mathcal {L}(z))$ . As T and ${\hbar }^{-2}B_i(\partial _z)\otimes B_j(-\partial _z)$ are homogeneous operators of degree zero on $ {L}(\widehat {Y_{\hbar }\mathfrak {g}}{\vphantom {Y_{\hbar }\mathfrak {g}}}^{\scriptscriptstyle {(2)}}_z)$ , it follows from (3.19) that

$$ \begin{align*} \mathcal{R}^0(z)\in 1+{\hbar} {L}(\widehat{Y_{\hbar}\mathfrak{g}}{\vphantom{Y_{\hbar}\mathfrak{g}}}^{\scriptscriptstyle{(2)}}_z)_{-1}\subset \widehat{Y_{\hbar}\mathfrak{g}}{\vphantom{Y_{\hbar}\mathfrak{g}}}^{\scriptscriptstyle{(2)}}_z. \end{align*} $$

We are now in a position to introduce the universal R-matrix of the Yangian. Set $\mathcal {R}^+(z)=\mathcal {R}_{21}^-(-z)^{-1}$ and define

$$ \begin{align*} \mathcal{R}(z):=\mathcal{R}^+(z)\mathcal{R}^0(z)\mathcal{R}^-(z)\in 1+z^{-1}Y_{\hbar}(\mathfrak{g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]. \end{align*} $$

The following result is the content of Theorem 7.4 of [Reference Gautam, Toledano Laredo, Wendlandt, Alekseev, Frenkel, Rosso, Webster and Yakimov17].

Theorem 3.8 $\mathcal {R}(z)$ is the unique formal series in $ 1+z^{-1}Y_{\hbar }(\mathfrak {g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt] $ satisfying the intertwiner equation

(3.20)

in $Y_{\hbar }(\mathfrak {g})^{\otimes 2}[z;z^{-1}]\!\kern-1.2pt]$ , in addition to the cabling identities

$$ \begin{align*} \Delta\otimes \mathbf{1} (\mathcal{R}(z))&= \mathcal{R}_{13}(z)\mathcal{R}_{23}(z),\\ \mathbf{1}\otimes \Delta (\mathcal{R}(z))&= \mathcal{R}_{13}(z)\mathcal{R}_{12}(z) \end{align*} $$

in $Y_{\hbar }(\mathfrak {g})^{\otimes 3}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]$ . Moreover, $\mathcal {R}(z)$ has the following properties:

1. (1) It is unitary: $\mathcal {R}(z)^{-1}=\mathcal {R}_{21}(-z)$ .

2. (2) For any $a,b\in \mathbb {C}$ , one has

   $$ \begin{align*} (\tau_a\otimes \tau_b)\mathcal{R}(z)=\mathcal{R}(z+a-b). \end{align*} $$

3. (3) $\mathcal {R}(z)$ is a homogeneous, degree zero, element of $ {L}(\widehat {Y_{\hbar }\mathfrak {g}}{\vphantom {Y_{\hbar }\mathfrak {g}}}^{\scriptscriptstyle {(2)}}_z)$ , with

   $$ \begin{align*} \mathcal{R}(z)-1\in {\hbar} {L}(\widehat{Y_{\hbar}\mathfrak{g}}{\vphantom{Y_{\hbar}\mathfrak{g}}}^{\scriptscriptstyle{(2)}}_z)_{-1}= {\hbar} z^{-1}\widehat{Y_{\hbar}\mathfrak{g}}{\vphantom{Y_{\hbar}\mathfrak{g}}}^{\scriptscriptstyle{(2)}}_z \end{align*} $$
   and semiclassical limit given by
   $$ \begin{align*} q^{\otimes 2} {\hbar}^{-1}({\mathcal{R}(z)-1})=\frac{\Omega_{\mathfrak{g}}}{z+t-w}\in (U(\mathfrak{g}[t])\otimes U(\mathfrak{g}[w]))[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]. \end{align*} $$

The series $\mathcal {R}(z)$ is called the universal R-matrix of the Yangian, and is related to the element $\mathcal {R}^D(z)\in Y_{\hbar }(\mathfrak {g})^{\otimes 2}[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt]$ introduced by Drinfeld in Theorem 3 of [Reference Drinfel’d6] by $\mathcal {R}(z)=\mathcal {R}^D(-z)^{-1}$ (see Section 1.1 and Corollary 7.4 of [Reference Gautam, Toledano Laredo, Wendlandt, Alekseev, Frenkel, Rosso, Webster and Yakimov17]).

The final result of this section shows that, in particular, $\mathcal {R}(z)$ is invariant under the Chevalley involution $\omega $ of Section 3.3.

Corollary 3.10 The universal R-matrix $\mathcal {R}(z)$ satisfies

$$ \begin{align*} (\omega\otimes \omega)\mathcal{R}(z)=\mathcal{R}(z)\quad \text{ and } \quad (\varsigma\otimes \varsigma)\mathcal{R}(z)=\mathcal{R}_{21}(z)=(\unicode{x3ba}\otimes \unicode{x3ba})\mathcal{R}(z). \end{align*} $$

Proof Set $\mathcal {R}^\omega (z):=(\omega \otimes \omega )\mathcal {R}(z)$ and $\mathcal {R}^\varsigma (z):=(\varsigma \otimes \varsigma )\mathcal {R}(z)$ . Applying the anti-automorphisms $\omega \otimes \omega $ and $\varsigma \otimes \varsigma $ to the intertwiner equation (8.3), while making use of the relations of Lemma 3.3, we find that

for all $x\in Y_{\hbar }(\mathfrak {g})$ . Hence, $\mathcal {R}^\omega (z)$ and $\mathcal {R}^\varsigma (-z)^{-1}=\mathcal {R}_{21}^{\varsigma }(z)$ are both solutions of (8.3). One verifies similarly that these both satisfy the cabling identities, and hence coincide with $\mathcal {R}(z)$ by the uniqueness statement of Theorem 3.8. Since $\unicode{x3ba} =\varsigma \circ \omega $ , this completes the proof of the proposition.

4.1 The Yangian double

The definition of the Yangian double $\mathrm {D}Y_{\hbar }\mathfrak {g}$ is obtained by allowing the second index of the generators in Definition 6.1 to take values in $\mathbb {Z}$ , while working in the category of topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebras:

Definition 4.1 The Yangian double $\mathrm {D}Y_{\hbar }\mathfrak {g}$ is the unital, associative $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra topologically generated by $\{x_{ir}^\pm , h_{ir}\}_{i\in \mathbf {I},r\in \mathbb {Z}}$ , subject to the relations (3.1)–(3.6) of Definition 6.1. In terms of generating series

$$ \begin{align*} \mathcal{X}_i^\pm(u)=\sum_{r\in \mathbb{Z}}x_{ir}^\pm u^{-r-1} \quad \text{ and }\quad \mathcal{H}_i(u)=\sum_{r\in \mathbb{Z}}h_{ir}u^{-r-1}, \end{align*} $$

these defining relations can be expressed as follows, for $i,j\in \mathbf {I}$ :

$$ \begin{gather*} [\mathcal{H}_{i}(u),\mathcal{H}_j(v)] =0,\\ [h_{i0},\mathcal{X}_j^\pm(u)]=\pm 2d_{ij}\mathcal{X}_j^\pm(u), \\ \left(u-v\mp{\hbar} d_{ij}\right)\mathcal{H}_i(u)\mathcal{X}_j^{\pm}(v)=\left(u-v\pm{\hbar} d_{ij}\right)\mathcal{X}_j^{\pm}(v)\mathcal{H}_i(u), \\ \left(u-v\mp{\hbar} d_{ij}\right)\mathcal{X}_{i}^{\pm}(u)\mathcal{X}_{j}^{\pm}(v) =\left(u-v\pm{\hbar} d_{ij}\right) \mathcal{X}_{j}^{\pm}(v)\mathcal{X}_{i}^{\pm}(u) , \\ [\mathcal{X}_i^+(u),\mathcal{X}_j^-(v)] =\delta_{ij} u^{-1}\delta(v/u)\mathcal{H}_i(v), \\ \sum_{\pi \in S_{m}} \left[\mathcal{X}_i^{\pm}(u_{\pi(1)}), \left[\mathcal{X}_i^{\pm}(u_{\pi(2)}), \ldots, \left[\mathcal{X}_i^{\pm}(u_{\pi(m)}),\mathcal{X}_j^{\pm}(v)\right] \cdots\right]\right] = 0, \end{gather*} $$

where $\delta (u)=\sum _{r\in \mathbb {Z}}u^r\in \mathbb {C}[\kern-1.2pt\![u^{\pm 1}]\!\kern-1.2pt]$ is the formal delta function and in the last relation $i\neq j$ and $m=1-a_{ij}$ .

$\mathrm {D}Y_{\hbar }\mathfrak {g}$ is $\mathbb {Z}$ -graded as a topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra, with grading induced by the degree assignment $\deg x_{ir}^\pm =\deg h_{ir}=r$ for all $i\in \mathbf {I}$ and $r\in \mathbb {Z}$ . That is, if $\mathrm {D}Y_{\hbar }{\mathfrak {g}}_k$ denotes the closure of the subspace of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ spanned over the complex numbers by monomials in $x_{ir}^\pm $ , $h_{ir}$ and ${\hbar }$ of total degree k, then

$$ \begin{align*} {D}Y_{{\hbar}}\mathfrak{g}:=\bigoplus_{k\in \mathbb{Z}}\mathrm{D}Y_{\hbar}{\mathfrak{g}}_k \end{align*} $$

is a dense, $\mathbb {Z}$ -graded $\mathbb {C}[{\hbar }]$ -subalgebra of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ satisfying the conditions of Lemma 2.4. In particular, in the notation of Section 2.2, one has $ {D}Y_{{\hbar }}\mathfrak {g}=\mathrm {D}Y_{\hbar }{\mathfrak {g}}_{\mathbb {Z}}$ .

Remark 4.2 Let $ {D}Y_{{\hbar }}\mathfrak {g}^\jmath $ denote the $\mathbb {C}[{\hbar }]$ -algebra generated by $\{x_{ir}^\pm ,h_{ir}\}_{i\in \mathbf {I},r\in \mathbb {Z}}$ , subject to the defining relations (3.1)–(3.6). Then $ {D}Y_{{\hbar }}\mathfrak {g}^\jmath $ is a $\mathbb {Z}$ -graded $\mathbb {C}[{\hbar }]$ -algebra, and there is a natural algebra homomorphism

$$ \begin{align*} \jmath: {D}Y_{{\hbar}}\mathfrak{g}^\jmath\to {D}Y_{{\hbar}}\mathfrak{g}\subset\mathrm{D}Y_{\hbar}\mathfrak{g}. \end{align*} $$

The kernel of $\jmath $ is the graded ideal $\cap _{n\in \mathbb {N}}{\hbar }^n {D}Y_{{\hbar }}\mathfrak {g}^\jmath $ , and $\mathrm {D}Y_{\hbar }\mathfrak {g}$ can be recovered as the ${\hbar }$ -adic completion of $ {D}Y_{{\hbar }}\mathfrak {g}^\jmath $ (see [Reference Wendlandt41, Proposition 2.7]). Thus, $\jmath ( {D}Y_{{\hbar }}\mathfrak {g}^\jmath )$ is a dense, $\mathbb {Z}$ -graded $\mathbb {C}[{\hbar }]$ -subalgebra of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ . It is, however, a proper subalgebra of $ {D}Y_{{\hbar }}\mathfrak {g}$ as the graded components $\jmath ( {D}Y_{{\hbar }}\mathfrak {g}^\jmath _k)$ are not closed in $\mathrm {D}Y_{\hbar }\mathfrak {g}$ . Rather, one has

$$ \begin{align*} \mathrm{D}Y_{\hbar}{\mathfrak{g}}_k=\varprojlim_n({D}Y_{{\hbar}}{\mathfrak{g}}_k^\jmath/{\hbar}^n {D}Y_{{\hbar}}{\mathfrak{g}}_{k-n}^\jmath) \quad \forall\; k\in \mathbb{Z}. \end{align*} $$

The above definition implies that $\mathrm {D}Y_{\hbar }\mathfrak {g}$ is a $\mathbb {Z}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra deformation of the enveloping algebra $U(\mathfrak {t})$ , where we recall that $\mathfrak {t}=\mathfrak {g}[t^{\pm 1}]$ . Analogously to the Yangian case recalled in Section 3.2, the identification $\mathrm {D}Y_{\hbar }\mathfrak {g}/{\hbar }\mathrm {D}Y_{\hbar }\mathfrak {g} \cong U(\mathfrak {t})$ is induced by the graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra epimorphism $\mathrm {D}Y_{\hbar }\mathfrak {g}\twoheadrightarrow U(\mathfrak {t})$ given by

$$ \begin{align*} x_{ir}^\pm \mapsto x_i^\pm t^r, \quad h_{ir}\mapsto h_it^r \quad \forall\; i\in \mathbf{I}\;\text{ and }\; r\in \mathbb{Z}. \end{align*} $$

The Poincaré–Birkhoff–Witt theorem for $\mathrm {D}Y_{\hbar }\mathfrak {g}$ , established in Theorem 6.2 of [Reference Wendlandt41], asserts that $\mathrm {D}Y_{\hbar }\mathfrak {g}$ is a topologically free $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module, and thus a flat deformation of $U(\mathfrak {t})$ :

The notation for the generators of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ may seem, on the surface, to conflict with the notation used for generators in the Yangian associated with $\mathfrak {g}$ . However, there is a natural $\mathbb {Z}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra homomorphism

(4.1) $$ \begin{align} \imath:Y_{\hbar}\mathfrak{g} \to \mathrm{D}Y_{\hbar}\mathfrak{g} \end{align} $$

sending each generator of $Y_{\hbar }(\mathfrak {g})\subset Y_{\hbar }\mathfrak {g}$ to the corresponding element of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ , denoted with the same symbol. By Corollary 4.4 of [Reference Wendlandt41], $\imath $ is injective, and we shall henceforth identify $Y_{\hbar }\mathfrak {g}$ with $\imath (Y_{\hbar }\mathfrak {g})$ .

4.2 Automorphisms and root vectors

To each $i\in \mathbf {I}$ , we may associate series $\dot {x}_i^\pm (u)$ and $\dot {h}_i(u)$ in $\mathrm {D}Y_{\hbar }\mathfrak {g}[\kern-1.2pt\![u]\!\kern-1.2pt]$ by setting

$$ \begin{align*} \dot{x}_i^\pm(u):= x_i^\pm(u)-\mathcal{X}_i^\pm(u) \quad \text{ and }\quad \dot{h}_i^\pm(u):= h_i^\pm(u)-\mathcal{H}_i^\pm(u). \end{align*} $$

The following lemma is then a straightforward consequence of the defining relations of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ , where $\omega $ and $\varsigma $ are as in Lemma 3.3.

Lemma 4.4 There are unique extensions of the anti-automorphisms $\omega $ and $\varsigma $ of $Y_{\hbar }(\mathfrak {g})$ to anti-automorphisms of the $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra $\mathrm {D}Y_{\hbar }\mathfrak {g}$ such that, for each $i\in \mathbf {I}$ ,

$$ \begin{gather*} \omega(\dot{x}_i^\pm(u))=\dot{x}_{i}^\mp(u), \quad \omega(\dot{h}_i(u))=\dot{h}_i(u),\\ \varsigma(\dot{x}_i^\pm(u))=\dot{x}_i^\pm(-u), \quad \varsigma(\dot{h}_i(u))=\dot{h}_i(-u). \end{gather*} $$

Following the terminology from Section 3.3, we shall refer to the involution $\omega $ as the Chevalley involution of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ .

The adjoint action of $\mathfrak {h}\subset \mathrm {D}Y_{\hbar }\mathfrak {g}$ on $\mathrm {D}Y_{\hbar }\mathfrak {g}$ gives rise to a topological Q-grading on the $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra $\mathrm {D}Y_{\hbar }\mathfrak {g}$ (cf. Corollary 6.5 and [Reference Wendlandt41, Section 3.1]) with graded components given by the weight spaces

$$ \begin{align*} \mathrm{D}Y_{\hbar}{\mathfrak{g}}_\beta:=\{ x\in \mathrm{D}Y_{\hbar}\mathfrak{g}: [h,x]=\beta(h)x \quad \forall\; h\in \mathfrak{h}\} \quad \forall \; \beta \in Q. \end{align*} $$

That is to say, each of these subspaces is a closed $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -submodule of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ , and the direct sum

$$ \begin{align*} \mathrm{D}Y_{\hbar}{\mathfrak{g}}_Q:=\bigoplus_{\beta\in Q} \mathrm{D}Y_{\hbar}{\mathfrak{g}}_\beta \end{align*} $$

is a Q-graded dense $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -subalgebra of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ whose subspace topology coincides with its ${\hbar }$ -adic topology. Here, we have borrowed, and modified appropriately, the terminology of Section 2.2. It should be emphasized that the word topological is key in this statement, as the Q-graded algebra $\mathrm {D}Y_{\hbar }{\mathfrak {g}}_Q$ is a proper subset of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ .

We now introduce root vectors in $\mathrm {D}Y_{\hbar }\mathfrak {g}$ of arbitrary degree, following the procedure used in Section 3.4. Recall from (3.11) that to each positive root $\beta \in \Delta ^+$ , we attached an index $i(\beta )\in \mathbf {I}$ and an element $\mathbf {X}^\beta \in U({\mathfrak {n}}_{+})_{\beta -\alpha _{i(\beta )}}$ . For each $k\in \mathbb {Z}$ , we then set

$$ \begin{align*} x_{\beta,k}^+:=\mathbf{X}^\beta\cdot x_{i(\beta),k}^+\in \mathrm{D}Y_{\hbar}{\mathfrak{g}}_\beta\quad \text{ and }\quad x_{\beta,k}^-:=\omega(x_{\beta,k}^+)\in \mathrm{D}Y_{\hbar}{\mathfrak{g}}_{-\beta}, \end{align*} $$

where $\mathfrak {g}$ operates on $\mathrm {D}Y_{\hbar }\mathfrak {g}$ via the adjoint action. For $k\in \mathbb {N}$ , these elements are identical to those introduced below (3.11). Moreover, we have

$$ \begin{align*} x_\beta^\pm t^r =x_{\beta,r}^\pm\quad\mod {\hbar} \quad \forall\; \beta\in \Delta^+,\, r\in \mathbb{Z}. \end{align*} $$

It shall be convenient for us to organize the above elements into generating series $x_\beta ^\pm (u)\in \mathrm {D}Y_{\hbar }{\mathfrak {g}}_{\pm \beta }[\kern-1.2pt\![u^{-1}]\!\kern-1.2pt]$ and $\dot {x}_\beta ^\pm (u)\in \mathrm {D}Y_{\hbar }{\mathfrak {g}}_{\pm \beta }[\kern-1.2pt\![u]\!\kern-1.2pt]$ by setting

$$ \begin{align*} x_\beta^\pm(u):=\sum_{r\in \mathbb{N}} x_{\beta,r}^\pm u^{-r-1}\quad \text{ and }\quad \dot{x}_\beta^\pm(u):=-\sum_{r\in \mathbb{N}} x_{\beta,-r-1}^\pm u^r \quad \forall\quad \beta \in \Delta^+. \end{align*} $$

4.3 The formal shift operator

We now shift our attention to recalling some of the main constructions of [Reference Wendlandt41], subject to our standing assumption that $\mathfrak {g}$ is a finite-dimensional simple Lie algebra. To begin, we introduce a number of relevant spaces built from the Yangian $Y_{\hbar }(\mathfrak {g})$ , following Sections 4.1 and 4.2 of [Reference Wendlandt41] and Section 3.6 above. First, let

$$ \begin{align*} \widehat{Y_{\hbar}\mathfrak{g}}=\prod_{k\in \mathbb{N}} Y_{\hbar}(\mathfrak{g})_k \end{align*} $$

denote the formal completion of $Y_{\hbar }(\mathfrak {g})$ with respect to its $\mathbb {N}$ -grading. This is a topologically free $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra containing $Y_{\hbar }\mathfrak {g}$ as a subalgebra (see [Reference Gautam and Toledano Laredo15, Proposition 6.3] or [Reference Wendlandt41, Lemma 4.1]).

Next, let $ {L}\widehat {Y_{\hbar }\mathfrak {g}}_z$ and $\widehat {Y_{\hbar }\mathfrak {g}}_z$ denote the subspaces $ {L}(\widehat {Y_{\hbar }\mathfrak {g}}{\vphantom {Y_{\hbar }\mathfrak {g}}}^{\scriptscriptstyle {(1)}}_z)$ and $\widehat {Y_{\hbar }\mathfrak {g}}{\vphantom {Y_{\hbar }\mathfrak {g}}}^{\scriptscriptstyle {(1)}}_z$ of the space of Laurent series $Y_{\hbar }(\mathfrak {g})[z;z^{-1}]\!\kern-1.2pt]$ introduced in Section 3.6. That is,

$$ \begin{align*} \widehat{Y_{\hbar}\mathfrak{g}}_z=\prod_{k\in \mathbb{N}}Y_{\hbar}(\mathfrak{g})_k z^{-k} \subset Y_{\hbar}(\mathfrak{g})[\kern-1.2pt\![z^{-1}]\!\kern-1.2pt], \end{align*} $$

and $ {L}\widehat {Y_{\hbar }\mathfrak {g}}_z$ is the $\mathbb {Z}$ -graded subalgebra of $Y_{\hbar }(\mathfrak {g})[z;z^{-1}]\!\kern-1.2pt]$ over $\mathbb {C}[{\hbar }]$ defined by

$$ \begin{align*} {L}\widehat{Y_{\hbar}\mathfrak{g}}_z=\bigoplus_{n\in \mathbb{Z}} z^n \widehat{Y_{\hbar}\mathfrak{g}}_z. \end{align*} $$

The following lemma, established in [Reference Wendlandt41, Proposition 4.2], provides a characterization of the ${\hbar }$ -adic completion of $ {L}\widehat {Y_{\hbar }\mathfrak {g}}_z$ .

Lemma 4.5 The ${\hbar }$ -adic completion $\mathrm {L}\widehat {Y_{\hbar }\mathfrak {g}}_z$ of $ {L}\widehat {Y_{\hbar }\mathfrak {g}}_z$ is the subspace of $Y_{\hbar }\mathfrak {g}[\kern-1.2pt\![z^{\pm 1}]\!\kern-1.2pt]$ consisting of formal series

$$ \begin{align*} \sum_{k\in \mathbb{Z}} z^k f_k(z), \quad f_k(z)\in \widehat{Y_{\hbar}\mathfrak{g}}_z \end{align*} $$

with the property that, for each $n\in \mathbb {N}$ , $\exists \ N_n\in \mathbb {N}$ such that

$$ \begin{align*} f_k(z)\in ({\hbar}/z)^n \widehat{Y_{\hbar}\mathfrak{g}}_z \quad \forall \quad |k|\geq N_n. \end{align*} $$

Moreover, $\mathrm {L}\widehat {Y_{\hbar }\mathfrak {g}}_z$ is a topologically free $\mathbb {Z}$ -graded $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra with

$$ \begin{align*} (\mathrm{L}\widehat{Y_{\hbar}\mathfrak{g}}_z)_{\mathbb{Z}}={L}\widehat{Y_{\hbar}\mathfrak{g}}_z. \end{align*} $$

The last statement of the lemma employs the notation from Lemma 2.4 and follows from the fact that $ {L}\widehat {Y_{\hbar }\mathfrak {g}}_z$ is a torsion-free $\mathbb {Z}$ -graded $\mathbb {C}[{\hbar }]$ -algebra and that each subspace $z^k\widehat {Y_{\hbar }\mathfrak {g}}_z$ is closed in $Y_{\hbar }\mathfrak {g}[\kern-1.2pt\![z^{\pm 1}]\!\kern-1.2pt]$ , equipped with the ${\hbar }$ -adic topology, and thus in $\mathrm {L}\widehat {Y_{\hbar }\mathfrak {g}}_z$ .

Next, recall that $\tau _z$ is the formal shift homomorphism of the Yangian introduced in (3.8), which we may view as a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra homomorphism

$$ \begin{align*} \tau_z:Y_{\hbar}\mathfrak{g}\hookrightarrow Y_{\hbar}\mathfrak{g}[\kern-1.2pt\![z]\!\kern-1.2pt]. \end{align*} $$

In addition, we set $\partial _z^{(n)}=\frac {1}{n!}\partial _z^n$ for each $n\in \mathbb {N}$ , where $\partial _z$ is the partial derivative operator with respect to z. The following theorem, which is a combination of a special case of Theorems 4.3 and 6.2 of [Reference Wendlandt41], introduces the so-called formal shift operator $\Phi _z$ on $\mathrm {D}Y_{\hbar }\mathfrak {g}$ .

Theorem 4.6 There is a unique homomorphism of $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebras

$$ \begin{align*} \Phi_z:\mathrm{D}Y_{\hbar}\mathfrak{g}\to \mathrm{L}\widehat{Y_{\hbar}\mathfrak{g}}_z \end{align*} $$

with the property that $\Phi _z\circ \imath =\tau _z$ . Moreover:

1. (1) $\Phi _z$ is injective, and satisfies

   $$ \begin{align*} \Phi_z(\dot{x}_\beta^\pm(u))&=\sum_{n\in \mathbb{N}}(-1)^{n}u^n \partial_z^{(n)} x_\beta^\pm(-z) \quad \forall \;\beta\in \Delta^+,\\ \Phi_z(\dot{h}_i(u))&=\sum_{n\in \mathbb{N}}(-1)^{n}u^n \partial_z^{(n)} h_i(-z) \quad \forall \; i\in \mathbf{I}. \end{align*} $$

2. (2) The restriction of $\Phi _z$ to $ {D}Y_{{\hbar }}\mathfrak {g}$ is a $\mathbb {Z}$ -graded $\mathbb {C}[{\hbar }]$ -algebra homomorphism

   $$ \begin{align*} \Phi_z|_{{D}Y_{{\hbar}}\mathfrak{g}}: {D}Y_{{\hbar}}\mathfrak{g}\to {L}\widehat{Y_{\hbar}\mathfrak{g}}_z=\bigoplus_{n\in \mathbb{Z}}z^n \widehat{Y_{\hbar}\mathfrak{g}}_z \subset Y_{\hbar}(\mathfrak{g})[z;z^{-1}]\!\kern-1.2pt]. \end{align*} $$

By [Reference Wendlandt41, Proposition 4.2(4)], the evaluation map

(4.2) $$ \begin{align} \mathscr{E}\mathscr{v}:\mathrm{L}{\widehat{Y_{\hbar}\mathfrak{g}}}_z\to \widehat{Y_{\hbar}\mathfrak{g}},\quad f(z)\mapsto f(1), \end{align} $$

is an epimorphism of $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebras. We may thus compose $\Phi _z$ with $\mathscr{E}\mathscr{v}$ to obtain a $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra homomorphism

$$ \begin{align*} \Phi:=\mathscr{E}\mathscr{v}\circ \Phi_z:\mathrm{D}Y_{\hbar}\mathfrak{g}\to \widehat{Y_{\hbar}\mathfrak{g}}. \end{align*} $$

By Theorem 6.2 of [Reference Wendlandt41], this homomorphism is injective. One of the main results of [Reference Wendlandt41] is that $\Phi $ induces an isomorphism between completions of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ of $Y_{\hbar }(\mathfrak {g})$ . To make this precise, let $\mathcal {J}\subset \mathrm {D}Y_{\hbar }\mathfrak {g}$ denote the kernel of the composition

$$ \begin{align*} \mathrm{D}Y_{\hbar}\mathfrak{g}\xrightarrow{{\hbar}\mapsto 0} U(\mathfrak{g}[t^{\pm 1}])\xrightarrow{t\mapsto 1} U(\mathfrak{g}), \end{align*} $$

and define $\widehat {\mathrm {D}Y_{\hbar }\mathfrak {g}}$ to be the completion of $\mathrm {D}Y_{\hbar }\mathfrak {g}$ with respect to the $\mathcal {J}$ -adic filtration

$$ \begin{align*} \mathrm{D}Y_{\hbar}\mathfrak{g}=\mathcal{J}^0\supset \mathcal{J} \supset \mathcal{J}^2\supset \cdots \supset \mathcal{J}^n \supset \cdots. \end{align*} $$

We then have the following analogue of [Reference Gautam and Toledano Laredo15, Theorem 6.2], which is a special case of Theorem 5.5 in [Reference Wendlandt41].

Theorem 4.8 $\Phi $ is injective and induces an isomorphism of $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebras

with inverse $\Gamma $ uniquely extending the embedding $\imath \circ \tau _{-1}:Y_{\hbar }(\mathfrak {g})\to \mathrm {D}Y_{\hbar }\mathfrak {g}$ .

Remark 4.9 One subtle consequence of this result is that the natural homomorphism

$$ \begin{align*} \mathrm{D}Y_{\hbar}\mathfrak{g}\to \widehat{\mathrm{D}Y_{\hbar}\mathfrak{g}} \end{align*} $$

is injective. Indeed, its composition with the isomorphism $\widehat {\Phi }$ recovers the injection $\Phi $ . Henceforth, we shall freely make use of this fact and view $\mathrm {D}Y_{\hbar }\mathfrak {g}$ as a subalgebra of $\widehat {\mathrm {D}Y_{\hbar }\mathfrak {g}}$ . We further note that the subspace topology on $\mathrm {D}Y_{\hbar }\mathfrak {g}$ , with respect to the ${\hbar }$ -adic topology on $\widehat {\mathrm {D}Y_{\hbar }\mathfrak {g}}$ , coincides with the ${\hbar }$ -adic topology on $\mathrm {D}Y_{\hbar }\mathfrak {g}$ . Indeed, as $\widehat {\mathrm {D}Y_{\hbar }\mathfrak {g}}\cong \widehat {Y_{\hbar }\mathfrak {g}}$ is torsion-free, to see this it suffices to show that

$$ \begin{align*} {\hbar}\widehat{\mathrm{D}Y_{\hbar}\mathfrak{g}}\cap \mathrm{D}Y_{\hbar}\mathfrak{g}={\hbar} \mathrm{D}Y_{\hbar}\mathfrak{g}. \end{align*} $$

This, however, follows immediately from the injectivity of the semiclassical limit of $\Phi $ , established in [Reference Wendlandt41, Theorem 6.2]. In particular, this discussion implies that $\mathrm {D}Y_{\hbar }\mathfrak {g}$ is a closed subspace of the topological $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module $\widehat {\mathrm {D}Y_{\hbar }\mathfrak {g}}$ . Similarly, one deduces that $\Phi _z(\mathrm {D}Y_{\hbar }\mathfrak {g})$ is a closed subspace of $\mathrm {L}\widehat {Y_{\hbar }\mathfrak {g}}_z$ .

To conclude this preliminary section on $\mathrm {D}Y_{\hbar }\mathfrak {g}$ , we introduce the auxiliary $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -algebra homomorphism

(4.3) $$ \begin{align} \Gamma_z:= \Gamma\circ \mathscr{E}\mathscr{v}: \mathrm{L}\widehat{Y_{\hbar}\mathfrak{g}}_z\to \widehat{\mathrm{D}Y_{\hbar}\mathfrak{g}} \end{align} $$

which has the property that $\Gamma _z|_{\mathrm {Im}(\Phi _z)}=\Phi _z^{-1}$ . This homomorphism shall play a prominent role in the main result of Section 7 and its proof (see Theorem 7.5).

5.1 Quantum duality

To provide context, let us first briefly recall the general construction of the dual of a quantized enveloping algebra, following [Reference Drinfel’d7, Section 7] and [Reference Gavarini19, Section 4.4] (see also [Reference Etingof and Kazhdan11] and [Reference Appel and Toledano Laredo1, Section 2.19], for example).

Suppose that $U_{\hbar }\mathfrak {b}$ is a quantization of a finite-dimensional Lie bialgebra $(\mathfrak {b},\delta _{\mathfrak {b}})$ , where we follow the terminology and notation from Section 2.4. One would then like to introduce a notion of duality which sends $U_{\hbar }\mathfrak {b}$ to a quantization of the Lie bialgebra dual $({\mathfrak {b}}^\ast , \delta _{{\mathfrak {b}}^\ast }=[\,,\,]_{\mathfrak {b}}^\ast )$ to $(\mathfrak {b},\delta _{\mathfrak {b}})$ . The first crucial observation is that $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -linear dual $U_{\hbar }{\mathfrak {b}}^\ast =\mathrm {Hom}_{\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]}(U_{\hbar }\mathfrak {b},\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt])$ of $U_{\hbar }\mathfrak {b}$ is not itself a quantized enveloping algebra (see Lemma 2.1 of [Reference Gavarini19], in addition to [Reference Drinfel’d7, Section 7] and [Reference Appel and Toledano Laredo1, Section 2.19]). The correct notion of duality within the category of quantized enveloping algebras was introduced in [Reference Drinfel’d7, Section 7]. One considers the $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -submodule

$$ \begin{align*} U_{\hbar}{\mathfrak{b}}^\prime:=\{x\in U_{\hbar}\mathfrak{b}:(\mathbf{1}-\varepsilon)^{\otimes n}\Delta^n(x)\in {\hbar}^n U_{\hbar}\mathfrak{b} ^{\otimes n}\; \forall\; n\in \mathbb{N}\}\subset U_{\hbar}\mathfrak{b}, \end{align*} $$

where $\varepsilon $ and $\Delta $ are the counit and coproduct, respectively, on the topological Hopf algebra $U_{\hbar }\mathfrak {b}$ , and all notation is as in Section 5.2 below. Then, by [Reference Gavarini19, Proposition 3.6], $U_{\hbar }{\mathfrak {b}}^\prime $ is a quantized formal series Hopf algebra, with semiclassical limit isomorphic as an algebra to the completion of the symmetric algebra $\mathsf {S}(\mathfrak {b})=\bigoplus _{n\in \mathbb {N}} \mathsf {S}^n(\mathfrak {b})$ with respect to its standard grading. In particular, this means that although $U_{\hbar }{\mathfrak {b}}^\prime $ is not in general a topological Hopf algebra over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ in the sense of Section 2.1, it is a topological Hopf algebra with respect to the $\mathbf {J}_{\mathfrak {b}}$ -adic topology, where

$$ \begin{align*} \mathbf{J}_{\mathfrak{b}}={\hbar} U_{\hbar}\mathfrak{b}\cap U_{\hbar}{\mathfrak{b}}^\prime=\varepsilon|_{U_{\hbar}{\mathfrak{b}}^\prime}^{-1}({\hbar}\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]). \end{align*} $$

The subspace $ U_{\hbar }{\mathfrak {b}}^{\circ }\subset (U_{\hbar }{\mathfrak {b}}^\prime )^\ast $ consisting of continuous linear forms with respect to this topology is then a quantization of $({\mathfrak {b}}^\ast , \delta _{{\mathfrak {b}}^\ast })$ . This is the quantized enveloping algebra dual of $U_{\hbar }\mathfrak {b}$ .

Remark 5.1 Here, we note that $U_{\hbar }{\mathfrak {b}}^{\circ }$ can be equivalently defined as the ${\hbar }$ -adic completion of the $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ -module

$$ \begin{align*} (U_{\hbar}{\mathfrak{b}}^\ast)^\times=\sum_{n\in \mathbb{N}} {\hbar}^{-n} \mathfrak{m}^n_{\mathfrak{b}} \subset \mathbb{C}(\!({\hbar})\!)\otimes_{\mathbb{C}[\kern-1.2pt\![{\hbar}]\!\kern-1.2pt]}U_{\hbar}{\mathfrak{b}}^\ast, \end{align*} $$

where ${\mathfrak {m}}_{\mathfrak {b}}:=\{f\in U_{\hbar }{\mathfrak {b}}^\ast : f(1)\in {\hbar }\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]\}$ . That this produces a topological Hopf algebra which can be identified with $U_{\hbar }{\mathfrak {b}}^{\circ }$ is a nontrivial result, which is part of the quantum duality principle. This was first announced in [Reference Drinfel’d7, Section 7], and proven in detail in [Reference Gavarini19] (see Theorem 1.6 therein). We will not, however, need this equivalent formulation in the present article.

In our setting, $\mathfrak {b}={\mathfrak {t}}_{\scriptscriptstyle {+}}=\mathfrak {g}[t]$ is not finite-dimensional, but rather an $\mathbb {N}$ -graded Lie bialgebra $\mathfrak {b}=\bigoplus _{n\in \mathbb {N}}\mathfrak {b}_n$ with finite-dimensional graded components. As $Y_{\hbar }\mathfrak {g}=U_{\hbar }\mathfrak {b}$ is a homogeneous quantization of ${\mathfrak {t}}_{\scriptscriptstyle {+}}$ , the above construction remains valid, provided the notion of duality is adjusted so as to respect the underlying gradings. In fact, one can replace $U_{\hbar }{\mathfrak {b}}^\prime $ with an $\mathbb {N}$ -graded topological Hopf algebra ${\dot {\mathsf {Y}}}_{\hbar }\mathfrak {g}$ over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ of finite type, and $U_{\hbar }{\mathfrak {b}}^\circ $ with the restricted dual of ${\dot {\mathsf {Y}}}_{\hbar }\mathfrak {g}$ , as defined in Section 2.3. The topological Hopf algebra ${\dot {\mathsf {Y}}}_{\hbar }\mathfrak {g}$ and its $\mathbb {C}[{\hbar }]$ -form ${\dot {\mathsf {Y}}}_{\hbar }(\mathfrak {g})=({\dot {\mathsf {Y}}}_{\hbar }\mathfrak {g})_{\mathbb {N}}$ (see Section 2.2) are the focus of the present section.

5.2 The Drinfeld–Gavarini subalgebra

Let us define $\Delta ^n$ for any $n\in \mathbb {N}$ by setting $\Delta ^0=\varepsilon $ , $\Delta ^1=\mathbf {1}=\mathbf {1}_{Y_{\hbar }(\mathfrak {g})}$ and

$$ \begin{align*} \Delta^n:=(\Delta\otimes \mathbf{1}^{\otimes (n-2)})\circ \Delta^{n-1}:Y_{\hbar}(\mathfrak{g})\to Y_{\hbar}(\mathfrak{g})^{\otimes n} \end{align*} $$

for all $n\geq 2$ . We then define the $\mathbb {C}[{\hbar }]$ -submodule ${\dot {\mathsf {Y}}}_{\hbar }(\mathfrak {g})\subset Y_{\hbar }(\mathfrak {g})$ by

$$ \begin{align*} {\dot{\mathsf{Y}}}_{\hbar}(\mathfrak{g}):=\{x\in Y_{\hbar}(\mathfrak{g}):(\mathbf{1}-\varepsilon)^{\otimes n} \Delta^n(x)\in {\hbar}^nY_{\hbar}(\mathfrak{g})^{\otimes n} \; \forall \; n\in \mathbb{N}\}. \end{align*} $$

By Lemma 3.2 and Proposition 3.5 of [Reference Kassel and Turaev27] (see also [Reference Gavarini19, Proposition 2.6], [Reference Gavarini20, Theorem 3.5], and [Reference Finkelberg and Tsymbaliuk13, Lemma A.1]), ${\dot {\mathsf {Y}}}_{\hbar }(\mathfrak {g})$ is a subalgebra of $Y_{\hbar }(\mathfrak {g})$ which is commutative modulo ${\hbar }$ :