2.1 Locally presentable categories

Let k be a field. All categories and functors we will consider are k-linear. Throughout this paper, we work with locally presentable categories (refer to [Reference Adámek and Rosický1] and [Reference Brandenburg, Chirvasitu and Johnson-Freyd15, Section 2] for more details). Here are the main examples:

• ○ If $\mathcal {C}$ is a small category, the category of presheaves $\mathrm {Fun}(\mathcal {C}^{\mathrm {op}}, \mathrm {Vect})$ is locally presentable. For instance, this applies to the category $\mathrm {LMod}_A$ of (left) modules over a k-algebra A.

• ○ If $\mathcal {C}$ is a small category that admits finite colimits, the ind-completion $\mathrm {Ind}(\mathcal {C})$ (see [Reference Kashiwara and Schapira51, Chapter 6] for what it means) is locally presentable.

• ○ If C is a k-coalgebra, the category of C-comodules $\mathrm {CoMod}_C$ is locally presentable (see [Reference Wischnewsky91, Corollary 9], noting that a Grothendieck category is locally presentable). In fact, $\mathrm {CoMod}_C$ is the ind-completion of the category of finite-dimensional C-comodules (see [Reference Rivano71, Corollaire 2.2.2.3]).

• ○ If $\mathcal {C}, \mathcal {D}$ are locally presentable categories, the category $\mathrm {Fun}^{\mathrm {L}}(\mathcal {C}, \mathcal {D})$ of colimit-preserving functors from $\mathcal {C}$ to $\mathcal {D}$ is locally presentable.

It turns out that many examples of locally presentable categories are, in fact, presheaf categories.

We denote by $\mathcal {C}^{\mathrm {cp}}\subset \mathcal {C}$ the full subcategory of compact projective objects.

Proposition 2.2. Suppose $\mathcal {C}$ has enough compact projectives. Then the functor

$$\begin{align*}\mathcal{C}\longrightarrow \mathrm{Fun}((\mathcal{C}^{\mathrm{cp}})^{\mathrm{op}}, \mathrm{Vect})\end{align*}$$

given by $x\mapsto (y\mapsto \mathrm {Hom}_{\mathcal {C}}(y, x))$ is an equivalence.

Locally presentable categories naturally form a symmetric monoidal 2-category $\mathrm {Pr}^{\mathrm {L}}$ [Reference Bird13]:

• ○ Its objects are locally presentable categories.

• ○ Its 1-morphisms are colimit-preserving functors.

• ○ Its 2-morphisms are natural transformations.

• ○ The tensor product is uniquely determined by the following property: for $\mathcal {C},\mathcal {D},\mathcal {E}\in \mathrm {Pr}^{\mathrm {L}}$ a colimit-preserving functor $\mathcal {C}\otimes \mathcal {D}\rightarrow \mathcal {E}$ is the same as a bifunctor $\mathcal {C}\times \mathcal {D}\rightarrow \mathcal {E}$ preserving colimits in each variables.

• ○ The unit is $\mathrm {Vect}\in \mathrm {Pr}^{\mathrm {L}}$ .

An important fact about locally presentable categories is that a colimit-preserving functor between locally presentable categories admits a right adjoint. We will now write a formula for the adjoint, assuming the source category has enough compact projectives. Let us first recall the notion of a coend (see [Reference Loregian60] for more details on coends).

Definition 2.3. Suppose $\mathcal {C}$ and $\mathcal {D}$ are locally presentable categories. The coend of a bifunctor $F\colon \mathcal {C}\times \mathcal {C}^{\mathrm {op}}\rightarrow \mathcal {D}$ is the coequaliser

We will use the following Yoneda-like property of coends (see [Reference Loregian60, Proposition 2.2.1]).

Proposition 2.4. For any functors $F\colon \mathcal {C}\rightarrow \mathcal {D}, G\colon \mathcal {C}^{\mathrm {op}}\rightarrow \mathcal {D}$ , we have natural isomorphisms

$$\begin{align*}\int^{x\in\mathcal{C}} \mathrm{Hom}_{\mathcal{C}}(x, y)\otimes F(x)\cong F(y),\qquad \int^{x\in\mathcal{C}} \mathrm{Hom}_{\mathcal{C}}(y, x)\otimes G(x)\cong G(y).\end{align*}$$

The following is an immediate corollary.

Proposition 2.5. Suppose $F\colon \mathcal {C}\rightarrow \mathcal {D}$ is a colimit-preserving functor of locally presentable categories, where $\mathcal {C}$ has enough compact projectives. Then the right adjoint is given by the coend

$$\begin{align*}F^{\mathrm{R}}(x) = \int^{y\in\mathcal{C}^{\mathrm{cp}}} \mathrm{Hom}_{\mathcal{D}}(F(y), x)\otimes y.\end{align*}$$

The counit of the adjunction $FF^{\mathrm {R}}(x)\rightarrow x$ is given by the evaluation map $\mathrm {Hom}_{\mathcal {D}}(F(y), x)\otimes F(y)\rightarrow x$ ; the unit of the adjunction $z\rightarrow F^{\mathrm {R}} F(z)$ is given by including the identity map $\mathrm {id}\colon F(z)\rightarrow F(z)$ in the coend.

2.2 Cp-rigidity

By convention, all monoidal categories $\mathcal {C}$ that we consider in this paper are locally presentable such that the tensor product bifunctor $\mathcal {C}\times \mathcal {C}\rightarrow \mathcal {C}$ preserves colimits in each variable. So, by the universal property of the tensor product in $\mathrm {Pr}^{\mathrm {L}}$ , it descends to a colimit-preserving functor

$$\begin{align*}T\colon \mathcal{C}\otimes\mathcal{C}\longrightarrow \mathcal{C}.\end{align*}$$

We denote by $\mathcal {C}^{\otimes \mathrm {op}}$ the same category with the opposite monoidal structure.

We will consider rigid monoidal categories in the text. Since we work with large categories, we cannot expect all objects to be dualisable (as in the category of all vector spaces); instead, we will restrict our attention to compact projective objects.

Proof. We have

$$\begin{align*}\mathrm{Hom}_{\mathcal{C}}(x\otimes y, -) \cong \mathrm{Hom}_{\mathcal{C}}(x, (-)\otimes y^\vee). \end{align*}$$

By assumption, the tensor product preserves colimits in each variable, so $(-)\otimes y^\vee $ is colimit-preserving. Since x is compact projective, $\mathrm {Hom}_{\mathcal {C}}(x, -)$ is colimit-preserving. Therefore, $\mathrm {Hom}_{\mathcal {C}}(x\otimes y, -)$ is also colimit-preserving.

If $\mathcal {C}$ is cp-rigid, the tensor product functor $T\colon \mathcal {C}\otimes \mathcal {C}\rightarrow \mathcal {C}$ admits a colimit-preserving right adjoint $T^{\mathrm {R}}\colon \mathcal {C}\rightarrow \mathcal {C}\otimes \mathcal {C}$ (see, e.g., [Reference Brochier, Jordan and Snyder18, Section 5.3]). It has the following explicit formula.

Proposition 2.8. Suppose $\mathcal {C}$ is a cp-rigid monoidal category. Then

$$\begin{align*}T^{\mathrm{R}}(y) \cong \int^{x\in\mathcal{C}^{\mathrm{cp}}} (y\otimes x^\vee)\boxtimes x.\end{align*}$$

Proof. By Theorem 2.5, the right adjoint is

$$\begin{align*}T^{\mathrm{R}}(y) \cong \int^{x_1,x_2\in\mathcal{C}^{\mathrm{cp}}} \mathrm{Hom}_{\mathcal{C}}(x_1\otimes x_2, y)\otimes (x_1\boxtimes x_2).\end{align*}$$

Since compact projective objects in $\mathcal {C}$ are dualisable, we can rewrite it as

$$ \begin{align*} T^{\mathrm{R}}(y)&\cong \int^{x_1,x_2\in\mathcal{C}^{\mathrm{cp}}} \mathrm{Hom}_{\mathcal{C}}(x_1, y\otimes x_2^\vee)\otimes (x_1\boxtimes x_2) \\ &\cong \int^{x_2\in\mathcal{C}^{\mathrm{cp}}} (y\otimes x_2^\vee)\boxtimes x_2, \end{align*} $$

where in the last isomorphism, we have used Theorem 2.4.

Consider $\mathcal {C}\otimes \mathcal {C}$ as a $\mathcal {C}\otimes \mathcal {C}^{\otimes \mathrm {op}}$ -module category via the left action on the first factor and the right action on the second factor. By [Reference Brochier, Jordan and Snyder18, Proposition 4.1], $T^{\mathrm {R}}$ is a functor of $\mathcal {C}\otimes \mathcal {C}^{\otimes \mathrm {op}}$ -module categories. This can be expressed in the following isomorphism.

Proposition 2.9. Suppose $\mathcal {C}$ is as before. Then $T^{\mathrm {R}}\colon \mathcal {C}\rightarrow \mathcal {C}\otimes \mathcal {C}$ is a functor of $\mathcal {C}\otimes \mathcal {C}^{\otimes \mathrm {op}}$ -module categories. Concretely, for any object $y\in \mathcal {C}$ , there is a natural isomorphism

$$\begin{align*}\int^{x\in\mathcal{C}^{\mathrm{cp}}} (y\otimes x^\vee)\boxtimes x\cong \int^{x\in\mathcal{C}^{\mathrm{cp}}} x^\vee\boxtimes (x\otimes y)\end{align*}$$

that is given for a compact projective $y\in \mathcal {C}$ by

$$\begin{align*}x^\vee\boxtimes(x\otimes y)\xrightarrow{\mathrm{coev}_y\otimes \mathrm{id}} (y\otimes y^\vee\otimes x^\vee)\boxtimes (x\otimes y)\xrightarrow{\pi_{x\otimes y}} \int^{x\in\mathcal{C}^{\mathrm{cp}}} (y\otimes x^\vee)\boxtimes x.\end{align*}$$

The key property of cp-rigid monoidal categories is that they are canonically self-dual objects of $\mathrm {Pr}^{\mathrm {L}}$ .

Theorem 2.11. Let $\mathcal {C}$ be a cp-rigid monoidal category with a compact projective unit. The evaluation and coevaluation pairings

(11) $$ \begin{align} \mathrm{ev}&\colon \mathcal{C}\otimes\mathcal{C}\xrightarrow{T}\mathcal{C}\xrightarrow{\mathrm{Hom}_{\mathcal{C}}(\mathbf{1}, -)}\mathrm{Vect} \end{align} $$

(12) $$ \begin{align} \mathrm{coev}&\colon \mathrm{Vect}\xrightarrow{(-)\otimes\mathbf{1}}\mathcal{C}\xrightarrow{T^{\mathrm{R}}}\mathcal{C}\otimes \mathcal{C}. \end{align} $$

establish the self-duality of $\mathcal {C}$ as an object of the symmetric monoidal bicategory $\mathrm {Pr}^{\mathrm {L}}$ .

Corollary 2.13. Let $\mathcal {C}$ be a cp-rigid monoidal category with a compact projective unit and $\mathcal {D}$ any monoidal category. Then the functor

(13) $$ \begin{align} \mathcal{D}\otimes \mathcal{C}\longrightarrow \mathrm{Fun}^{\mathrm{L}}(\mathcal{C}, \mathcal{D}) \end{align} $$

given by

$$\begin{align*}d\boxtimes c\mapsto (c'\mapsto \mathrm{ev}(c, c')\otimes d)\end{align*}$$

is an equivalence.

2.3 Duoidal categories

Let us now study the monoidal properties of the equivalence given by equation (13). The functor category $\mathrm {Fun}^{\mathrm {L}}(\mathcal {C}, \mathcal {D})$ has a natural monoidal structure given by the Day convolution [Reference Day24] defined by

(14) $$ \begin{align} (F\otimes_{\mathrm{Day}} G)(x) = \int^{x_1,x_2\in\mathcal{C}^{\mathrm{cp}}} \mathrm{Hom}_{\mathcal{C}}(x_1\otimes x_2, x)\otimes F(x_1)\otimes G(x_2) \end{align} $$

with the unit functor

$$\begin{align*}x\mapsto \mathrm{Hom}_{\mathcal{C}}(\mathbf{1}_{\mathcal{C}}, x)\otimes \mathbf{1}_{\mathcal{D}}.\end{align*}$$

Proposition 2.14. The equivalence given by equation (13) upgrades to a monoidal equivalence

$$\begin{align*}\mathcal{D}\otimes\mathcal{C}^{\otimes\mathrm{op}}\xrightarrow{\sim} \mathrm{Fun}^{\mathrm{L}}(\mathcal{C}, \mathcal{D}),\end{align*}$$

where we equip $\mathrm {Fun}^{\mathrm {L}}(\mathcal {C}, \mathcal {D})$ with the Day convolution monoidal structure.

Proof. Clearly, the units are compatible since $\mathbf {1}_{\mathcal {D}}\boxtimes \mathbf {1}_{\mathcal {C}}$ is sent to the functor $(x\mapsto \mathrm {ev}(\mathbf {1}_{\mathcal {C}}, x)\otimes \mathbf {1}_{\mathcal {D}})$ .

Now consider two objects $d_1\boxtimes c_1, d_2\boxtimes c_2\in \mathcal {D}\otimes \mathcal {C}$ . Their Day convolution is computed by

$$ \begin{align*} &((d_1\boxtimes c_1)\otimes_{\mathrm{Day}} (d_2\boxtimes c_2))(x) \\ = &\int^{x_1,x_2\in\mathcal{C}^{\mathrm{cp}}} \mathrm{Hom}_{\mathcal{C}}(x_1\otimes x_2, x)\otimes (d_1\otimes d_2)\otimes \mathrm{Hom}_{\mathcal{C}}(\mathbf{1}, c_1\otimes x_1)\otimes \mathrm{Hom}_{\mathcal{C}}(\mathbf{1}, c_2\otimes x_2). \end{align*} $$

So, we have to exhibit a natural isomorphism

$$\begin{align*}\mathrm{Hom}_{\mathcal{C}}(\mathbf{1}, c_2\otimes c_1\otimes x)\cong \int^{x_1,x_2\in\mathcal{C}^{\mathrm{cp}}} \mathrm{Hom}_{\mathcal{C}}(x_1\otimes x_2, x)\otimes \mathrm{Hom}_{\mathcal{C}}(\mathbf{1}, c_1\otimes x_1)\otimes \mathrm{Hom}_{\mathcal{C}}(\mathbf{1}, c_2\otimes x_2).\end{align*}$$

By assumption, $\mathcal {C}$ is generated by compact projectives, so it is enough to define this isomorphism on those.

The right-hand side is

$$ \begin{align*} &\quad \int^{x_1,x_2} \mathrm{Hom}(x_1\otimes x_2, x)\otimes \mathrm{Hom}(\mathbf{1}, c_1\otimes x_1)\otimes \mathrm{Hom}(\mathbf{1}, c_2\otimes x_2) \\ &\cong \int^{x_1,x_2} \mathrm{Hom}(x_1\otimes x_2, x)\otimes \mathrm{Hom}(c_1^\vee, x_1)\otimes \mathrm{Hom}(c_2^\vee, x_2) \\ &\cong \mathrm{Hom}(c_1^\vee\otimes c_2^\vee, x) \\ &\cong \mathrm{Hom}(\mathbf{1}, c_2\otimes c_1\otimes x), \end{align*} $$

where we have used Theorem 2.4 in the third line.

We will now examine the monoidal properties of the self-duality pairings given by equations (11) and (12).

Proposition 2.15. The functors

$$\begin{align*}\mathrm{ev}\colon \mathcal{C}^{\otimes\mathrm{op}}\otimes \mathcal{C}\longrightarrow \mathrm{Vect},\qquad \mathrm{coev}\colon \mathrm{Vect}\rightarrow \mathcal{C}\otimes \mathcal{C}^{\otimes\mathrm{op}}\end{align*}$$

have a natural lax monoidal structure.

Proof. We begin with the evaluation functor. The unit map $k\rightarrow \mathrm {ev}(\mathbf {1}, \mathbf {1})=\mathrm {Hom}_{\mathcal {C}}(\mathbf {1}, \mathbf {1})$ is given by the inclusion of the identity. Suppose $c_1\boxtimes c_2,d_1\boxtimes d_2\in \mathcal {C}^{\otimes \mathrm {op}}\otimes \mathcal {C}$ are two compact projective objects. Then we define $\mathrm {ev}(c_1, c_2)\otimes \mathrm {ev}(d_1, d_2)\rightarrow \mathrm {ev}(d_1\otimes c_1, c_2\otimes d_2)$ via the commutative diagram

Next we consider the coevaluation functor. A lax monoidal structure on $\mathrm {coev}$ is the same as an algebra structure on $\mathrm {coev}(k)=T^{\mathrm {R}}(\mathbf {1})$ , which, in turn, is provided by Theorem 2.10.

Now suppose $\mathcal {C},\mathcal {D}$ are cp-rigid monoidal categories with compact projective units and $\mathcal {E}$ any monoidal category. Then the composition functor

$$\begin{align*}\mathrm{Fun}^{\mathrm{L}}(\mathcal{D}, \mathcal{E})\otimes \mathrm{Fun}^{\mathrm{L}}(\mathcal{C}, \mathcal{D})\longrightarrow \mathrm{Fun}^{\mathrm{L}}(\mathcal{C}, \mathcal{E})\end{align*}$$

has a natural lax monoidal structure with respect to the Day convolution.

Proposition 2.16. Suppose $\mathcal {C},\mathcal {D},\mathcal {E}$ are as above. The diagrams

and

of lax monoidal functors with respect to the Day convolution commute up to a monoidal natural isomorphism.

Recall the following notion (see [Reference Aguiar and Mahajan2, Definition 6.1], where it is called a 2-monoidal category).

Example 2.19. Consider the category $\mathcal {C}\otimes \mathcal {C}$ . It carries a convolution monoidal structure $\circ $ defined by

$$\begin{align*}(M_1\boxtimes M_2)\circ (N_1\boxtimes N_2) = \mathrm{ev}(M_2, N_1)\otimes M_1\boxtimes N_2\end{align*}$$

whose unit is $I = \mathrm {coev}(k)\in \mathcal {C}\otimes \mathcal {C}$ . It also carries a pointwise monoidal structure $\mathcal {C}\otimes \mathcal {C}^{\otimes \mathrm {op}}$ whose unit is $J = \mathbf {1}_{\mathcal {C}}\boxtimes \mathbf {1}_{\mathcal {C}}$ . It follows from Theorem 2.15 that the two monoidal structures are compatible, so $\mathcal {C}\otimes \mathcal {C}$ becomes a duoidal category. Moreover, $\mathcal {C}$ is naturally a module category over $\mathcal {C}\otimes \mathcal {C}$ with respect to convolution

$$\begin{align*}(\mathcal{C}\otimes \mathcal{C})\otimes \mathcal{C}\longrightarrow \mathcal{C}\end{align*}$$

and is given by

$$\begin{align*}(c_1\boxtimes c_2)\boxtimes d\mapsto \mathrm{ev}(c_2, d)\otimes c_1.\end{align*}$$

We are now ready to relate the two duoidal structures. The following statement combines Theorems 2.14 and 2.16.

Theorem 2.20. Suppose $\mathcal {C}$ is a cp-rigid monoidal category with a compact projective unit. The equivalence given by equation (13)

$$\begin{align*}\mathcal{C}\otimes \mathcal{C}\longrightarrow \mathrm{Fun}^{\mathrm{L}}(\mathcal{C}, \mathcal{C})\end{align*}$$

given by $c_1\boxtimes c_2\mapsto (d\mapsto \mathrm {ev}(c_2, d)\otimes c_1)$ upgrades to an equivalence of duoidal categories, where the two monoidal structures are the convolution product and the pointwise monoidal structure on $\mathcal {C}\otimes \mathcal {C}^{\otimes \mathrm {op}}$ while the two monoidal structures on $\mathrm {Fun}^{\mathrm {L}}(\mathcal {C}, \mathcal {C})$ are the composition of functors and Day convolution. This equivalence intertwines $\mathcal {C}$ as a $\mathcal {C}\otimes \mathcal {C}$ -module category with respect to convolution and $\mathcal {C}$ as a $\mathrm {Fun}^{\mathrm {L}}(\mathcal {C}, \mathcal {C})$ -module category with respect to composition of functors.

2.4 Bimodules and lax monoidal functors

Suppose $f\colon A\rightarrow B$ is a homomorphism of algebras. Then B becomes an $(A, B)$ -bimodule with a distinguished element given by $1\in B$ . Conversely, the data of an $(A, B)$ -bimodule M with a distinguished element $1_M\in M$ such that the action map $B\rightarrow M$ is an isomorphism, is the same as the data of a homomorphism $A\rightarrow B$ . In this section, we will describe a similar construction on the categorical level. Recall from [Reference Etingof, Gelaki, Nikshych and Ostrik34, Chapters 7.1,7.2] the notion of a module category over a monoidal category.

Suppose $\mathcal {C}$ and $\mathcal {D}$ are monoidal categories and $\mathcal {M}$ a $(\mathcal {C}, \mathcal {D})$ bimodule category together with a distinguished object $\mathrm {Dist}\in \mathcal {M}$ . The action functors of $\mathcal {C}$ and $\mathcal {D}$ on $\mathrm {Dist}\in \mathcal {M}$ define colimit-preserving functors

$$\begin{align*}\mathrm{act}_{\mathcal{C}}\colon \mathcal{C}\longrightarrow \mathcal{M},\qquad \mathrm{act}_{\mathcal{D}}\colon \mathcal{D}\longrightarrow \mathcal{M},\end{align*}$$

which we write as $x\mapsto x\otimes \mathrm {Dist}$ and $y\mapsto \mathrm {Dist}\otimes y$ , respectively. By the adjoint functor theorem, these admit right adjoints that we denote by $\mathrm {act}^{\mathrm {R}}_{\mathcal {C}}$ and $\mathrm {act}^{\mathrm {R}}_{\mathcal {D}}$ . The counit of the adjunction defines a natural morphism

$$\begin{align*}\epsilon\colon \mathrm{Dist}\otimes\mathrm{act}^{\mathrm{R}}_{\mathcal{D}}(m)\rightarrow m\end{align*}$$

for $m\in \mathcal {M}$ . Moreover, $\mathrm {act}_{\mathcal {D}}\colon \mathcal {D}\rightarrow \mathcal {M}$ is a functor of right $\mathcal {D}$ -module categories, so $\mathrm {act}^{\mathrm {R}}_{\mathcal {D}}\colon \mathcal {M}\rightarrow \mathcal {D}$ is a lax $\mathcal {D}$ -module functor: that is, we have a natural morphism

$$\begin{align*}\phi\colon\mathrm{act}^{\mathrm{R}}_{\mathcal{D}}(m)\otimes y\longrightarrow \mathrm{act}^{\mathrm{R}}_{\mathcal{D}}(m\otimes y)\end{align*}$$

satisfying an associativity axiom.

Consider the functor

$$\begin{align*}F_{\mathcal{C}\mathcal{D}}=\mathrm{act}^{\mathrm{R}}_{\mathcal{D}}\circ\mathrm{act}_{\mathcal{C}}\colon \mathcal{C}\longrightarrow \mathcal{D}.\end{align*}$$

Proposition 2.21. The morphisms

$$\begin{align*}\mathbf{1}_{\mathcal{D}}\rightarrow \mathrm{act}^{\mathrm{R}}_{\mathcal{D}}\circ \mathrm{act}_{\mathcal{D}}(\mathbf{1}_{\mathcal{D}})\cong \mathrm{act}^{\mathrm{R}}_{\mathcal{D}}\circ \mathrm{act}_{\mathcal{C}}(\mathbf{1}_{\mathcal{C}})\end{align*}$$

and

$$ \begin{align*} \mathrm{act}^{\mathrm{R}}_{\mathcal{D}}(x\otimes\mathrm{Dist})\otimes\mathrm{act}^{\mathrm{R}}_{\mathcal{D}}(y\otimes\mathrm{Dist})&\rightarrow \mathrm{act}^{\mathrm{R}}_{\mathcal{D}}(x\otimes \mathrm{Dist}\otimes \mathrm{act}^{\mathrm{R}}_{\mathcal{D}}(y\otimes \mathrm{Dist})) \\ &\rightarrow \mathrm{act}^{\mathrm{R}}_{\mathcal{D}}(x\otimes y\otimes \mathrm{Dist}) \end{align*} $$

define the structure of a lax monoidal functor on $F_{\mathcal {C}\mathcal {D}}$ .

Proof. Let us prove the associativity condition. For brevity, denote $a^{\mathrm {R}}=\mathrm {act}^{\mathrm {R}}_{\mathcal {D}}$ , $D=\mathrm {Dist}$ . We have to show that the diagram

is commutative. Using naturality and the associativity condition for the lax module structure on $\mathrm {act}^{\mathrm {R}}$ , the above diagram is reduced to

The top segment commutes by naturality of $\phi $ . The middle segment commutes since $\epsilon $ is a natural transformation of $\mathcal {D}$ -module functors. The bottom segment commutes by naturality of $\epsilon $ .

Unitality is proven analogously.

Note that in the above construction, we may freely replace $\mathcal {C}$ and $\mathcal {D}$ , so we similarly obtain a lax monoidal functor

$$\begin{align*}F_{\mathcal{D}\mathcal{C}}\colon \mathcal{D}\longrightarrow \mathcal{C}.\end{align*}$$

2.5 Tannaka reconstruction for bialgebras

Recall the Tannaka reconstruction results for bialgebras; refer to [Reference Deligne25, Reference Rivano71] for the commutative case and [Reference Ulbrich85, Reference Ulbrich86, Reference Schauenburg75] for the general case.

Let $B\in \mathrm {Vect}$ be a bialgebra. Then $\mathcal {C}=\mathrm {CoMod}_B$ , the category of (left) B-comodules, is locally presentable. Moreover, it is equipped with a conservative and colimit-preserving monoidal forgetful functor $F\colon \mathcal {C}\rightarrow \mathrm {Vect}$ , which admits a colimit-preserving right adjoint $F^{\mathrm {R}}\colon \mathrm {Vect}\rightarrow \mathcal {C}$ sending V to the cofree B-comodule $B\otimes V$ cogenerated by V. There is a converse to this statement.

Proposition 2.24. Suppose $\mathcal {C}$ is a monoidal category with a colimit-preserving monoidal forgetful functor $F\colon \mathcal {C}\rightarrow \mathrm {Vect}$ , which admits a colimit-preserving right adjoint $F^{\mathrm {R}}\colon \mathrm {Vect}\rightarrow \mathcal {C}$ . Then $B=FF^{\mathrm {R}}(k)$ is a bialgebra and F factors as

$$\begin{align*}\mathcal{C}\longrightarrow \mathrm{CoMod}_B.\end{align*}$$

Moreover, the latter functor is an equivalence if and only if F is conservative and preserves equalisers.

Remark 2.25. A more familiar statement of Tannaka reconstruction is obtained by passing to compact objects in the above statement. Namely, for a small abelian monoidal category $\mathcal {C}^{\mathrm {c}}$ with a biexact tensor product and a monoidal functor

$$\begin{align*}F\colon \mathcal{C}^{\mathrm{c}}\longrightarrow \mathrm{Vec}\end{align*}$$

to the category of finite-dimensional vector spaces, there is a canonical bialgebra B (the bialgebra of coendomorphisms of F; see [Reference Etingof, Gelaki, Nikshych and Ostrik34, Section 1.10]) such that F factors through

$$\begin{align*}\mathcal{C}^{\mathrm{c}}\longrightarrow \mathrm{CoMod}^{\mathrm{fd}}_B\end{align*}$$

through the category of finite-dimensional B-comodules. Moreover, the latter functor is an equivalence if and only if F is exact and faithful. Refer to [Reference Etingof, Gelaki, Nikshych and Ostrik34, Section 5.4] for more details.

Let us now be more explicit. Consider the setup of Theorem 2.24, where $\mathcal {C}$ is a monoidal category with enough compact projectives and a compact projective unit. Since $F^{\mathrm {R}}$ preserves colimits, F preserves compact projective objects. In particular, for $y\in \mathcal {C}^{\mathrm {cp}}$ , the vector space $F(y)$ is finite-dimensional. So, by Theorem 2.5, the bialgebra B is

(15) $$ \begin{align} B = \int^{y\in\mathcal{C}^{\mathrm{cp}}} F(y)^\vee\otimes F(y). \end{align} $$

For $y\in \mathcal {C}^{\mathrm {cp}}$ let us denote by

$$\begin{align*}\pi_y\colon F(y)^\vee \otimes F(y)\rightarrow B\end{align*}$$

the natural projection. For $y,z\in \mathcal {C}$ , denote by

$$\begin{align*}J_{y, z}\colon F(y)\otimes F(z)\xrightarrow{\sim} F(y\otimes z)\end{align*}$$

the monoidal structure on F (the unit isomorphism will be implicit). The bialgebra structure on B is given on generators as follows:

• ○ The coproduct is

  $$\begin{align*}F(y)^\vee\otimes F(y)\xrightarrow{\mathrm{id}\otimes\mathrm{coev}_{F(y)}\otimes \mathrm{id}}F(y)^\vee\otimes F(y)\otimes F(y)^\vee\otimes F(y)\xrightarrow{\pi_y\otimes \pi_y} B\otimes B.\end{align*}$$

• ○ The counit is

  $$\begin{align*}F(y)^\vee\otimes F(y)\xrightarrow{\mathrm{ev}_{F(y)}} k.\end{align*}$$

• ○ The product is

  $$ \begin{align*} (F(y)^\vee\otimes F(y))\otimes (F(z)^\vee\otimes F(z))&\cong (F(y)\otimes F(z))^\vee\otimes F(y)\otimes F(z)\\ &\xrightarrow{(J_{y,z}^{-1})^\vee\otimes J_{y,z}} F(y\otimes z)^\vee\otimes F(y\otimes z)\\ &\xrightarrow{\pi_{y\otimes z}} B. \end{align*} $$

• ○ The unit is

  $$\begin{align*}k\cong F(\mathbf{1})^\vee\otimes F(\mathbf{1})\xrightarrow{\pi_{\mathbf{1}}} B.\end{align*}$$

It will also be useful to think about $\pi _y$ as elements

$$\begin{align*}T_y\in B\otimes \mathrm{End}(F(y)).\end{align*}$$

The following statement is immediate from the above formulas.

Theorem 2.26. The bialgebra B is spanned, as a k-vector space, by the matrix coefficients of $T_y$ for $y\in \mathcal {C}^{\mathrm {cp}}$ , subject to the relation

(16) $$ \begin{align} F(f) \circ T_x = T_y \circ F(f) \end{align} $$

for every $f\colon x\rightarrow y$ . Moreover:

• ○ For $y\in \mathcal {C}^{\mathrm {cp}}$ , we have

  (17) $$ \begin{align} \Delta(T_y) = T_y\otimes T_y. \end{align} $$

• ○ For $y\in \mathcal {C}^{\mathrm {cp}}$ , we have

  (18) $$ \begin{align} \epsilon(T_y) = \mathrm{id}_{F(y)}\in\mathrm{End}(F(y)). \end{align} $$

• ○ Suppose $x,y\in \mathcal {C}^{\mathrm {cp}}$ are two objects. Then

  (19) $$ \begin{align} J^{-1}_{x,y}T_{x\otimes y}J_{x,y}=(T_x\otimes \mathrm{id}_{F(y)})(\mathrm{id}_{F(x)}\otimes T_y) \end{align} $$
  as elements of $B\otimes \mathrm {End}(F(x\otimes y))\cong B\otimes \mathrm {End}(F(x)\otimes F(y))$ .

• ○ $T_{\mathbf {1}}\in B\otimes \mathrm {End}(F(\mathbf {1}))\cong B$ is the unit.

Let us now study what happens when $\mathcal {C}$ is in addition equipped with a braiding.

Definition 2.27. Suppose $\mathcal {C}$ is a braided monoidal category and $F\colon \mathcal {C}\rightarrow \mathrm {Vect}$ a monoidal functor. For $x,y\in \mathcal {C}$ , the $\boldsymbol {R}$ -matrix is

$$\begin{align*}R_{x,y}\colon F(x)\otimes F(y)\xrightarrow{J_{x,y}} F(x\otimes y)\xrightarrow{F(\sigma_{x, y})} F(y\otimes x)\xrightarrow{J_{y,x}^{-1}} F(y)\otimes F(x) \xrightarrow{\sigma_{F(x), F(y)}^{-1}} F(x)\otimes F(y).\end{align*}$$

It will be convenient to use the standard matrix notation for R-matrices acting on several variables: given $x,y,z\in \mathcal {C}$ , we denote

$$\begin{align*}R_{12} = R_{x,y}\otimes \mathrm{id}\end{align*}$$

as an element of $\mathrm {End}(F(x)\otimes F(y)\otimes F(z))$ and similarly for $R_{13}$ and $R_{23}$ . We let the transposed R-matrix $R_{21}$ be

$$\begin{align*}F(x)\otimes F(y)\xrightarrow{\sigma^{-1}} F(y)\otimes F(x)\xrightarrow{R_{y, x}} F(y)\otimes F(x)\xrightarrow{\sigma} F(x)\otimes F(y).\end{align*}$$

We also denote

$$\begin{align*}T_1 = T_x\otimes \mathrm{id},\qquad T_2 = \mathrm{id}\otimes T_y\end{align*}$$

as elements of $B\otimes \mathrm {End}(F(x)\otimes F(y))$ .

Proposition 2.28. Suppose $x,y,z\in \mathcal {C}$ . Then the R-matrix satisfies the Yang–Baxter equation

(20) $$ \begin{align} R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12} \end{align} $$

in $\mathrm {End}(F(x)\otimes F(y)\otimes F(z))$ . Moreover, T satisfies the FRT relation

(21) $$ \begin{align} R_{12}T_1T_2 = T_2T_1 R_{12} \end{align} $$

in $B\otimes \mathrm {End}(F(x)\otimes F(y))$ .

Proof. Denote

$$\begin{align*}\check{R}_{x,y}\colon F(x)\otimes F(y)\xrightarrow{J_{x,y}} F(x\otimes y)\xrightarrow{F(\sigma_{x, y})} F(y\otimes x)\xrightarrow{J_{y,x}^{-1}} F(y)\otimes F(x).\end{align*}$$

Then the Yang–Baxter equation (20) is equivalent to the braid equation

$$\begin{align*}\check{R}_{12}\check{R}_{23}\check{R}_{12} = \check{R}_{23}\check{R}_{12}\check{R}_{23},\end{align*}$$

which holds in any braided monoidal category.

By equation (16), we have $F(\sigma _{x,y})T_{x\otimes y} = T_{y\otimes x} F(\sigma _{x,y})$ . The relation given by (19) and the equality

$$\begin{align*}F(\sigma_{x,y}) = J_{y,x} \hat{R}_{x,y} J^{-1}_{x,y} \end{align*}$$

imply equation (21).

2.6 Coend algebras and reflection equation

Let $\mathcal {C}$ be a cp-rigid monoidal category. Recall the formula for the right adjoint $T^{\mathrm {R}}\colon \mathcal {C}\rightarrow \mathcal {C}\otimes \mathcal {C}$ for the tensor product functor $\mathcal {C}\otimes \mathcal {C}\rightarrow \mathcal {C}$ from Theorem 2.8.

Definition 2.30. The canonical coend is the object $\mathcal {F}\in \mathcal {C}$ defined by

(22) $$ \begin{align} \mathcal{F} = T T^{\mathrm{R}}(\mathbf{1}) = \int^{x\in\mathcal{C}^{\mathrm{cp}}} x^\vee\otimes x. \end{align} $$

For $x\in \mathcal {C}^{\mathrm {cp}}$ , let us denote by

$$\begin{align*}\pi_x\colon x^\vee\otimes x\rightarrow \mathcal{F}\end{align*}$$

the natural projection.

Now, suppose in addition that $\mathcal {C}$ is braided monoidal. Then $\mathcal {F}$ admits a structure of a braided Hopf algebra (see, e.g., [Reference Lyubashenko and Majid64, Reference Lyubashenko63, Reference Shimizu78]). Explicitly, the algebra structure is given on generators as follows:

• ○ The product is

  $$ \begin{align*} (x^\vee\otimes x)\otimes (y^\vee\otimes y)&\xrightarrow{\sigma_{x^\vee\otimes x, y^\vee}} y^\vee\otimes x^\vee\otimes x\otimes y\\ &\cong (x\otimes y)^\vee\otimes x\otimes y\\ &\xrightarrow{\pi_{x\otimes y}}\mathcal{F}. \end{align*} $$

• ○ The unit is

  $$\begin{align*}\mathbf{1}\xrightarrow{\pi_{\mathbf{1}}} \mathcal{F}.\end{align*}$$

Consider a monoidal functor $F\colon \mathcal {C}\rightarrow \mathrm {Vect}$ . The projections $\pi _x$ give rise to elements

$$\begin{align*}K_x\in F(\mathcal{F})\otimes \mathrm{End}(F(x)).\end{align*}$$

Comparing the formulas in equations (15) and (22), we see that there is an isomorphism of vector spaces

$$\begin{align*}F(\mathcal{F})\cong B.\end{align*}$$

In particular, as before, $F(\mathcal {F})$ is spanned, as a k-vector space, by the matrix coefficients of $K_x$ for $x\in \mathcal {C}^{\mathrm {cp}}$ subject to the relation in equation (16) for every $f\colon x\rightarrow y$ . As before, $K_{\mathbf {1}}\in F(\mathcal {F})$ is the unit. However, the multiplication is different. The following was proved in [Reference Majid66, Reference Donin, Kulish and Mudrov26].

Proposition 2.31. Suppose $x,y\in \mathcal {C}^{\mathrm {cp}}$ are two objects. Then the reflection equation

(23) $$ \begin{align} R_{21}K_1R_{12}K_2 = K_2R_{21}K_1R_{12} \end{align} $$

holds in $F(\mathcal {F})\otimes \mathrm {End}(F(x)\otimes F(y))$ .

Example 2.33. Suppose H is a Hopf algebra and consider $\mathcal {C}=\mathrm {LMod}_H$ . Then the coend algebra $\mathcal {F}$ is a Drinfeld twist of the restricted dual Hopf algebra

$$\begin{align*}H^\circ=\int^{V\in\mathrm{LMod}_H^{\mathrm{cp}}} V^\vee\otimes V;\end{align*}$$

see [Reference Donin and Mudrov27, Definition 4.12].

3.1 General definition

We will now present a general categorical definition that encompasses categories of both classical and quantum Harish-Chandra bimodules. We refer to section 3.3 for a relationship to the usual Harish-Chandra bimodules. This formalism is closely related to the theory of dynamical extensions of monoidal categories introduced in [Reference Donin and Mudrov28]; see Theorem 3.2.

Throughout this section, we fix a cp-rigid monoidal category $\mathcal {C}$ . Recall from [Reference Etingof, Gelaki, Nikshych and Ostrik34, Definition 7.13.1] that the Drinfeld centre $\mathrm {Z}_{\mathrm {Dr}}(\mathcal {C})$ is the braided monoidal category consisting of pairs $(z, \tau )$ , where $z\in \mathcal {C}$ and

$$\begin{align*}\tau_x\colon x\otimes z\xrightarrow{\sim} z\otimes x\end{align*}$$

is a natural isomorphism satisfying standard compatibilities. The monoidal structure is given by

$$\begin{align*}(z, \tau)\otimes (z', \tau') = (z\otimes z', \tilde{\tau}),\end{align*}$$

where $\tilde {\tau }$ is the composite

$$\begin{align*}x\otimes z\otimes z'\xrightarrow{\tau_x\otimes \mathrm{id}_{z'}} z\otimes x\otimes z'\xrightarrow{\mathrm{id}\otimes \tau^{\prime}_x} z\otimes z'\otimes x,\end{align*}$$

where we omit associators. We refer to [Reference Etingof, Gelaki, Nikshych and Ostrik34, Proposition 8.5.1] for the braided monoidal structure on $\mathrm {Z}_{\mathrm {Dr}}(\mathcal {C})$ .

Definition 3.1. Let $(\mathcal {L}, \tau )$ be a commutative algebra in $\mathrm {Z}_{\mathrm {Dr}}(\mathcal {C})$ . The category of Harish-Chandra bimodules is

$$\begin{align*}\mathrm{HC}(\mathcal{C}, \mathcal{L}) = \mathrm{LMod}_{\mathcal{L}}(\mathcal{C}).\end{align*}$$

When there is no confusion, we simply denote $\mathrm {HC}=\mathrm {HC}(\mathcal {C}, \mathcal {L})$ .

If H is a Hopf algebra, recall that the Drinfeld centre $\mathrm {Z}_{\mathrm {Dr}}(\mathrm {LMod}_H)$ is equivalent to the category of Yetter–Drinfeld modules over H (see [Reference Kassel52, Section XIII.5]). This gives rise to the following important example.

Proposition 3.3. Suppose H is a Hopf algebra, and consider $\mathcal {C}=\mathrm {LMod}_H$ . A commutative algebra $\mathcal {L}$ in $\mathrm {Z}_{\mathrm {Dr}}(\mathrm {LMod}_H)$ is the same as an H-algebra $\mathcal {L}$ equipped with a left H-coaction $\delta \colon \mathcal {L}\rightarrow H\otimes \mathcal {L}$ , a map of H-algebras, denoted by $x\mapsto x_{(-1)}\otimes x_{(0)}$ satisfying

$$\begin{align*}xy = y_{(0)} (S^{-1}(y_{(-1)}))\triangleright x,\qquad x,y\in \mathcal{L}.\end{align*}$$

The corresponding isomorphism $\tau _M\colon M\otimes \mathcal {L}\rightarrow \mathcal {L}\otimes M$ is given by

$$\begin{align*}m\otimes x\mapsto x_{(0)}\otimes (S^{-1}(x_{(-1)})\triangleright m).\end{align*}$$

Proof. The compatibility of $\tau _M$ with the monoidal structure on $\mathrm {LMod}_H$ follows from the coassociativity and counitality of the H-coaction. The compatibility of $\tau _M$ with the algebra structure on $\mathcal {L}$ is equivalent to the equation

$$\begin{align*}x_{(0)}y_{(0)}\otimes S^{-1}(y_{(-1)}) S^{-1}(x_{(-1)}) \triangleright m = (xy)_{(0)}\otimes S^{-1}((xy)_{(-1)})\triangleright m, \end{align*}$$

which follows from the condition that $\mathcal {L}\rightarrow H\otimes \mathcal {L}$ is an algebra map. The commutativity of the multiplication on $\mathcal {L}\in \mathrm {Z}_{\mathrm {Dr}}(\mathrm {LMod}_H)$ is

$$\begin{align*}xy = y_{(0)} (S^{-1}(y_{(-1)}))\triangleright x.\end{align*}$$

Remark 3.4. The inverse morphism $\mathcal {L}\otimes M\rightarrow M\otimes \mathcal {L}$ is given by

$$\begin{align*}x\otimes m\mapsto x_{(-1)}\triangleright m\otimes x_{(0)}.\end{align*}$$

Example 3.5. Consider a Hopf algebra H, and let $\mathcal {L}=H$ . Consider the adjoint action of H on $\mathcal {L}$

$$\begin{align*}h\otimes x\mapsto h_{(1)} x S(h_{(2)})\end{align*}$$

for $h\in H$ and $x\in \mathcal {L}$ . Consider the H-coaction $\mathcal {L}\rightarrow H\otimes \mathcal {L}$ given by the coproduct on H. Then

$$\begin{align*}S^{-1}(y)\triangleright x = S^{-1}(y_{(2)})xy_{(1)}.\end{align*}$$

In particular,

$$ \begin{align*} y_{(2)} (S^{-1}(y_{(1)}))\triangleright x &= y_{(3)} S^{-1}(y_{(2)})x y_{(1)} \\ &= \epsilon(y_{(2)}) x y_{(1)} \\ &= xy, \end{align*} $$

which shows that $(\mathcal {L}, \tau )$ is a commutative algebra in $\mathrm {Z}_{\mathrm {Dr}}(\mathrm {LMod}_H)$ .

Since $\mathcal {L}$ is a commutative algebra in $\mathrm {Z}_{\mathrm {Dr}}(\mathcal {C})$ , the category $\mathrm {HC}$ has a natural monoidal structure given by the relative tensor product: given left $\mathcal {L}$ -modules $M,N\in \mathcal {C}$ , we may turn M into a right $\mathcal {L}$ -module using $\tau _M$ , and then the tensor product is given by $M\otimes _{\mathcal {L}} N$ . We also have an adjunction

where $\mathrm {free}\colon \mathcal {C}\rightarrow \mathrm {HC}$ is the monoidal functor $x\mapsto \mathcal {L}\otimes x$ given by the free left $\mathcal {L}$ -module and $\mathrm {forget}\colon \mathrm {HC}\rightarrow \mathcal {C}$ is given by forgetting the $\mathcal {L}$ -module structure.

Observe that $\mathcal {L}^{\mathrm {op}}$ is an algebra in $\mathcal {C}^{\otimes \mathrm {op}}$ . Moreover, it lifts to a commutative algebra in $\mathrm {Z}_{\mathrm {Dr}}(\mathcal {C}^{\otimes \mathrm {op}})$ if we consider the inverse isomorphism $\tau _x$ .

The following construction explains why $\mathrm {HC}$ deserves to be called the category of bimodules. There is a natural monoidal functor

(24) $$ \begin{align} \mathrm{bimod}\colon \mathrm{HC}\longrightarrow {}_{\mathcal{L}}\mathrm{BMod}_{\mathcal{L}}(\mathcal{C}) \end{align} $$

given by sending a left $\mathcal {L}$ -module M to the $\mathcal {L}$ -bimodule, where the right $\mathcal {L}$ -action is obtained via $\tau _M$ . It realises $\mathrm {HC}$ as a full subcategory of ${}_{\mathcal {L}}\mathrm {BMod}_{\mathcal {L}}(\mathcal {C})$ consisting of objects $M\in {}_{\mathcal {L}}\mathrm {BMod}_{\mathcal {L}}(\mathcal {C})$ such that the right and left actions are related by $\tau _M$ .

Let us now analyse the categorical properties of $\mathrm {HC}$ .

Proof. The functor $\mathrm {free}\colon \mathcal {C}\rightarrow \mathrm {HC}$ has a colimit-preserving right adjoint $\mathrm {forget}\colon \mathrm {HC}\rightarrow \mathcal {C}$ . So, $\mathrm {free}(V)\in \mathrm {HC}$ is compact projective if $V\in \mathcal {C}^{\mathrm {cp}}$ .

The category $\mathrm {HC}$ is generated by $\mathrm {free}(V)$ for $V\in \mathcal {C}$ since $\mathrm {forget}$ is conservative. But since $\mathcal {C}$ has enough compact projectives, we may restrict to $V\in \mathcal {C}^{\mathrm {cp}}$ .

Since $\mathcal {C}$ is cp-rigid, the objects $V\in \mathcal {C}^{\mathrm {cp}}$ are dualisable. Since $\mathrm {free}\colon \mathcal {C}\rightarrow \mathrm {HC}$ is monoidal, the objects $\mathrm {free}(V)\in \mathrm {HC}$ are also dualisable. But we have just shown that such objects are the generating compact projective objects, while by [Reference Brochier, Jordan and Snyder18, Proposition 4.1], it is enough to check cp-rigidity on the generating compact projective objects.

The unit of $\mathrm {HC}$ is $\mathcal {L}$ viewed as a free left $\mathcal {L}$ -module of rank 1, so

$$\begin{align*}\mathrm{Hom}_{\mathrm{HC}}(\mathcal{L}, -)\cong \mathrm{Hom}_{\mathcal{C}}(\mathbf{1}_{\mathcal{C}}, \mathrm{forget}(-)),\end{align*}$$

which shows that $\mathcal {L}$ is compact projective if and only if $\mathbf {1}_{\mathcal {C}}\in \mathcal {C}$ is.

3.2 Quantum moment maps

Recall that for an algebra $A\in \operatorname {\mathrm {Rep}}(G)$ , a quantum moment map is a map $\mu \colon \mathrm {U}\mathfrak {g}\rightarrow A$ such that the infinitesimal $\mathfrak {g}$ -action on A is given by $[\mu (x), -]$ for $x\in \mathfrak {g}$ . The following version of this definition in our setting was introduced in [Reference Safronov73, Definition 3.1].

Definition 3.8. Let $A\in \mathcal {C}$ be an algebra. A quantum moment map is an algebra map $\mu \colon \mathcal {L}\rightarrow A$ such that the diagram

(25)

commutes.

The following is [Reference Safronov73, Definition 3.10].

Definition 3.11. Suppose $\epsilon \colon \mathcal {L}\rightarrow \mathbf {1}_{\mathcal {C}}$ is a morphism of algebras in $\mathcal {C}$ and A is an algebra equipped with a quantum moment map. The Hamiltonian reduction of $\boldsymbol {A}$ is

$$\begin{align*}\mathrm{Hom}_{\mathcal{C}}(\mathbf{1}_{\mathcal{C}}, A\otimes_{\mathcal{L}} \mathbf{1})\cong \mathrm{Hom}_{\mathrm{LMod}_A(\mathcal{C})}(A\otimes_{\mathcal{L}} \mathbf{1}_{\mathcal{C}}, A\otimes_{\mathcal{L}} \mathbf{1}_{\mathcal{C}}).\end{align*}$$

A canonical example of an algebra with a quantum moment map we will use is the following. Let $T_{\mathrm {HC}}\colon \mathrm {HC}\otimes \mathrm {HC}\rightarrow \mathrm {HC}$ be the tensor product functor. By Theorem 2.15, the object $T^{\mathrm {R}}(\mathbf {1}_{\mathrm {HC}})\in \mathrm {HC}\otimes \mathrm {HC}^{\otimes \mathrm {op}}$ is an algebra. Identifying $\mathrm {HC}(\mathcal {C}, \mathcal {L})^{\otimes \mathrm {op}}\cong \mathrm {HC}(\mathcal {C}^{\otimes \mathrm {op}}, \mathcal {L}^{\mathrm {op}})$ using Theorem 3.6, we see that

$$\begin{align*}(\mathrm{forget}\otimes \mathrm{forget})(T_{\mathrm{HC}}^{\mathrm{R}}(\mathbf{1}_{\mathrm{HC}}))\in\mathcal{C}\otimes \mathcal{C}^{\otimes\mathrm{op}}\end{align*}$$

is an algebra equipped with a quantum moment map from $\mathcal {L}\boxtimes \mathcal {L}^{\mathrm {op}}$ .

Definition 3.12. Let $\mathcal {C}, \mathrm {HC}$ be as before. The algebra $\mathcal {D}\in \mathcal {C}\otimes \mathcal {C}^{\otimes \mathrm {op}}$ is

$$\begin{align*}\mathcal{D} = (\mathrm{forget}\otimes \mathrm{forget})(T_{\mathrm{HC}}^{\mathrm{R}}(\mathbf{1}_{\mathrm{HC}})).\end{align*}$$

We denote the canonical quantum moment map by

(26) $$ \begin{align} \mu\colon \mathcal{L}\boxtimes \mathcal{L}^{\mathrm{op}}\longrightarrow T_{\mathcal{C}}^{\mathrm{R}}(\mathcal{L}). \end{align} $$

Proposition 3.13. We have an equivalence

$$\begin{align*}\mathcal{D}\cong \int^{x\in\mathcal{C}^{\mathrm{cp}}} (\mathcal{L}\otimes x^\vee)\boxtimes x\cong \int^{x\in\mathcal{C}^{\mathrm{cp}}} x^\vee\boxtimes (x\otimes \mathcal{L}),\end{align*}$$

where the latter isomorphism is provided by Theorem 2.9. The algebra structure is given by

$$ \begin{align*} ((\mathcal{L}\otimes x^\vee)\boxtimes x)\otimes((\mathcal{L}\otimes y^\vee)\boxtimes y)&\cong (\mathcal{L}\otimes x^\vee\otimes \mathcal{L}\otimes y^\vee)\boxtimes (y\otimes x) \\ &\xrightarrow{\mathrm{id}\otimes \tau_{x^\vee}\otimes \mathrm{id}} (\mathcal{L}\otimes \mathcal{L}\otimes x^\vee\otimes y^\vee)\boxtimes (y\otimes x) \\ &\xrightarrow{m\otimes \mathrm{id}} (\mathcal{L}\otimes (y\otimes x)^\vee)\boxtimes (y\otimes x) \\ &\xrightarrow{\pi_{y\otimes x}} \int^{x\in\mathcal{C}^{\mathrm{cp}}} (\mathcal{L}\otimes x^\vee)\boxtimes x. \end{align*} $$

The two quantum moment maps $\mathcal {L},\mathcal {L}^{\mathrm {op}}\rightarrow \mathcal {D}$ are given by

$$\begin{align*}\mathcal{L}\cong (\mathcal{L}\otimes \mathbf{1})\boxtimes \mathbf{1}\xrightarrow{\pi_{\mathbf{1}}} \mathcal{D}\end{align*}$$

and

$$\begin{align*}\mathcal{L}^{\mathrm{op}}\cong \mathbf{1}\boxtimes (\mathbf{1}\otimes \mathcal{L})\xrightarrow{\pi_{\mathbf{1}}} \mathcal{D}.\end{align*}$$

Proof. Since $\mathrm {free}\colon \mathcal {C}\rightarrow \mathrm {HC}$ is a monoidal functor, by adjunction, we get a natural isomorphism

(27) $$ \begin{align} (\mathrm{forget}\otimes \mathrm{forget}) \circ T_{\mathrm{HC}}^{\mathrm{R}}\cong T_{\mathcal{C}}^{\mathrm{R}}\circ \mathrm{forget}, \end{align} $$

where $T_{\mathcal {C}}\colon \mathcal {C}\otimes \mathcal {C}\rightarrow \mathcal {C}$ is the tensor product functor. In particular, applying equation (27) to $\mathbf {1}_{\mathrm {HC}}$ , we get isomorphisms

$$\begin{align*}\mathcal{D}\cong \int^{x\in\mathcal{C}^{\mathrm{cp}}} (\mathcal{L}\otimes x^\vee)\boxtimes x\cong \int^{x\in\mathcal{C}^{\mathrm{cp}}} x^\vee\boxtimes (x\otimes \mathcal{L})\end{align*}$$

using Theorem 2.8.

We have a natural isomorphism

$$\begin{align*}(\mathrm{forget}\circ\mathrm{free}\otimes \mathrm{id})\circ T_{\mathcal{C}}^{\mathrm{R}}(-)\cong T_{\mathcal{C}}^{\mathrm{R}}(\mathrm{forget}\circ\mathrm{free}(-))\end{align*}$$

given by Theorem 2.9 that gives rise to an algebra isomorphism

$$\begin{align*}\mathcal{D}\cong(\mathrm{forget}\circ\mathrm{free}\otimes\mathrm{id})\circ T_{\mathcal{C}}^{\mathrm{R}}(\mathbf{1}),\end{align*}$$

which gives the required formula.

Example 3.14. Suppose H is a Hopf algebra, $\mathcal {L}$ is a commutative algebra in $\mathrm {Z}_{\mathrm {Dr}}(\mathrm {LMod}_H)$ (see Theorem 3.3) and $\mathcal {C}=\mathrm {LMod}_H$ . Let

$$\begin{align*}H^\circ = \int^{V\in\mathrm{LMod}_H^{\mathrm{cp}}} V^\vee\otimes V\end{align*}$$

be the restricted dual Hopf algebra. By construction, $\mathcal {L}$ is an H-comodule algebra, and $H^\circ $ is an H-module algebra (via the left H-action). Then $\mathcal {D}$ is the smash product algebra generated by $\mathcal {L}$ and $H^\circ $ with the additional relation

$$\begin{align*}hl = l_{(0)} (S^{-1}(l_{(-1)})\triangleright h)\end{align*}$$

for $h\in H^\circ $ and $l\in \mathcal {L}$ .

3.3 Classical Harish-Chandra bimodules

Let G be a reductive group over a characteristic zero field k, and denote by $\mathfrak {g}$ its Lie algebra. Let $\operatorname {\mathrm {Rep}}(G)$ be the ind-completion of the category of finite-dimensional representations. The category $\operatorname {\mathrm {Rep}}(G)$ is semisimple, so it has enough compact projectives and its unit is compact projective.

Suppose $V\in \operatorname {\mathrm {Rep}}(G)$ is a G-representation. For $x\in \mathrm {U}\mathfrak {g}$ and $v\in V$ , we denote by $x\triangleright v$ the induced $\mathrm {U}\mathfrak {g}$ -action on V. Consider the natural isomorphism

(28) $$ \begin{align} \tau_V\colon V\otimes \mathrm{U}\mathfrak{g}\longrightarrow \mathrm{U}\mathfrak{g}\otimes V \end{align} $$

given by

$$\begin{align*}v\otimes x\mapsto x\otimes v - 1\otimes xv \end{align*}$$

for $x\in \mathfrak {g}$ . It follows from Theorem 3.3 that $(\mathrm {U}\mathfrak {g}, \tau )$ defines a commutative algebra in $\mathrm {Z}_{\mathrm {Dr}}(\operatorname {\mathrm {Rep}}(G))$ .

Definition 3.15. The category of classical Harish-Chandra bimodules is

$$\begin{align*}\mathrm{HC}(G) = \mathrm{HC}(\operatorname{\mathrm{Rep}}(G), \mathrm{U}\mathfrak{g}).\end{align*}$$

The following easy lemma (see [Reference Safronov73, Example 3.4]) shows that the definition of a quantum moment map we gave in Theorem 3.8 coincides with the classical notion of a quantum moment map.

In the same way, the quantum Hamiltonian reduction from Theorem 3.11 coincides with the usual definition

$$\begin{align*}A/\!\!/ G = (A/A\mu(\mathfrak{g}))^G\end{align*}$$

of the reduced algebra.

Given a variety X equipped with a G-action, the algebra of differential operators $\mathrm {D}(X)$ carries a quantum moment map $\mu \colon \mathrm {U}\mathfrak {g}\rightarrow \mathrm {D}(X)$ , which sends $\mathfrak {g}\subset \mathrm {U}\mathfrak {g}$ to vector fields on X generating the infinitesimal action. For instance, $\mathrm {D}(G)$ carries a moment map

(29) $$ \begin{align} \mu\colon \mathrm{U}\mathfrak{g}\otimes \mathrm{U}\mathfrak{g}^{\mathrm{op}}\rightarrow \mathrm{D}(G) \end{align} $$

coming from the left and right G-action on itself. Let us explain how it arises in our context.

By the Peter–Weyl theorem, we have an isomorphism of algebras

$$\begin{align*}\mathcal{O}(G)\cong \int^{V\in\operatorname{\mathrm{Rep}}^{\mathrm{fd}}(G)} V^\vee\boxtimes V\in\operatorname{\mathrm{Rep}}(G)\otimes \operatorname{\mathrm{Rep}}(G),\end{align*}$$

where $\mathcal {O}(G)$ carries a $G\times G$ -action coming from the left and right G-action on itself. Using this, we can also describe the algebra $\mathcal {D}$ from Theorem 3.12.

In the abelian case, the category of Harish-Chandra bimodules has a straightforward description. Suppose H is a split torus; let $\mathfrak {h}$ be its Lie algebra and $\Lambda =\mathrm {Hom}(H, \mathbb {G}_{\mathrm {m}})$ the character lattice. Then $\operatorname {\mathrm {Rep}}(H)$ is equivalent to the category of $\Lambda $ -graded vector spaces, and $\mathrm {HC}(H)$ is equivalent to the category of $\Lambda $ -graded $\mathrm {Sym}(\mathfrak {h})$ -modules $\oplus _{\lambda \in \Lambda } M(\lambda )$ .

Given $\lambda \in \Lambda $ , we consider the translation functor $\lambda ^*\colon \mathrm {LMod}_{\mathrm {Sym}(\mathfrak {h})}\rightarrow \mathrm {LMod}_{\mathrm {Sym}(\mathfrak {h})}$ . Then the monoidal structure $\otimes ^{\mathrm {HC}}$ on $\mathrm {HC}(H)$ is given by

$$\begin{align*}M\otimes^{\mathrm{HC}} N = \bigoplus_{\lambda\in\Lambda} \lambda^*(M)\otimes N(\lambda).\end{align*}$$

Suppose $V\in \operatorname {\mathrm {Rep}}(H)$ . Given a vector $v\in V$ of weight $\mu \in \Lambda $ and $f\in \mathcal {O}(\mathfrak {h}^*)\cong \mathrm {U}\mathfrak {h}$ , the map given by equation (28) is given by

$$\begin{align*}v\otimes f(\lambda)\mapsto f(\lambda - \mu)\otimes v\end{align*}$$

for $\lambda \in \mathfrak {h}^*$ . It is convenient to write it as

$$\begin{align*}v\otimes f(\lambda)\mapsto f(\lambda - h)\otimes v,\end{align*}$$

where h is understood as acting on $v\in V$ . Similarly, given a collection of representations $V_1, \dots , V_n\in \operatorname {\mathrm {Rep}}(H)$ and vectors $v_i\in V_i$ , we denote

$$\begin{align*}f(\lambda - h^{(i)}) v_1\otimes \dots v_n = f(\lambda - \mu_i) v_1\otimes \dots v_n\end{align*}$$

if $v_i$ has weight $\mu _i\in \Lambda $ .

3.4 Quantum groups

In this section, we fix our conventions for quantum groups. Fix $k=\mathbf {C}$ . Let G be a connected reductive group, $B, B_-\subset G$ a pair of opposite Borel subgroups and $H=B\cap B_-$ a Cartan subgroup. Denote by $\Lambda =\mathrm {Hom}(H, \mathbb {G}_{\mathrm {m}})$ its weight lattice and $\Lambda ^\vee =\mathrm {Hom}(\mathbb {G}_{\mathrm {m}}, H)$ the coweight lattice; we denote by $\langle -, -\rangle \colon \Lambda ^\vee \times \Lambda \rightarrow \mathbf {Z}$ the canonical pairing. For two simple roots $\alpha _i,\alpha _j\in \Lambda $ , denote by $\alpha _i\cdot \alpha _j\in \mathbf {Z}$ the corresponding entry of the symmetrised Cartan matrix. Choose an integer $d\in \mathbf {Z}$ and a symmetric bilinear form $(-, -)\colon \Lambda \times \Lambda \rightarrow \frac {1}{d}\mathbf {Z}$ such that $(\alpha _i, \alpha _j) = \alpha _i\cdot \alpha _j$ . Given a complex number $q^{1/d}\in \mathbf {C}^\times $ , we have the exponentiated pairing

$$\begin{align*}\Pi\colon \Lambda\times \Lambda\longrightarrow \mathbf{C}^\times\end{align*}$$

given by $\lambda ,\mu \mapsto q^{-(\lambda , \mu )}$ . Our assumption is that $q^{1/d}$ is not a root of unity.

We denote by $\mathrm {U}_q(\mathfrak {g})$ the quantum group defined as in [Reference Lusztig62] with a slight modification that its Cartan part is $\mathrm {U}_q(\mathfrak {h})=k[\Lambda ]$ with Cartan generators $K_\mu $ for $\mu \in \Lambda $ (note that the Cartan part in [Reference Lusztig62] is $k[\Lambda ^\vee ]$ ). We denote by $\mathrm {U}_q(\mathfrak {b})\subset \mathrm {U}_q(\mathfrak {g})$ the quantum Borel subalgebra, $\mathrm {U}_q(\mathfrak {n}), \mathrm {U}_q(\mathfrak {n}_-)\subset \mathrm {U}_q(\mathfrak {g})$ the quantum nilpotent subalgebras and $\mathrm {U}_q^{>0}(\mathfrak {n}),\mathrm {U}_q^{<0}(\mathfrak {n}_-)$ their augmentation ideals. For each simple root $\alpha $ , we denote by $\{E_\alpha , K_\alpha , F_\alpha \}$ the corresponding generators of the $\mathrm {U}_q(\mathfrak {sl}_2)$ -subalgebra (they are denoted by $E_i, \tilde {K}_i, F_i$ in [Reference Lusztig62, Section 3.1.1]).

We have the corresponding categories obtained from this data:

• ○ $\operatorname {\mathrm {Rep}}_q(H)$ is the braided monoidal category of $\Lambda $ -graded vector spaces with the braiding given by $\Pi \tau $ , where $\tau $ is the map exchanging the tensor factors.

• ○ $\operatorname {\mathrm {Rep}}_q(G)$ is the ind-completion of the braided monoidal category of finite-dimensional $\Lambda $ -graded vector spaces with a $\mathrm {U}_q(\mathfrak {g})$ -module structure such that for every vector $x_\lambda $ of weight $\lambda \in \Lambda $ , we have $K_\mu x_\lambda = q^{(\mu , \lambda )} x_\lambda $ . The braiding is given by $\Theta \circ \Pi \circ \tau $ , where $\Theta \in \mathrm {U}_q(\mathfrak {n}_-)\widehat {\otimes }\mathrm {U}_q(\mathfrak {n})$ is the so-called quasi R-matrix. Refer to [Reference Lusztig62, Section 32] for more details.

• ○ $\operatorname {\mathrm {Rep}}_q(B)$ is the ind-completion of the monoidal category of finite-dimensional $\Lambda $ -graded vector spaces with a compatible $\mathrm {U}_q(\mathfrak {b})$ -module structure.

Definition 3.19. A $\mathrm {U}_q(\mathfrak {g})$ -module M is integrable if it lies in the image of the forgetful functor

$$\begin{align*}\operatorname{\mathrm{Rep}}_q(G)\longrightarrow \mathrm{LMod}_{\mathrm{U}_q(\mathfrak{g})}.\end{align*}$$

Equivalently, an integrable $\mathrm {U}_q(\mathfrak {g})$ -module is a locally finite type 1 $\mathrm {U}_q(\mathfrak {g})$ -module. We introduce an analogous definition for $\mathrm {U}_q(\mathfrak {b})$ -modules.

Denote by $\mathcal {O}_q(G)\in \operatorname {\mathrm {Rep}}_q(G)$ the coend algebra from Theorem 2.30.

The algebra $\mathrm {U}_q(\mathfrak {g})$ with respect to the adjoint $\mathrm {U}_q(\mathfrak {g})$ -action on itself $x, y\mapsto x_{(1)}yS(x_{(2)})$ is not locally finite, and we denote by

$$\begin{align*}\mathrm{U}_q(\mathfrak{g})^{\mathrm{lf}}\subset\mathrm{U}_q(\mathfrak{g})\end{align*}$$

the largest locally finite submodule.

Example 3.21. Consider $\mathrm {U}_q(\mathfrak {sl}_2)$ with the generators $E,K,F$ and relations

$$\begin{align*}KE = q^2 EK,\qquad KF = q^{-2} FK,\qquad EF-FE=\frac{K-K^{-1}}{q-q^{-1}}.\end{align*}$$

Then $\mathrm {U}_q(\mathfrak {sl}_2)^{\mathrm {lf}}$ is the subalgebra generated by $EK^{-1}$ , F and $K^{-1}$ .

It is easy to see that $\mathrm {U}_q(\mathfrak {g})^{\mathrm {lf}}\subset \mathrm {U}_q(\mathfrak {g})$ is a subalgebra, but note that it is not a subcoalgebra. Nevertheless, the following is shown in [Reference Joseph50, Theorem 7.1.6].

Proposition 3.22. $\mathrm {U}_q(\mathfrak {g})^{\mathrm {lf}}\subset \mathrm {U}_q(\mathfrak {g})$ is a left coideal: that is, the coproduct restricts to a map

$$\begin{align*}\Delta\colon \mathrm{U}_q(\mathfrak{g})^{\mathrm{lf}}\longrightarrow\mathrm{U}_q(\mathfrak{g})\otimes \mathrm{U}_q(\mathfrak{g})^{\mathrm{lf}}. \end{align*}$$

3.5 Quantum Harish-Chandra bimodules

For $V\in \operatorname {\mathrm {Rep}}_q(G)$ , $v\in V$ an $x\in \mathrm {U}_q(\mathfrak {g})$ , we denote by $x\triangleright v$ the $\mathrm {U}_q(\mathfrak {g})$ -action. For $x\in \mathrm {U}_q(\mathfrak {g})^{\mathrm {lf}}$ , we denote by $\Delta (x) = x_{(1)}\otimes x_{(2)}$ the coproduct on $\mathrm {U}_q(\mathfrak {g})$ , where we note that $x_{(2)}\in \mathrm {U}_q(\mathfrak {g})^{\mathrm {lf}}$ by Theorem 3.22. We define the natural isomorphism

(30) $$ \begin{align} \tau_V\colon V\otimes \mathrm{U}_q(\mathfrak{g})^{\mathrm{lf}}\longrightarrow \mathrm{U}_q(\mathfrak{g})^{\mathrm{lf}}\otimes V \end{align} $$

by

$$\begin{align*}v\otimes x\mapsto x_{(2)}\otimes S^{-1}(x_{(1)})\triangleright v.\end{align*}$$

Consider $\mathrm {U}_q(\mathfrak {g})^{\mathrm {lf}}\in \operatorname {\mathrm {Rep}}_q(G)$ with respect to the adjoint action. It follows from Theorem 3.3 that $(\mathrm {U}_q(\mathfrak {g})^{\mathrm {lf}}, \tau )$ is a commutative algebra in $\mathrm {Z}_{\mathrm {Dr}}(\operatorname {\mathrm {Rep}}_q(G))$ .

Definition 3.24. The category of quantum Harish-Chandra bimodules is

$$\begin{align*}\mathrm{HC}_q(G) = \mathrm{HC}(\operatorname{\mathrm{Rep}}_q(G), \mathrm{U}_q(\mathfrak{g})^{\mathrm{lf}}).\end{align*}$$

In this case, the algebra $\mathcal {D}$ from Theorem 3.12 is the algebra of quantum differential operators $\mathcal {D}_q(G)$ on G (see [Reference Backelin and Kremnizer5, Section 4.1], where it is denoted by $\mathcal {D}_q^{fin}$ ).

As in the case of classical Harish-Chandra bimodules, in the abelian case, the category $\mathrm {HC}_q(G)$ has a straightforward description. Let H be a torus and $\Lambda $ its weight lattice. Then $\mathrm {U}_q(\mathfrak {h})^{\mathrm {lf}} = \mathrm {U}_q(\mathfrak {h}) = \mathcal {O}(H)$ and $\mathrm {HC}_q(H)$ is equivalent to the category of $\Lambda $ -graded $\mathcal {O}(H)$ -modules. There is a homomorphism

$$\begin{align*}\Lambda\longrightarrow H\end{align*}$$

whose dual map $\mathcal {O}(H)=k[\Lambda ]\rightarrow \mathcal {O}(\Lambda )$ on the level of functions is

$$\begin{align*}K_\mu\mapsto \left(\lambda \mapsto q^{(\mu, \lambda)}\right)\end{align*}$$

for $\mu ,\lambda \in \Lambda $ . In particular, $\Lambda $ acts by translations on H, and we denote the induced functor by

$$\begin{align*}(q^\lambda)^*\colon \mathrm{LMod}_{\mathcal{O}(H)}\rightarrow \mathrm{LMod}_{\mathcal{O}(H)}.\end{align*}$$

The monoidal structure $\otimes ^{\mathrm {HC}}$ on $\mathrm {HC}_q(H)$ is given by

$$\begin{align*}M\otimes^{\mathrm{HC}} N = \bigoplus_{\lambda\in\Lambda} (q^\lambda)^*(M)\otimes N(\lambda).\end{align*}$$

Suppose $V\in \operatorname {\mathrm {Rep}}(H)$ , $v\in V$ and $f\in \mathcal {O}(H)\cong \mathrm {U}_q(\mathfrak {h})$ . Then the map given by equation (30) is

$$\begin{align*}v\otimes f(\lambda)\mapsto f(\lambda q^{-h}) \otimes v\end{align*}$$

for $\lambda \in H$ .

3.6 Harish-Chandra bimodules and bialgebroids

Let us again consider the general setup of section 3.1, where $\mathcal {C}$ is a cp-rigid monoidal category with a compact projective unit. In particular, $\mathrm {HC}$ is also a cp-rigid monoidal category with a compact projective unit. Our goal in this section is to describe a Tannaka reconstruction result for monoidal forgetful functors to $\mathrm {HC}$ .

Recall from Theorem 2.19 that the category $\mathrm {HC}\otimes \mathrm {HC}$ carries two monoidal structures: the pointwise monoidal structure on $\mathrm {HC}\otimes \mathrm {HC}^{\otimes \mathrm {op}}$ and the convolution product. We will call the latter the Takeuchi product in this setting.

Definition 3.27. The Takeuchi product $\times _{\mathcal {L}}$ is the monoidal structure on $\mathrm {HC}\otimes \mathrm {HC}$ given by

$$\begin{align*}(M_1\boxtimes M_2)\times_{\mathcal{L}} (N_1\boxtimes N_2) = \mathrm{ev}(M_2, N_1)\otimes (M_1\boxtimes N_2)\end{align*}$$

with the unit $\mathrm {coev}(k)=\mathcal {D}\in \mathrm {HC}\otimes \mathrm {HC}$ .

Example 3.28. Consider the setup of Theorem 3.15. An object of $\mathrm {HC}(G)\otimes \mathrm {HC}(G)\cong \mathrm {HC}(G\times G)$ is a $\mathrm {U}\mathfrak {g}\otimes (\mathrm {U}\mathfrak {g})^{\mathrm {op}}$ -bimodule with a certain integrability condition. For a $(\mathrm {U}\mathfrak {g})^{\mathrm {op}}$ -bimodule M and a $\mathrm {U}\mathfrak {g}$ -bimodule N, the Takeuchi product is the subspace

$$\begin{align*}M\times_{\mathrm{U}\mathfrak{g}} N\subset M\otimes_{\mathrm{U}\mathfrak{g}} N\end{align*}$$

of elements $\sum _i m_i\otimes n_i$ satisfying

$$\begin{align*}\sum_i m_i x\otimes n_i = m_i\otimes n_ix\end{align*}$$

for every $x\in \mathrm {U}\mathfrak {g}$ ; see [Reference Takeuchi82].

We will now formulate the notion of bialgebroids in the category of Harish-Chandra bimodules. Recall that the algebra $\mathcal {D}\cong T^{\mathrm {R}}(\mathcal {L})\in \mathcal {C}\otimes \mathcal {C}^{\mathrm {op}}$ carries a natural quantum moment map given by equation (26).

Theorem 3.32. Suppose B is a Harish-Chandra bialgebroid. The functor $\bot \colon \mathrm {HC}\rightarrow \mathrm {HC}$ given by

$$\begin{align*}\bot(M) = B\times_{\mathcal{L}} M\end{align*}$$

defines a lax monoidal comonad. Conversely, let $\bot \colon \mathrm {HC}\rightarrow \mathrm {HC}$ be a colimit-preserving lax monoidal comonad on $\mathrm {HC}$ . Then $\bot (-)\cong B\times _{\mathcal {L}}(-)$ for some Harish-Chandra bialgebroid B.

Proof. Recall from [Reference Aguiar and Mahajan2, Definition 6.25] that a bimonoid in a duoidal category is an algebra with respect to one monoidal structure and a coalgebra with respect to the other monoidal structure, both compatible in a natural way. A coalgebra in $(\mathrm {Fun}^{\mathrm {L}}(\mathrm {HC}, \mathrm {HC}), \circ )$ is a colimit-preserving comonad on $\mathrm {HC}$ , and a bimonoid in $\mathrm {Fun}^{\mathrm {L}}(\mathrm {HC}, \mathrm {HC})$ is the same as lax monoidal comonad on $\mathrm {HC}$ .

A colimit-preserving lax monoidal comonad on $\mathrm {HC}$ is the same as a bimonoid in the duoidal category $\mathrm {Fun}^{\mathrm {L}}(\mathrm {HC}, \mathrm {HC})$ . By Theorem 2.20, we have an equivalence of duoidal categories $\mathrm {HC}\otimes \mathrm {HC}\cong \mathrm {Fun}^{\mathrm {L}}(\mathrm {HC}, \mathrm {HC})$ . So, $\bot $ corresponds to an object $B\in \mathrm {HC}\otimes \mathrm {HC}$ , which is both an algebra in $\mathrm {HC}\otimes \mathrm {HC}^{\otimes \mathrm {op}}$ as well as a coalgebra in $(\mathrm {HC}\otimes \mathrm {HC}, \times _{\mathcal {L}})$ , both in a compatible way.

By Theorem 3.6, we have an equivalence of monoidal categories

$$\begin{align*}\mathrm{HC}(\mathcal{C}, \mathcal{L})\otimes \mathrm{HC}(\mathcal{C}, \mathcal{L})^{\otimes\mathrm{op}}\cong \mathrm{HC}(\mathcal{C}, \mathcal{L})\otimes \mathrm{HC}(\mathcal{C}^{\otimes\mathrm{op}}, \mathcal{L}^{\mathrm{op}}),\end{align*}$$

so by Theorem 3.10, the data of an algebra $B\in \mathrm {HC}\otimes \mathrm {HC}^{\otimes \mathrm {op}}$ boils down to an algebra $B\in \mathcal {C}\otimes \mathcal {C}^{\otimes \mathrm {op}}$ equipped with a quantum moment map $\mathcal {L}\boxtimes \mathcal {L}^{\mathrm {op}}\rightarrow B$ .

The data of a comonad boils down to a coalgebra $(B, \Delta , \epsilon )$ in $(\mathrm {HC}\otimes \mathrm {HC}, \times _{\mathcal {L}})$ . The counit is given by a map of algebras $\epsilon \colon B\rightarrow \mathrm {coev}(k)$ in $\mathrm {HC}\otimes \mathrm {HC}^{\otimes \mathrm {op}}$ . Identifying algebras in $\mathrm {HC}\otimes \mathrm {HC}^{\otimes \mathrm {op}}$ with algebras in $\mathcal {C}\otimes \mathcal {C}^{\otimes \mathrm {op}}$ equipped with quantum moment maps by Theorem 3.10, the counit is the same as a map $\varepsilon \colon B\rightarrow T^{\mathrm {R}}(\mathcal {L})\cong \mathcal {D}$ of algebras in $\mathcal {C}\otimes \mathcal {C}^{\otimes \mathrm {op}}$ compatible with quantum moment maps from $\mathcal {L}\boxtimes \mathcal {L}^{\mathrm {op}}$ .

The definition of representations of Harish-Chandra bialgebroids is straightforward.

Equivalently, by Theorem 3.32 a B-comodule is a coalgebra over the comonad $\bot (M) = B\times _{\mathcal {L}} M$ .

Example 3.34. Consider the category of Harish-Chandra bimodules $\mathrm {HC}(H)$ for a split torus H as in Theorem 3.30, and let B be a Harish-Chandra bialgebroid in $\mathrm {HC}(H)$ . Then a B-comodule is a $\mathcal {O}(\mathfrak {h}^*)$ -module

$$\begin{align*}M = \bigoplus_{\alpha\in\Lambda} M_\alpha\end{align*}$$

together with a coaction map

$$\begin{align*}M_\alpha\longrightarrow \bigoplus_{\beta\in\Lambda} B_{\alpha\beta}\otimes_{\mathcal{O}(\mathfrak{h}^*)} M_\beta,\end{align*}$$

where B is considered a right $\mathcal {O}(\mathfrak {h}^*)$ -module via the left action of $t\colon \mathcal {O}(\mathfrak {h}^*)\rightarrow B$ . We require this coaction map to be compatible with the $\mathcal {O}(\mathfrak {h}^*)$ -actions on both sides, where the $\mathcal {O}(\mathfrak {h}^*)$ -action on the right is via the left multiplication by $s\colon \mathcal {O}(\mathfrak {h}^*)\rightarrow B$ and coassociative and counital in the obvious way.

We obtain a Tannaka recognition statement for Harish-Chandra bialgebroids.

Theorem 3.35. Suppose $\mathcal {D}$ is a monoidal category with a monoidal functor $F\colon \mathcal {D}\rightarrow \mathrm {HC}$ , which admits a colimit-preserving right adjoint $F^{\mathrm {R}}\colon \mathrm {HC}\rightarrow \mathcal {D}$ . Then there is a Harish-Chandra bialgebroid B such that $(F\circ F^{\mathrm {R}})(-)\cong B\times _{\mathcal {L}}(-)$ and F factors through a monoidal functor

$$\begin{align*}\mathcal{D}\longrightarrow \mathrm{CoMod}_B(\mathrm{HC}).\end{align*}$$

If F is conservative and preserves equalisers, the above functor is an equivalence.

Proof. Since F is monoidal, $F^{\mathrm {R}}$ is lax monoidal. Therefore, $\bot = FF^{\mathrm {R}}$ is a colimit-preserving lax monoidal comonad on $\mathrm {HC}$ . By Theorem 3.32, there is a Harish-Chandra bialgebroid B such that $\bot (-)\cong B\times _{\mathcal {L}}(-)$ . By the standard monadic arguments, F factors through

$$\begin{align*}\mathcal{D}\longrightarrow \mathrm{CoMod}_\bot(\mathrm{HC})\cong \mathrm{CoMod}_B(\mathrm{HC}),\end{align*}$$

which is monoidal (see [Reference Szlachányi81, Proposition 3.5] for the dual statement). If F is conservative and preserves equalisers, by the Barr–Beck theorem [Reference MacLane65, Theorem VI.7.1] the above functor is an equivalence.

4.1 Dynamical twists

Consider a Hopf algebra H, a commutative algebra $\mathcal {L}\in \mathrm {Z}_{\mathrm {Dr}}(\mathrm {LMod}_H)$ (see Theorem 3.3) with a coaction map $\delta \colon \mathcal {L}\rightarrow H\otimes \mathcal {L}$ (denoted by $x\mapsto x_{(-1)}\otimes x_{(0)}$ ) and a cp-rigid monoidal category $\mathcal {C}$ together with a forgetful functor $F\colon \mathcal {C}\rightarrow \mathrm {LMod}_H$ , which we assume sends compact projective objects in $\mathcal {C}$ to finite-dimensional H-modules. It will be convenient to introduce the right H-coaction

$$\begin{align*}\delta^R\colon \mathcal{L}\longrightarrow \mathcal{L}\otimes H\end{align*}$$

by

$$\begin{align*}x\mapsto x_{(0)}\otimes S^{-1}(x_{(-1)}).\end{align*}$$

Proposition 4.1. A monoidal structure with a strict unit map on the composite

$$\begin{align*}\mathcal{C} \longrightarrow \mathrm{LMod}_H \xrightarrow{\mathrm{free}} \mathrm{HC}(\mathrm{LMod}_H, \mathcal{L})\end{align*}$$

is the same as a natural collection of elements $J_{X, Y}\in \mathcal {L}\otimes \mathrm {Hom}(F(X)\otimes F(Y), F(X\otimes Y))$ for $X,Y\in \mathcal {C}^{\mathrm {cp}}$ satisfying

• ○ The elements $J_{X, Y}$ are H-invariant.

• ○ For a triple $X,Y,Z\in \mathcal {C}^{\mathrm {cp}}$ , the equation

  $$\begin{align*}J_{X\otimes Y, Z}\circ(J_{X, Y}\otimes \mathrm{id}_Z) = J_{X, Y\otimes Z}\circ \delta^R_X(J_{Y, Z})\end{align*}$$
  holds, where $\delta ^R_X$ means the H-factor in $\delta ^R$ acts on X.

• ○ For any $X\in \mathcal {C}^{\mathrm {cp}}$ , we have $J_{\mathbf {1}, X}=J_{X, \mathbf {1}} = 1\otimes \mathrm {id}_{F(X)}$ .

Proof. Recall that the monoidal structure on the functor $\mathrm {free}\colon \mathrm {LMod}_H \rightarrow \mathrm {HC}(\mathrm {LMod}_H, \mathcal {L})$ is given by the natural isomorphism

$$ \begin{align*} (\mathcal{L}\otimes X) \otimes_{\mathcal{L}} (\mathcal{L}\otimes Y) &\xrightarrow{\sim} \mathcal{L}\otimes X \otimes Y\\ a\otimes x \otimes b \otimes y &\mapsto ab_{(0)} \otimes S^{-1}(b_{(-1)}) \triangleright x \otimes y \end{align*} $$

for any $X,Y\in \mathrm {LMod}_H$ . So, the monoidal structure on $\mathcal {C}\rightarrow \mathrm {HC}(\mathrm {LMod}_H, \mathcal {L})$ is given by

$$\begin{align*}(\mathcal{L}\otimes F(X))\otimes_{\mathcal{L}} (\mathcal{L}\otimes F(Y))\cong \mathcal{L}\otimes F(X)\otimes F(Y)\xrightarrow{\sim} \mathcal{L}\otimes F(X\otimes Y),\end{align*}$$

where the first isomorphism is given by the monoidal structure on $\mathrm {free}$ and the second isomorphism is

$$\begin{align*}l\otimes a\otimes b\mapsto (l\otimes \mathrm{id}_{F(X\otimes Y)})J_{X, Y}(a\otimes b).\end{align*}$$

The composite is automatically a map of $\mathcal {L}$ -modules, and the compatibility with the H-action is the H-invariance condition on $J_{X, Y}$ .

The associativity condition for the monoidal structure on F is that for compact projective objects $X,Y,Z\in \mathcal {C}^{\mathrm {cp}}$ , the diagram

commutes. Considering the image of $(1\otimes a)\otimes (1\otimes b)\otimes (1\otimes c)$ under these maps, we get the second equation.

The unitality condition for the monoidal structure on F is equivalent to the last equation.

Let us now introduce a universal version of the previous statement. Suppose A is another Hopf algebra with a map of algebras $H\rightarrow A$ . We assume $\mathcal {C}=\mathrm {LMod}_A$ , and the forgetful functor

$$\begin{align*}F\colon \mathrm{LMod}_A\longrightarrow \mathrm{LMod}_H\end{align*}$$

is given by restriction of modules. Denote by $(A, \Delta , \varepsilon )$ the coalgebra structure on A.

Definition 4.2. A dynamical twist is an invertible element

$$\begin{align*}J=J^0 \otimes J^1 \otimes J^2 \in \mathcal{L} \otimes A \otimes A\end{align*}$$

satisfying

1. 1. The invariance condition

   $$\begin{align*}h_{(1)} \triangleright J^0 \otimes h_{(2)} J^1 \otimes h_{(3)} J^2 = J^0\otimes J^1 h_{(1)} \otimes J^2 h_{(2)}\end{align*}$$
   for every $h\in H$ ;

2. 2. The shifted cocycle equation

   $$\begin{align*}((\mathrm{id}\otimes \Delta \otimes \mathrm{id})J) (J \otimes 1) = ((\mathrm{id} \otimes \mathrm{id}\otimes \Delta)J)(J_{(0)}^0 \otimes S^{-1}(J_{(-1)}^0) \otimes J^1 \otimes J^2); \end{align*}$$

3. 3. The normalisation condition

   $$\begin{align*}(\mathrm{id} \otimes \varepsilon \otimes \mathrm{id}) J = 1\otimes 1\otimes 1 = (\mathrm{id} \otimes \mathrm{id} \otimes \varepsilon) J.\end{align*}$$

Example 4.4. Suppose $\mathfrak {h}$ is an abelian Lie algebra, and consider $H=\mathcal {L}=\mathrm {U}\mathfrak {h}$ as in section 3.3. Then a dynamical twist is a function $J\colon \mathfrak {h}^*\rightarrow A\otimes A$ . The invariance condition is that $J(\lambda )$ is $\mathfrak {h}$ -invariant with respect to the adjoint action (the zero-weight condition). The shifted cocycle equation is

$$\begin{align*}((\Delta\otimes\mathrm{id}) J(\lambda))J_{12}(\lambda) = ((\mathrm{id}\otimes\Delta) J(\lambda)) J_{23}(\lambda-h^{(1)}).\end{align*}$$

Proof. By Theorem 4.1, the monoidal structure is specified by a collection of elements

$$\begin{align*}J_{X, Y}\in \mathcal{L}\otimes \mathrm{End}(X\otimes Y),\qquad X,Y\in\mathrm{LMod}_A.\end{align*}$$

By naturality, these are uniquely determined by the elements

$$\begin{align*}J = J_{A, A}(1_A\otimes 1_A)\in\mathcal{L}\otimes A\otimes A.\end{align*}$$

Two dynamical twists may be related by a gauge transformation.

Definition 4.6. A gauge transformation is an invertible H-invariant element $G\in \mathcal {L} \otimes A$ satisfying the normalisation condition

$$\begin{align*}(\mathrm{id} \otimes \varepsilon) (G) = 1 \otimes 1. \end{align*}$$

Given a dynamical twist J and a gauge transformation G, we obtain a new dynamical twist by the formula

(31) $$ \begin{align} J^G = (\mathrm{id} \otimes \Delta)G\cdot J\cdot ((\delta^R\otimes\mathrm{id})G)^{-1}(G \otimes 1)^{-1}. \end{align} $$

Example 4.7. Consider the pair $H=\mathcal {L}=\mathrm {U}\mathfrak {h}$ , as in Theorem 4.4. Then a gauge transformation is a zero-weight function $G\colon \mathfrak {h}^*\rightarrow A^\times $ satisfying $\varepsilon (G(\lambda )) = 1$ . Given a dynamical twist $J\colon \mathfrak {h}^*\rightarrow A\otimes A$ , its gauge transformation is

$$\begin{align*}J^G(\lambda) = (\mathrm{id}\otimes \Delta)G(\lambda) \cdot J(\lambda) \cdot (G_2(\lambda-h^{(1)}))^{-1}(G_1(\lambda))^{-1}.\end{align*}$$

Proposition 4.8. Suppose $J_1, J_2$ are two dynamical twists that give rise to monoidal structures on the functor $\mathrm {LMod}_A\rightarrow \mathrm {HC}(\mathrm {LMod}_H, \mathcal {L})$ by Theorem 4.5. The data of a gauge transformation between them is a monoidal natural isomorphism

4.2 Dynamical FRT and reflection equation algebras

Let us describe the Harish-Chandra bialgebroid B from Theorem 3.35 explicitly. Let $\mathcal {D}$ be a cp-rigid monoidal category and $F\colon \mathcal {D}\rightarrow \mathrm {HC}$ a monoidal functor that admits a colimit-preserving right adjoint $F^{\mathrm {R}}$ .

By Theorem 2.5, the functor $FF^{\mathrm {R}}$ can be calculated as

$$ \begin{align*} FF^{\mathrm{R}}(x) &= \int^{y\in \mathcal{D}^{\mathrm{cp}}} \mathrm{Hom}_{\mathrm{HC}}(F(y)^\vee, x) \otimes F(y)^\vee \\ &\cong \int^{y\in\mathcal{D}^{\mathrm{cp}}} \mathrm{Hom}_{\mathrm{HC}}(\mathcal{L}, F(y)\otimes_{\mathcal{L}} x)\otimes F(y)^\vee. \end{align*} $$

Recalling the definition of the Takeuchi product from Theorem 3.27, we obtain that $FF^{\mathrm {R}}(x)\cong B\times _{\mathcal {L}} x$ , where the Harish-Chandra bialgebroid B is

$$\begin{align*}B\cong \int^{y\in\mathcal{D}^{\mathrm{cp}}} F(y)^\vee\boxtimes F(y)\in\mathrm{HC}\otimes\mathrm{HC}.\end{align*}$$

As in section 2.5, denote by

$$\begin{align*}\pi_y\colon F(y)^\vee\boxtimes F(y)\longrightarrow B\end{align*}$$

the natural projections. The Harish-Chandra bialgebroid structure is given on generators as follows:

1. 1. The coproduct

   $$\begin{align*}B\longrightarrow B\times_{\mathcal{L}} B\cong \int^{(y,z)\in \mathcal{D}^{\mathrm{cp}} \times \mathcal{D}^{\mathrm{cp}}} \mathrm{Hom}_{\mathrm{HC}}(\mathcal{L}, F(y)\otimes_{\mathcal{L}} F(z)^\vee)\otimes F(y)^\vee\boxtimes F(z)\end{align*}$$
   is
   $$\begin{align*}F(y)^\vee\boxtimes F(y)\xrightarrow{\mathrm{coev}\otimes \mathrm{id}} \mathrm{Hom}_{\mathrm{HC}}(\mathcal{L}, F(y)\otimes_{\mathcal{L}} F(y)^\vee)\otimes F(y)^\vee\boxtimes F(y)\xrightarrow{\pi_{y, y}} B\times_{\mathcal{L}} B.\end{align*}$$

2. 2. The counit

   $$\begin{align*}B\longrightarrow T^{\mathrm{R}}_{\mathrm{HC}}(\mathcal{L}) = \int^{P\in\mathrm{HC}^{\mathrm{cp}}} P^\vee\boxtimes P\end{align*}$$
   is the projection
   $$\begin{align*}F(y)^\vee\boxtimes F(y) \rightarrow T^{\mathrm{R}}_{\mathrm{HC}}(\mathcal{L}).\end{align*}$$

3. 3. The product is the composite

   $$ \begin{align*} (F(y)^\vee \boxtimes F(y)) \otimes_{\mathcal{L}\otimes \mathcal{L}} (F(z)^\vee \boxtimes F(z)) &= (F(y)^\vee\otimes_{\mathcal{L}} F(z)^\vee)\boxtimes (F(z)\otimes_{\mathcal{L}} F(y)) \\ &\cong (F(z)\otimes_{\mathcal{L}} F(y))^\vee\boxtimes (F(z)\otimes_{\mathcal{L}} F(y)) \\ &\xrightarrow{(J^{-1}_{z,y})^\vee\boxtimes J_{z,y}} F(z\otimes y)^\vee\boxtimes F(z\otimes y) \\ &\xrightarrow{\pi_{z\otimes y}} B. \end{align*} $$

4. 4. The quantum moment map is

   $$\begin{align*}\mathcal{L} \boxtimes \mathcal{L}^{\mathrm{op}} \cong F(\mathbf{1}) \boxtimes F(\mathbf{1}) \xrightarrow{\pi_{\mathbf{1}}} B. \end{align*}$$

We will now concentrate on the case $\mathcal {C}=\operatorname {\mathrm {Rep}}(H)$ for H a split torus with weight lattice $\Lambda $ and $\mathcal {L}=\mathrm {U}\mathfrak {h}$ , so $\mathrm {HC}=\mathrm {HC}(H)$ . Moreover, we assume that the functor $F\colon \mathcal {D}\rightarrow \mathrm {HC}(H)$ factors as the composite

$$\begin{align*}\mathcal{D}\longrightarrow \operatorname{\mathrm{Rep}}(H)\xrightarrow{\mathrm{free}} \mathrm{HC}(H).\end{align*}$$

For an object $y\in \mathcal {D}$ , we denote its image in $\operatorname {\mathrm {Rep}}(H)$ by the same letter. In this case, the monoidal structure is given by a dynamical twist

$$\begin{align*}J_{y, z}(\lambda)\colon \mathfrak{h}^*\rightarrow \mathrm{End}(y\otimes z)\end{align*}$$

as in Theorem 4.1.

The projections

$$\begin{align*}\pi_y\colon (\mathrm{U}\mathfrak{h}\otimes y^\vee)\boxtimes (\mathrm{U}\mathfrak{h}\otimes y)\longrightarrow B\end{align*}$$

in $\mathrm {HC}(H\times H)$ may be encoded in elements

$$\begin{align*}T_y\in B\otimes \mathrm{End}(y).\end{align*}$$

Analogously to Theorem 2.26, we obtain the following explicit description of the bialgebroid B.

Theorem 4.9. The bialgebroid B is spanned, as an $\mathcal {O}(\mathfrak {h}^*)$ -bimodule, by the matrix coefficients of $T_y$ for $y\in \mathcal {D}^{\mathrm {cp}}$ subject to the relation

$$\begin{align*}F(j)\circ T_x = T_y\circ F(j)\end{align*}$$

for every $j\colon x\rightarrow y$ . Moreover, we have:

1. 1. $\Delta (T_y) = T_y\otimes T_y$ for every $y\in \mathcal {D}^{cp}$ .

2. 2. $\epsilon (T_y) = 1\otimes \mathrm {id}_y\in \mathrm {D}(H)\otimes \mathrm {End}(y)$ for every $y\in \mathcal {D}^{\mathrm {cp}}$ .

3. 3. For every $f\in \mathcal {O}(\mathfrak {h}^*)$ and $y\in \mathcal {D}^{\mathrm {cp}}$ ,

   $$ \begin{align*} s(f(\lambda)) T_y &= T_y s(f(\lambda + h)) \\ t(f(\lambda+h)) T_y &= T_y t(f(\lambda )). \end{align*} $$

4. 4. $J^t_{y, z}(\lambda )^{-1} T_{y\otimes z} J^s_{y, z}(\lambda ) = (T_y\otimes \mathrm {id})(\mathrm {id}\otimes T_z)$ , where by the superscripts, we mean the left multiplication with the $\mathcal {O}(\mathfrak {h}^*)$ -part by either the source $(s)$ or the target $(t)$ map.

5. 5. $T_{\mathbf {1}}\in B$ is the unit.

Definition 4.10. Suppose $\mathcal {D}$ is a braided monoidal category together with a forgetful functor $\mathcal {D}\rightarrow \operatorname {\mathrm {Rep}}(H)$ and a monoidal structure on the composite $\mathcal {D}\rightarrow \operatorname {\mathrm {Rep}}(H)\xrightarrow {\mathrm {free}} \mathrm {HC}(H)$ . For $x,y\in \mathcal {D}$ , define the morphism $\mathrm {U}\mathfrak {h}\otimes x\otimes y\rightarrow \mathrm {U}\mathfrak {h}\otimes y\otimes x$ by

$$\begin{align*}\check{R}_{x,y} \colon F(x) \otimes_{\mathrm{U}\mathfrak{h}} F(y) \xrightarrow{J_{x,y}} F(x\otimes y) \xrightarrow{F(\sigma_{x,y})} F(y\otimes x) \xrightarrow{J_{x,y}^{-1}} F(y) \otimes_{\mathrm{U}\mathfrak{h}} F(x).\end{align*}$$

The dynamical $\boldsymbol {R}$ -matrix is the map $R_{x, y}\colon \mathfrak {h}^*\rightarrow \mathrm {End}(x\otimes y)$ given by

$$\begin{align*}R_{x, y} = (\mathrm{id}_{\mathrm{U}\mathfrak{h}}\otimes \sigma_{x, y}^{-1})\circ \check{R}_{x, y}.\end{align*}$$

As in section 2.5, we use the standard notation $T_1 = T_x \otimes \mathrm {id}$ and $T_2=\mathrm {id}\otimes T_y$ and similarly for the R-matrix.

Proof. As in the proof of Theorem 2.28, the element $\hat {R}$ satisfies the braid relation

$$\begin{align*}\check{R}_{12} \check{R}_{23} \check{R}_{12} = \check{R}_{23} \check{R}_{12} \check{R}_{23}\end{align*}$$

in $\mathrm {U}\mathfrak {h}\otimes \mathrm {End}(x\otimes y\otimes z)$ . Observing that $\check {R} = \sigma \circ R$ , we get the dynamical Yang–Baxter equation.

To show the second part, recall from Theorem 4.9 that

$$\begin{align*}F(\sigma_{x,y}) T_{x\otimes y} = T_{y\otimes x} F(\sigma_{x,y}).\end{align*}$$

Decomposing $T_{x\otimes y}$ and $T_{y\otimes x}$ into $T_x$ and $T_y$ using property $(4)$ of the same theorem, we get the result.

5.1 Classical parabolic restriction

Let G be a reductive group over an algebraically closed field k of characteristic zero and $\mathfrak {g}$ its Lie algebra. Fix a Borel subgroup $B\subset G$ and denote $H = B / [B, B]$ ; their Lie algebras are denoted by $\mathfrak {b}$ and $\mathfrak {h}$ . We denote by N the kernel of $B\rightarrow H$ with Lie algebra $\mathfrak {n}$ . Let W be the Weyl group.

We will later use the Harish-Chandra isomorphism; see [Reference Humphreys49, Theorem 1.10].

Theorem 5.1. There is a unique homomorphism of algebras

$$\begin{align*}\mathrm{hc}\colon \mathrm{Z}(\mathrm{U}\mathfrak{g})\longrightarrow \mathrm{U}\mathfrak{h},\end{align*}$$

the Harish-Chandra homomorphism, such that for any $z\in \mathrm {Z}(\mathrm {U}\mathfrak {g})$ and $m\in M^{\mathrm {univ}}$ , we have

$$\begin{align*}z m = m \mathrm{hc}(z).\end{align*}$$

Definition 5.2. The universal category $\mathcal {O}$ is the category $\mathcal {O}^{\mathrm {univ}}$ of $(\mathrm {U}\mathfrak {g}, \mathrm {U}\mathfrak {h})$ -bimodules whose diagonal $\mathfrak {b}$ -action integrates to a B-action. The universal Verma module is

$$\begin{align*}M^{\mathrm{univ}} = \mathrm{U}\mathfrak{g}\otimes_{\mathrm{U}\mathfrak{b}} \mathrm{U}\mathfrak{h} \in \mathcal{O}^{\mathrm{univ}}.\end{align*}$$

We will now define an important bimodule structure on $\mathcal {O}^{\mathrm {univ}}$ :

(32) $$ \begin{align} \mathrm{HC}(G)\curvearrowright \mathcal{O}^{\mathrm{univ}} \curvearrowleft \mathrm{HC}(H). \end{align} $$

Both actions are given by the relative tensor products of bimodules. Given a $\mathrm {U}\mathfrak {g}$ -bimodule $X\in \mathrm {HC}(G)$ and a $(\mathrm {U}\mathfrak {g}, \mathrm {U}\mathfrak {h})$ -bimodule $M\in \mathcal {O}^{\mathrm {univ}}$ , $X\otimes _{\mathrm {U}\mathfrak {g}} M$ is an $(\mathrm {U}\mathfrak {g}, \mathrm {U}\mathfrak {h})$ -bimodule. Since the diagonal $\mathfrak {g}$ -action on X is integrable, so is the diagonal $\mathfrak {b}$ -action. Therefore, the diagonal $\mathfrak {b}$ -action on $X\otimes _{\mathrm {U}\mathfrak {g}} M$ is integrable. The $\mathrm {HC}(H)$ action is defined similarly.

Let

$$\begin{align*}\mathrm{act}_G\colon \mathrm{HC}(G)\longrightarrow \mathcal{O}^{\mathrm{univ}},\qquad \mathrm{act}_H\colon \mathrm{HC}(H)\longrightarrow \mathcal{O}^{\mathrm{univ}}\end{align*}$$

be the actions of $\mathrm {HC}(G)$ and $\mathrm {HC}(H)$ on the universal Verma module $M^{\mathrm {univ}}\in \mathcal {O}^{\mathrm {univ}}$ . Using Theorem 2.21, we obtain the following lax monoidal functors.

Definition 5.5. The parabolic restriction is the lax monoidal functor

$$\begin{align*}\mathrm{res}= \mathrm{act}^{\mathrm{R}}_H\circ \mathrm{act}_G\colon \mathrm{HC}(G)\longrightarrow \mathrm{HC}(H).\end{align*}$$

The parabolic induction is the lax monoidal functor

$$\begin{align*}\mathrm{ind} = \mathrm{act}^{\mathrm{R}}_G\circ \mathrm{act}_H\colon \mathrm{HC}(H)\longrightarrow \mathrm{HC}(G).\end{align*}$$

Let us now make these functors more explicit. Consider the functor

$$\begin{align*}(-)^N\colon \mathcal{O}^{\mathrm{univ}}\longrightarrow \mathrm{HC}(H),\end{align*}$$

which sends a $(\mathrm {U}\mathfrak {g}, \mathrm {U}\mathfrak {h})$ -bimodule to the subspace of highest-weight vectors with respect to the $\mathrm {U}\mathfrak {g}$ -action. It still has a remaining $\mathrm {U}\mathfrak {h}$ -bimodule structure, so it defines an object of $\mathrm {HC}(H)$ .

Proof. Identify $\mathcal {O}^{\mathrm {univ}}$ with the highest-weight $\mathrm {U}\mathfrak {g}$ -modules in the category $\operatorname {\mathrm {Rep}}(H)$ following Theorem 5.4. For $M\in \mathcal {O}^{\mathrm {univ}}$ and $X\in \mathrm {HC}(H)$ , we have

$$ \begin{align*} \mathrm{Hom}_{\mathcal{O}^{\mathrm{univ}}}(\mathrm{act}_H(X), M) &= \mathrm{Hom}_{\mathcal{O}^{\mathrm{univ}}}(\mathrm{U}\mathfrak{g}\otimes_{\mathrm{U}\mathfrak{b}} X, M) \\ &\cong \mathrm{Hom}_{{}_{\mathrm{U}\mathfrak{b}}\mathrm{BMod}_{\mathrm{U}\mathfrak{h}}}(X, M) \\ &\cong \mathrm{Hom}_{\mathrm{HC}(H)}(X, M^N). \end{align*} $$

So,

$$\begin{align*}\mathrm{res}(X)\cong (X/X\mathfrak{n})^N.\end{align*}$$

The lax monoidal structure on $\mathrm {res}$ can be described explicitly as follows. For $X,Y\in \mathrm {HC}(G)$ , the morphism

(33) $$ \begin{align} (X/X\mathfrak{n})^N \otimes_{\mathrm{U}\mathfrak{h}} (Y/Y/\mathfrak{n})^N \longrightarrow (X\otimes_{\mathrm{U}\mathfrak{g}} Y/(X\otimes_{\mathrm{U}\mathfrak{g}} Y)\mathfrak{n})^N \end{align} $$

is given by $[x] \otimes [y] \mapsto [x \otimes y]$ . This assignment is independent of the choice of a representative of $[x]$ since $[y]$ is N-invariant.

Recall that the coinduction functor

$$\begin{align*}\mathrm{coind}_B^G\colon \operatorname{\mathrm{Rep}}(B)\longrightarrow \operatorname{\mathrm{Rep}}(G)\end{align*}$$

is right adjoint to the obvious restriction functor $\operatorname {\mathrm {Rep}}(G)\rightarrow \operatorname {\mathrm {Rep}}(B)$ . Denote in the same way the functor

$$\begin{align*}\mathrm{coind}_B^G\colon \mathcal{O}^{\mathrm{univ}}\longrightarrow \mathrm{HC}(G)\end{align*}$$

of coinduction from B to G using the diagonal B-action.

Proof. For $M\in \mathcal {O}^{\mathrm {univ}}$ and $X\in \mathrm {HC}(G)$ , we have

$$ \begin{align*} \mathrm{Hom}_{\mathcal{O}^{\mathrm{univ}}}(\mathrm{act}_G(X), M) &= \mathrm{Hom}_{\mathcal{O}^{\mathrm{univ}}}(X\otimes_{\mathrm{U}\mathfrak{b}} \mathrm{U}\mathfrak{h}, M) \\ &\cong \mathrm{Hom}_{{}_{\mathrm{U}\mathfrak{g}}\mathrm{BMod}_{\mathrm{U}\mathfrak{b}}}(X, M). \end{align*} $$

Both X and M are $(\mathrm {U}\mathfrak {g}, \mathrm {U}\mathfrak {b})$ -bimodules whose diagonal $\mathfrak {b}$ -action integrates to a B-action: that is, they are objects of $\mathrm {LMod}_{\mathrm {U}\mathfrak {g}}(\operatorname {\mathrm {Rep}} B)$ . Moreover, X lies in the image of the forgetful functor

$$\begin{align*}\mathrm{HC}(G)=\mathrm{LMod}_{\mathrm{U}\mathfrak{g}}(\operatorname{\mathrm{Rep}} G)\rightarrow \mathrm{LMod}_{\mathrm{U}\mathfrak{g}}(\operatorname{\mathrm{Rep}} B).\end{align*}$$

But by definition, $\mathrm {coind}_B^G$ is the right adjoint to the forgetful functor $\operatorname {\mathrm {Rep}} G\rightarrow \operatorname {\mathrm {Rep}} B$ .

Let us now compute the values of $\mathrm {res}$ and $\mathrm {ind}$ on the units.

Proof. By Theorem 5.6, $\mathrm {res}(\mathrm {U}\mathfrak {g}) \cong (M^{\mathrm {univ}})^N$ , and we have to show that

$$\begin{align*}\mathrm{U}\mathfrak{h}\longrightarrow (M^{\mathrm{univ}})^N\end{align*}$$

is an isomorphism. Let

$$\begin{align*}M^{\mathrm{univ}, \mathrm{gen}} = \mathrm{U}\mathfrak{g}\otimes_{\mathrm{U}\mathfrak{b}}\mathrm{Frac}(\mathrm{U}\mathfrak{h}),\end{align*}$$

where $\mathrm {Frac}(\mathrm {U}\mathfrak {h})$ is the fraction field of $\mathrm {U}\mathfrak {h}$ . The map $M^{\mathrm {univ}}\rightarrow M^{\mathrm {univ}, \mathrm {gen}}$ is injective, and $(-)^N$ is left exact, so $(M^{\mathrm {univ}})^N\longrightarrow (M^{\mathrm {univ}, \mathrm {gen}})^N$ is injective. But the Verma module for generic highest weights is irreducible (see [Reference Humphreys49, Theorem 4.4]), so

$$\begin{align*}\mathrm{Frac}(\mathrm{U}\mathfrak{h})\longrightarrow (M^{\mathrm{univ}, \mathrm{gen}})^N\end{align*}$$

is an isomorphism. This implies the claim.

Corollary 5.10. The induced map

$$\begin{align*}\mathrm{res}\colon \mathrm{Z}(\mathrm{U}\mathfrak{g})=\mathrm{End}_{\mathrm{HC}(G)}(\mathrm{U}\mathfrak{g})\longrightarrow \mathrm{U}\mathfrak{h}=\mathrm{End}_{\mathrm{HC}(H)}(\mathrm{U}\mathfrak{h})\end{align*}$$

coincides with the Harish-Chandra homomorphism $\mathrm {hc}\colon \mathrm {Z}(\mathrm {U}\mathfrak {g})\rightarrow \mathrm {U}\mathfrak {h}$ .

Proof. The map

$$\begin{align*}\mathrm{act}_G\colon \mathrm{Z}(\mathrm{U}\mathfrak{g})=\mathrm{End}_{\mathrm{HC}(G)}(\mathrm{U}\mathfrak{g})\longrightarrow \mathrm{End}_{\mathcal{O}^{\mathrm{univ}}}(M^{\mathrm{univ}})\end{align*}$$

sends a central element $z\in \mathrm {Z}(\mathrm {U}\mathfrak {g})$ to the left action of $z\in \mathrm {Z}(\mathrm {U}\mathfrak {g})$ on $M^{\mathrm {univ}}$ . By Theorem 5.1, it is equal to the right action of $\mathrm {hc}(z)\in \mathrm {U}\mathfrak {h}$ on $M^{\mathrm {univ}}$ . To conclude, observe that the map

$$\begin{align*}\mathrm{U}\mathfrak{h}\longrightarrow (M^{\mathrm{univ}})^N\end{align*}$$

is an isomorphism of right $\mathrm {U}\mathfrak {h}$ -modules.

Proposition 5.11. Suppose G is connected and simply connected. Then there is an isomorphism

$$\begin{align*}\mathrm{ind}(\mathrm{U}\mathfrak{h})\cong \mathrm{U}\mathfrak{g}\otimes_{\mathrm{Z}(\mathrm{U}\mathfrak{g})} \mathrm{U}\mathfrak{h},\end{align*}$$

where the $\mathrm {Z}(\mathrm {U}\mathfrak {g})$ -action on $\mathrm {U}\mathfrak {h}$ is via the Harish-Chandra homomorphism $\mathrm {hc}$ .

Proof. By Theorem 5.8, $\mathrm {ind}(\mathrm {U}\mathfrak {h})\cong \mathrm {coind}_B^G(M^{\mathrm {univ}})$ . Identifying B-representations with G-equivariant quasi-coherent sheaves on $G/B$ , $M^{\mathrm {univ}}$ is sent to $(\pi _* \mathrm {D}_{G/N})^H$ , where $\pi \colon G/N\rightarrow G/B$ . Therefore,

$$\begin{align*}\mathrm{coind}_B^G(M^{\mathrm{univ}})\cong \mathrm{D}(G/N)^H.\end{align*}$$

The claim then follows from [Reference Šapovalov90, Reference van den Hombergh and de Vries87]; see also [Reference Miličić68, Lemma 3.1].

Note that the functor $\mathrm {res}$ preserves neither limits nor colimits, and it is merely lax monoidal. We will now show that after a localisation, it becomes exact and monoidal.

By construction,

$$\begin{align*}\mathrm{HC}(H)^{\mathrm{gen}} = \mathrm{HC}(\operatorname{\mathrm{Rep}}(H), (\mathrm{U}\mathfrak{h})^{\mathrm{gen}})\end{align*}$$

and similarly for $\mathcal {O}^{\mathrm {univ}, \mathrm {gen}}$ . Moreover, both $\mathrm {HC}(H)^{\mathrm {gen}}\subset \mathrm {HC}(H)$ and $\mathcal {O}^{\mathrm {univ}, \mathrm {gen}}\subset \mathcal {O}^{\mathrm {univ}}$ admit left adjoints given by localisation. Let

$$\begin{align*}M^{\mathrm{univ}, \mathrm{gen}} = \mathrm{U}\mathfrak{g}\otimes_{\mathrm{U}\mathfrak{b}} (\mathrm{U}\mathfrak{h})^{\mathrm{gen}}\end{align*}$$

be the universal Verma module with generic highest weights.

Choose a Borel subgroup $B_-\subset G$ opposite to B, with Lie algebra $\mathfrak {b}_-$ . Let

$$\begin{align*}M^{\mathrm{univ}}_- = \mathrm{U}\mathfrak{h}\otimes_{\mathrm{U}\mathfrak{b}_-} \mathrm{U}\mathfrak{g}\end{align*}$$

be the opposite universal Verma module.

Definition 5.13. The functor of $\mathfrak {n}_-$ -coinvariants

$$\begin{align*}(-)_{\mathfrak{n}_-}\colon \mathcal{O}^{\mathrm{univ}}\longrightarrow \mathrm{HC}(H)\end{align*}$$

is $M_{\mathfrak {n}_-} = M^{\mathrm {univ}}_-\otimes _{\mathrm {U}\mathfrak {g}} M$ .

We will now recall the extremal projector introduced in [Reference Asherova, Smirnov and Tolstoy4]; see also [Reference Zhelobenko94].

Example 5.15. Suppose $\mathfrak {g}=\mathfrak {sl}_2$ . The extremal projector in this case is (see, e.g., [Reference Khoroshkin and Ogievetsky53])

$$\begin{align*}P = \sum_{n=0}^\infty \frac{(-1)^n}{n!} g_n^{-1} f^n e^n,\end{align*}$$

where

$$\begin{align*}g_n = \prod_{j=1}^n (h+j+1).\end{align*}$$

We will now describe some applications of extremal projectors.

Proof. Take $M\in \mathcal {O}^{\mathrm {univ}, \mathrm {gen}}$ , and consider the composite

$$\begin{align*}\pi\colon M^N\longrightarrow M\longrightarrow M_{\mathfrak{n}_-}.\end{align*}$$

We will prove that it is an isomorphism.

Since the weights of the right $\mathrm {U}\mathfrak {h}$ -action on M are generic and the weights of the diagonal $\mathrm {U}\mathfrak {h}$ -action are integral, the weights of the left $\mathrm {U}\mathfrak {h}$ -action are also generic. Moreover, the left $\mathrm {U}\mathfrak {n}$ -action is locally nilpotent. In particular, the action of the extremal projector from Theorem 5.14

$$\begin{align*}P\colon M\longrightarrow M\end{align*}$$

is well-defined. It lands in N-invariants by the property $\mathfrak {n} P = 0$ . It factors through $\mathfrak {n}_-$ -coinvariants by the property $P\mathfrak {n}_- = 0$ . So, it gives a map

$$\begin{align*}P\colon M_{\mathfrak{n}_-}\longrightarrow M^N.\end{align*}$$

For $m\in M^N$ , we have $Pm = m$ since $P-1\in T(\mathfrak {g})\mathfrak {n}$ . In particular, $P\circ \pi = \mathrm {id}$ . For $m\in M$ , $[m] = [Pm]$ in $M_{\mathfrak {n}_-}$ since $P-1\in \mathfrak {n}_- T(\mathfrak {g})$ . In particular, $\pi \circ P = \mathrm {id}$ .

Proof. The unit of the adjunction $\mathrm {act}_H\dashv (-)^N$ between $\mathrm {HC}(H)^{\mathrm {gen}}$ and $\mathcal {O}^{\mathrm {univ}, \mathrm {gen}}$ is

$$\begin{align*}X\longrightarrow (\mathrm{act}_H(X))^N\xrightarrow{\sim} (M^{\mathrm{univ}}\otimes_{\mathrm{U}\mathfrak{h}} X)_{\mathfrak{n}_-}.\end{align*}$$

By the PBW isomorphism, this map is an isomorphism. In particular, $\mathrm {act}_H\colon \mathrm {HC}(H)^{\mathrm {gen}}\rightarrow \mathcal {O}^{\mathrm {univ}, \mathrm {gen}}$ is fully faithful.

Since the $\mathfrak {n}$ -action on $M\in \mathcal {O}^{\mathrm {univ}, \mathrm {gen}}$ is locally nilpotent, $M^N = 0$ if and only if $M=0$ . But $(-)^N\colon \mathcal {O}^{\mathrm {univ}, \mathrm {gen}}\rightarrow \mathrm {HC}(H)^{\mathrm {gen}}$ is exact by Theorem 5.16. Therefore, it is conservative. Since its left adjoint $\mathrm {act}_H$ is fully faithful, it is an equivalence.

Corollary 5.18. The composite

$$\begin{align*}\mathrm{res}^{\mathrm{gen}}\colon \mathrm{HC}(G)\xrightarrow{\mathrm{res}} \mathrm{HC}(H)\longrightarrow \mathrm{HC}(H)^{\mathrm{gen}}\end{align*}$$

is strongly monoidal and colimit-preserving.

We will now show that $\mathrm {res}^{\mathrm {gen}}$ gives rise to a dynamical twist. For this, according to Theorem 4.1, we have to show that $\mathrm {res}^{\mathrm {gen}}$ of a free Harish–Chandra bimodule is free: that is, we have to establish an isomorphism between $(V\otimes M^{\mathrm {univ}})^N$ and $V\otimes (\mathrm {U}\mathfrak {h})^{\mathrm {gen}}$ in $\mathrm {HC}(H)^{\mathrm {gen}}$ , for every $V\in \operatorname {\mathrm {Rep}}(G)$ .

Theorem 5.20. The morphism

$$\begin{align*}(V\otimes M^{\mathrm{univ}, \mathrm{gen}})^N\subset V\otimes M^{\mathrm{univ}, \mathrm{gen}} \longrightarrow V\otimes (\mathrm{U}\mathfrak{h})^{\mathrm{gen}},\end{align*}$$

where the second morphism is induced by the projection $M^{\mathrm {univ}, \mathrm {gen}}\rightarrow (\mathrm {U}\mathfrak {h})^{\mathrm {gen}}$ onto the highest weights, defines a natural isomorphism witnessing commutativity of the diagram