2.1. Frobenius structures

We recall some standard notions, mainly to fix notation. The reader can find more details about monoidal categories, algebra objects and dualizable objects in Sections 2.1, 7.8 and 2.10 of [Reference Etingof, Gelaki, Nikshych and Ostrik8].

Let $(\mathcal{C}, \otimes , \unicode{x1D7D9})$ be a monoidal category. Recall that an algebra (or algebra object) in $\mathcal{C}$ is a triple $(A,m,u)$ where $A\in \mathcal{C}$ is an object and $m$ and $u$ are maps $m\;:\;A\otimes A\to A$ (multiplication) and $u\;:\;\unicode{x1D7D9}\to A$ (unit) in $\mathcal{C}$ which satisfy the associativity and unitality conditions. We sometimes draw maps using string diagrams which go up the page, as follows:

The unit object is denoted by the empty diagram.

An object $X\in \mathcal{C}$ is said to be left dualizable if it has a left dual $X^\vee$ , so there exist maps $e_X\;:\;X^\vee \otimes X\to \unicode{x1D7D9}$ (evaluation) and $c_X\;:\;\unicode{x1D7D9}\to X\otimes X^\vee$ (coevaluation) satisfying the triangle identities.

Similarly, $X\in \mathcal{C}$ is right dualizable if it has a right dual ${}^\vee X$ , so there exist maps $e^r_X\;:\;X\otimes {}^\vee X\to \unicode{x1D7D9}$ and $c^r_X\;:\;\unicode{x1D7D9}\to {}^\vee X\otimes X$ satisfying the triangle identities. If $X$ is left dualizable, then ${}^\vee (X^\vee )\cong X$ .

Definition 2.1. Suppose $A$ is left dualizable. A $\text{Frobenius structure}$ on the algebra $(A,u,m)$ is a pair $(W,n)$ where $W\in \mathcal{C}$ and $n\;:\;A\to W$ is a map in $\mathcal{C}$ such that the map $\kappa =n\circ m\;:\;A\otimes A\to W$ is non-degenerate, that is, the map $ \varphi \;:\;A \stackrel {1_A\otimes c_A}{\longrightarrow } A\otimes A\otimes A^\vee \stackrel {\kappa \otimes 1_{A^\vee }}{\longrightarrow } W\otimes A^\vee$

is invertible.

Note that if $W=\unicode{x1D7D9}$ , then we recover the more restrictive definition of a Frobenius structure given in [Reference Fuchs and Stigner11 and Reference Street24]. We call this a classical Frobenius structure.

Remark 2.2. If $A$ is an algebra object, then the left dual $A^\vee$ is a categorical right $A$ -module with the usual action map

\begin{equation*} A^\vee \otimes A \stackrel {1_{A^\vee \otimes A}\otimes c_A}{\longrightarrow } A^\vee \otimes A\otimes A\otimes A^\vee \stackrel {1_{A^\vee }\otimes m\otimes 1_{A^\vee }}{\longrightarrow } A^\vee \otimes A\otimes A^\vee \stackrel {e_A\otimes 1_{A^\vee }}{\longrightarrow }A^\vee \end{equation*}

and applying $W\otimes -$ gives the structure of a categorical right $A$ -module to $W\otimes A^\vee$ . Then applying associativity of $m$ and using a triangle identity shows that $\varphi$ is automatically a morphism of right $A$ -modules.

The next result shows that our more general definition gives nothing new in the classical situation of finite-dimensional algebras over vector spaces.

An object $X\in \mathcal{C}$ is called invertible if it is left dualizable and $e_X$ and $c_X$ are both isomorphisms. Then $X^\vee$ is called the inverse object of $X$ . Note that the inverse of the triangle identities for $X$ show that $X^\vee$ has left dual $X$ with $e_{X^\vee }=c_X^{-1}$ and $c_{X^\vee }=e_X^{-1}$ . Therefore, ${^\vee X}\cong X^\vee$ .

We now give some results on Frobenius structures $(W,n)$ under the assumption that $W$ is invertible.

Proof. We have an isomorphism $e_W\;:\;V\otimes W\stackrel {\sim }{\to } \unicode{x1D7D9}$ , so we construct a map

\begin{equation*} f\;:\; V\otimes A\stackrel {1_V\otimes \varphi }{\longrightarrow } V\otimes W\otimes A^\vee \stackrel {e_W\otimes 1_{A^\vee }}{\longrightarrow }A^\vee \end{equation*}

which, using $e_W^{-1}=c_V$ , is an isomorphism with inverse

\begin{equation*} f^{-1}\;:\; A^\vee \stackrel {c_V\otimes 1_{A^\vee }}{\longrightarrow } V\otimes W\otimes A^\vee \stackrel {1_V\otimes \varphi ^{-1}}{\longrightarrow } V\otimes A.\end{equation*}

Thus, $A^\vee \cong V\otimes A$ . Consider the composition

\begin{equation*} \varphi _{\text{new}}\;:\;A\stackrel {\varphi }{\longrightarrow } W\otimes A^\vee \stackrel {1_W\otimes f^{-1}}{\longrightarrow } W\otimes V\otimes A. \end{equation*}

Using naturality of $\otimes$ and the triangle identities, we have

\begin{align*} (e_V\otimes 1_A)\circ \varphi _{\text{new}} &= (e_V\otimes 1_A)\circ (1_W\otimes 1_V\otimes \varphi ^{-1})\circ (1_W\otimes c_V\otimes 1_{A^\vee } )\circ \varphi \\ &= \varphi ^{-1}\circ (e_V\otimes 1_W\otimes 1_{A^\vee })\circ (1_W\otimes c_V\otimes 1_{A^\vee } )\circ \varphi \\ &= \varphi ^{-1}\circ (1_W\otimes 1_{A^\vee })\circ \varphi = 1_A \end{align*}

So $\varphi _{\text{new}}=(e_V\otimes 1_A)^{-1}=c_W\otimes 1_A$ .

Proof. The following diagrams illustrate the constructions used below:

Suppose $\text{(i)}$ holds. We know that $A^\vee \cong V\otimes A$ by Lemma 2.4. First, we define ${^\vee A}= A\otimes V$ and show that it is a right dual of $A$ . We construct the evaluation as

\begin{equation*} e^r_A\;:\;A\otimes {^\vee A} =A\otimes A\otimes V\stackrel {\varphi \otimes 1_{A\otimes V}}{\longrightarrow } W\otimes A^\vee \otimes A\otimes V \stackrel {1\otimes e_A\otimes 1}{\longrightarrow }W\otimes V\stackrel {e_V}{\longrightarrow }\unicode{x1D7D9} \end{equation*}

and the coevaluation as

\begin{equation*} c^r_A\;:\;\unicode{x1D7D9}\stackrel {c_A}{\longrightarrow }A\otimes A^\vee \stackrel {1\otimes c_V\otimes 1}{\longrightarrow } A\otimes V\otimes W\otimes A^\vee \stackrel {1\otimes \varphi ^{-1}}{\longrightarrow } A\otimes V\otimes A={^\vee A} \otimes A\end{equation*}

and it is straightforward to check that they satisfy the triangle identities. So, we have constructed $^\vee A$ explicitly and thus $A$ is right dualizable.

Now we show that $\varphi '$ is invertible. The maps $\varphi$ and $\varphi '$ are related by

\begin{equation*}\varphi '=(1_{^\vee A\otimes W}\otimes e_A) \circ (1_{^\vee A}\otimes \varphi \otimes 1_A)\circ (c^r_A\otimes 1_A).\end{equation*}

Using the description of $c^r_A$ above, we deduce that $\varphi '=1_A\otimes c_V\;:\;A\to A\otimes V\otimes W$ , giving a description of $\varphi '$ similar to that of Lemma 2.4 for $\varphi$ . Then because $V$ is the inverse of $W$ , we know that $c_V$ is invertible, and therefore $\varphi '$ is invertible.

For $\text{(ii)}\Rightarrow \text{(i)}$ , we can apply $\text{(i)}\Rightarrow \text{(ii)}$ in the monoidal category with opposite tensor product.

Sometimes, the following less symmetric definition is more convenient to work with.

Definition 2.6. Suppose $A$ is left dualizable. A $\text{left Frobenius structure}$ on the algebra $(A,u,m)$ is a pair $(V,n^\ell )$ where $V\in \mathcal{C}$ is a right dualizable object and

\begin{equation*}n^\ell \;:\;V\otimes A\to \unicode{x1D7D9}\end{equation*}

is a map in $\mathcal{C}$ such that

\begin{equation*} \varphi ^\ell \;:\;V\otimes A\stackrel {1_V\otimes 1_A\otimes c_A}{\longrightarrow } V\otimes A\otimes A\otimes A^\vee \stackrel {1_{V}\otimes m\otimes 1_{A^\vee }}{\longrightarrow }V\otimes A\otimes A^\vee \stackrel {n^\ell \otimes 1_{A^\vee }}{\longrightarrow } A^\vee \end{equation*}

is invertible in $\mathcal{C}$ .

The following result shows that Definitions 2.1 and 2.6 are equivalent.

One could also define a right Frobenius structure with a map $n^r\;:\;A\otimes V\to \unicode{x1D7D9}$ , and a result analogous to Lemma 2.7 holds.

2.2. Nakayama automorphisms

Classically, there are at least two ways to define the Nakayama automorphism. The first is Nakayama’s original definition: given a Frobenius ${\unicode{x1D55C}}$ -algebra $A$ with Frobenius form $n\;:\;A\to {\unicode{x1D55C}}$ , Nakayama showed that there exists an automorphism $\alpha \;:\;A\to A$ such that, for all $x,y\in A$ , we have $n(\alpha (x)y)=n(yx)$ [Reference Nakayama20, Theorem 1], now known as the Nakayama automorphism. One could mimic this definition for a Frobenius algebra object in a braided monoidal category, but we will have reason to consider tensor categories which do not admit a braiding, as the following example shows.

Composing the multiplication $m\;:\;A\otimes _{{\unicode{x1D55C}}} A\to A$ with the pairing $n\;:\;A\to {\unicode{x1D55C}}$ gives a bilinear form $\kappa \;:\;A\otimes _{{\unicode{x1D55C}}} A\to {\unicode{x1D55C}}$ . This induces an isomorphism $\kappa ^r\;:\;A\stackrel {\sim }{\to } A^*$ of right $A$ -modules sending $y\in A$ to the linear functional $x\mapsto \kappa (y,x)$ . One checks that $\kappa ^r$ is in fact an isomorphism of $A\text{-} A$ -bimodules if one twists the left action on $A^*$ by the inverse Nakayama automorphism $\alpha ^{-1}$ , and this gives the second definition of $\alpha$ .

We can construct the Nakayama automorphism directly as the composition

\begin{equation*}A\stackrel {\kappa ^r}{\longrightarrow } A^* \stackrel {((\kappa ^r)^*)^{-1}}{\longrightarrow }A^{**}\stackrel {\text{ev}^{-1}}{\longrightarrow }A.\end{equation*}

One could mimic this definition for a Frobenius algebra object in a rigid monoidal category. We will generalise this approach to our setting in Definition 2.9.

Let $A\in \mathcal{C}$ . Throughout, we assume that $(A,u,m)$ has Frobenius structure $(W,n)$ where $W$ is invertible with $W^\vee =V$ . So, we have an isomorphism $\varphi \;:\; A\to W\otimes A^\vee$ which induces $A^\vee \cong V\otimes A$ .

The category $\mathcal{C}$ is pivotal if there is an isomorphism of monoidal functors $\iota \;:\;-\to -^{\vee \vee }$ from the identity to the double dual: see [Reference Etingof, Gelaki, Nikshych and Ostrik8, Section 4.7]. We say that an object $A\in \mathcal{C}$ is pivotal if the full subcategory on the object $A$ is pivotal.

The isomorphism $A^\vee \cong V\otimes A$ induces isomorphisms

\begin{equation*} A^{\vee \vee }\cong (V\otimes A)^\vee \cong A^\vee \otimes V^\vee \cong V\otimes A \otimes V^\vee \end{equation*}

so if $A$ is pivotal, we should get an isomorphism between $V\otimes A$ and $A\otimes V$ . Even though this is not an automorphism, it will play the role of the Nakayama automorphism in our setup.

Definition 2.9. Suppose $A$ is pivotal. The $\text{Nakayama morphism}$ of $A$ is the map

\begin{equation*} \alpha \;:\; V\otimes A \to A\otimes V \end{equation*}

defined by

\begin{equation*} V\otimes A \stackrel {1_V\otimes \varphi }{\longrightarrow } V\otimes W\otimes A^\vee \stackrel {e_W\otimes (\varphi ^\vee )^{-1}}{\longrightarrow } A^{\vee \vee }\otimes V \stackrel {\iota _A^{-1}\otimes 1_V}{\to } A\otimes V.\end{equation*}

By abuse of notation, we write $\alpha ^2$ for the map

\begin{equation*} \alpha ^2\;:\;V\otimes V\otimes A\stackrel {1_V\otimes \alpha }{\to } V\otimes A\otimes V\stackrel {\alpha \otimes 1_V}{\to } A\otimes V\otimes V.\end{equation*}

Similarly, for any $n\geq 1$ , let $\alpha ^n$ denote the map

\begin{equation*} \alpha ^n\;:\; V^{\otimes n}\otimes A\to A\otimes V^{\otimes n}.\end{equation*}

Definition 2.10. We write $\alpha ^n=1$ if there exists an invertible map $\xi \;:\;V^{\otimes n}\stackrel {\sim }{\to } \unicode{x1D7D9}$ in $\mathcal{C}$ such that the composition

\begin{equation*} A\stackrel {\sim }{\to } \unicode{x1D7D9}\otimes A\stackrel {\xi ^{-1}\otimes 1_A}{\to } V^{\otimes n}\otimes A \stackrel {\alpha ^n}{\to } A\otimes V^{\otimes n} \stackrel {1_A\otimes \xi }{\to } A\otimes \unicode{x1D7D9}\stackrel {\sim }{\to } A \end{equation*}

is the identity on $A$ . We say $\alpha$ has order $n\in \{1,2,\ldots ,\infty \}$ if $n$ is minimal such that $\alpha ^n=1$ .

2.3. Transfer of structure

Let $\mathcal{C}$ and $\mathcal{D}$ be monoidal categories. Recall that a monoidal functor from $\mathcal{C}$ to $\mathcal{D}$ is a pair $(F,J)$ consisting of a functor $F\;:\;\mathcal{C}\to \mathcal{D}$ and isomorphisms $J_{C,C'}\;:\;F(C)\otimes F(C')\to F(C\otimes C')$ natural in both $C,C'\in \mathcal{C}$ . It should satisfy unitality and associativity conditions. Note that monoidal functors respect duals [Reference Etingof, Gelaki, Nikshych and Ostrik8, Exercise 2.10.6].

Fix a monoidal functor $(F,J)\;:\;\mathcal{C}\to \mathcal{D}$ and an algebra $(A,m,u)$ in $\mathcal{C}$ . Then we get an algebra $(B=F(A),m_B,u_B)$ in $\mathcal{D}$ with structure maps defined using $F$ and $J$ .

Proof. We need to show that the map $\varphi _B:B\to Y\otimes B^\vee$ is invertible. Applying the definitions, and using the unitality, naturality and associativity conditions on $J$ , we get that the following diagram commutes:

As $F(\varphi _A)$ and $J$ are invertible, so is $\varphi _B$ .

Keep the notation from above, and let $V\in \mathcal{C}$ and $X\in \mathcal{D}$ be the left duals of $W$ and $Y$ , respectively.

Lemma 2.12. Suppose $A\in \mathcal{C}$ and $B\in \mathcal{D}$ are both pivotal, with $\iota _B^{\mathcal{D}}=F(\iota _A^{\mathcal{C}})$ . Let $\alpha _A$ and $\alpha _B$ denote the Nakayama morphisms of $A$ and $B$ . Then the following diagram commutes:

Keep the conditions and notation of Lemma 2.12. Applying the same methods as above, together with naturality of tensor products, gives the following result:

2.4. Examples

Fix a field ${\unicode{x1D55C}}$ .

Example 2.14. Let $\mathcal{C}$ be a semisimple ${\unicode{x1D55C}}$ -category with two non-isomorphic objects, $\unicode{x1D7D9}$ and $X$ , and all their finite direct sums. Define a strict monoidal structure on $\mathcal{C}$ by $X\otimes X=\unicode{x1D7D9}$ . This structure is rigid, with $X^\vee =X={^\vee X}$ . Let $A=\unicode{x1D7D9}\oplus X$ . Define $u\;:\;\unicode{x1D7D9}\to A$ by inclusion and $m\;:\;A\otimes A\to A$ by inclusion for $\unicode{x1D7D9}\otimes \unicode{x1D7D9}$ , $\unicode{x1D7D9}\otimes X$ and $X\otimes \unicode{x1D7D9}$ , and by the zero map for $X\otimes X$ . Then, $(A,m,u)$ is an algebra in $\mathcal{C}$ .

Define a map $n\;:\;A\to X$ by projection, and note that $\kappa =n\circ m\;:\;A\otimes A\to X$ acts as the identity on $\unicode{x1D7D9}\otimes X$ and $X\otimes \unicode{x1D7D9}$ , and as zero on $\unicode{x1D7D9}\otimes \unicode{x1D7D9}$ and $X\otimes X$ . We claim that $(X,n)$ is a Frobenius structure. By properties of duals of direct sums, the map $c_A\;:\;\unicode{x1D7D9}\to A\otimes A$ has identity components to $\unicode{x1D7D9}\otimes \unicode{x1D7D9}$ and $X\otimes X$ and zero components to $\unicode{x1D7D9}\otimes X$ and $X\otimes \unicode{x1D7D9}$ . Therefore, the map $\varphi \;:\;A\stackrel {1\otimes c}{\longrightarrow }A\otimes A\otimes A\stackrel {\kappa \otimes \unicode{x1D7D9}}{\longrightarrow } X\otimes A$ is defined on each summand by the following diagram, where all arrows are isomorphisms:

We see that $\varphi$ is an isomorphism having summands $\unicode{x1D7D9}\stackrel {\sim }{\to } X\otimes X$ and $X\stackrel {\sim }{\to } X\otimes \unicode{x1D7D9}$ . Therefore, the dual map $\varphi ^\vee$ has summands $X\otimes X\stackrel {\sim }{\to } \unicode{x1D7D9}$ and $\unicode{x1D7D9}\otimes X\stackrel {\sim }{\to } X$ .

Consider the trivial pivotal structure, where we identify an object with its double dual using the identity map. Then the Nakayama morphism $X\otimes A\to A\otimes X$ is the direct sum of the following isomorphisms:

Example 2.15. Let $Q$ be the following quiver

and let $A={\unicode{x1D55C}} Q/(ab,ba)$ be its path algebra modulo paths of length $2$ , so $A$ has basis $\{e_1,e_2,a,b\}$ . Then $A$ is a Frobenius ${\unicode{x1D55C}}$ -algebra with Frobenius form $n\;:\;A\to {\unicode{x1D55C}}$ defined by $n(a)=n(b)=1$ and $n(e_1)=$ $n(e_2)=0$ . Its (classical) Nakayama automorphism is the map of ${\unicode{x1D55C}}$ -vector spaces $A\to A$ which interchanges $e_1\leftrightarrow e_2$ and $a\leftrightarrow b$ .

Let $S$ denote the subalgebra generated by $e_1$ and $e_2$ and let $\mathcal{D}=S\!\operatorname {-mod-}\! S$ be the category of $S\text{-} S$ -bimodules, as in Example 2.8 . Note that $A$ has left and right $S$ -actions and the multiplication is balanced over $S$ , so $A$ is an algebra object in $\mathcal{D}$ . However, all maps $A\to S$ in $\mathcal{D}$ must send $a$ and $b$ to zero, so one can deduce that there are no classical Frobenius structures on the algebra $A\in \mathcal{D}$ .

Let $W$ be a $2$ -dimensional vector space with basis $\{w_{12}, w_{21}\}$ . We give $W$ an $S\text{-} S$ -bimodule structure by $e_iw_{ij}e_j=w_{ij}$ . Note that $W$ is self-inverse: $W\otimes W\cong S$ , and therefore $W^\vee =W$ . Define $n\;:\;A\to W$ by $n(a)=w_{12}$ , $n(b)=w_{21}$ and $n(e_1)=n(e_2)=0$ . Then one can check that $(W,n)$ is a Frobenius structure on $A\in \mathcal{D}$ . Its Nakayama morphism is defined by:

\begin{align*} \alpha \;:\;W\otimes A &\to A\otimes W\\ w_{12}\otimes e_2 &\mapsto e_1\otimes w_{12}\\ w_{21}\otimes e_1 &\mapsto e_2\otimes w_{21}\\ w_{12}\otimes b &\mapsto a\otimes w_{21}\\ w_{21}\otimes a &\mapsto b\otimes w_{12} \end{align*}

On ignoring the $w_{ij}$ factors of the tensor product, one can see the same behaviour that appears in the (classical) Nakayama automorphism of the ${\unicode{x1D55C}}$ -algebra.

One can check that $A$ is a $\beta$ -Frobenius extension of $S$ (as defined below), where $\beta$ is the automorphism of $S$ which interchanges $e_1\leftrightarrow e_2$ , and so this example illustrates Proposition 2.18 . Also, there is a monoidal functor $\mathcal{C}\to \mathcal{D}$ from the monoidal category in Example 2.14 to the monoidal category here, sending $\unicode{x1D7D9}$ to $S$ and $X$ to the $S\text{-} S$ -bimodule with ${\unicode{x1D55C}}$ -vector space basis $\{a,b\}$ , so this example illustrates Proposition 2.11 .

We now give a class of examples which shows how our construction generalises Frobenius extensions of the second kind.

Let $\mathcal{C}=S\!\operatorname {-mod-}\! S$ be the monoidal category of $S\text{-} S$ -bimodules, with tensor product over $S$ . Note that $R$ is an algebra object in $\mathcal{C}$ .

3.1. Jones–Wenzl projections

Let $\mathbb{N}$ denote the natural numbers, which for us include zero. We consider planar diagrams $m\to n$ on a rectangle, with $m$ dots on the bottom and $n$ dots on the top, and non-crossing lines so that every dot is joined to exactly one line. We allow finitely many loops, which are not joined to any dots.

Recall that the idempotent completion of a category has pairs $(f,x)$ as objects, where $x$ is an object in our original category and $f\;:\;x\to x$ is an idempotent. Maps $(f,x)\to (g,y)$ are of the form $ g\varphi f$ , where $\varphi \;:\;x\to y$ is a map in our original category, and we sometimes just write them as $\phi \;:\;(f,x)\to (g,y)$ .

We define the following structures:

• • $\mathcal{D}$ is the category with $\operatorname {ob}\nolimits \mathcal{D}=\mathbb{N}$ and

  \begin{equation*} \mathcal{D}(m,n)=\begin{cases} \emptyset &\text{if } 2 \nmid m+n;\\ \text{diagrams $m\to n$ up to isotopy} &\text{otherwise}.\end{cases}\end{equation*}
  Composition is by vertical stacking, and we have a monoidal structure where $m_1\otimes m_2=m_1+m_2$ and morphisms are tensored by joining diagrams horizontally. Note that $\mathcal{D}$ is the free pivotal monoidal category on one self-dual generator: see [Reference Abramsky1, Proposition 1.1].

• • Let $\mathbb{C} \mathcal{D}$ denote the linearisation of $\mathcal{D}$ , so $\operatorname {ob}\nolimits \mathbb{C} \mathcal{D}=\operatorname {ob}\nolimits \mathcal{D}$ and $\mathbb{C} \mathcal{D}(m,n)$ is the $\mathbb{C}$ -vector space with basis $\mathcal{D}(m,n)$ . Composition in $\mathbb{C} \mathcal{D}$ is defined by linearity.

• • Fix $k\geq 1$ . Let $t=e^{\frac {\pi i}{2k+4}}$ be a fixed $4k+8$ th root of unity and let $q=t^2=e^{\frac {\pi i}{k+2}}$ .

• • Let $[i]=\frac {q^i-q^{-i}}{q-q^{-1}}\in \mathbb{R}$ denote the $i$ th quantum integer and write $\delta =-[2]=-q-q^{-1}$ .

• • Define $TL_k$ to be the additive and idempotent completion of the quotient of $\mathbb{C} T$ by the relation that every loop is equal to the constant $\delta$ . We sometimes write this as $TL$ , dropping the subscript $k$ . The category $TL$ inherits a monoidal structure from $\mathcal{D}$ .

Note that $q^{k+2}=-1$ , so $[k+2-j]=[j]$ . In particular, $[k+2]=[0]=0$ .

Let $1_1\;:\;1\to 1$ , $\cup \;:\;0\to 2$ and $\cap \;:\;2\to 0$ denote the unique such morphisms in $\mathcal{D}$ , then define

\begin{equation*} \cup _{i,n}=1_1^{\otimes i} \otimes \cup \otimes 1_1^{\otimes n-i}\;:\;n\to n+2 \end{equation*}

and

\begin{equation*} \cap _{i,n}=1_1^{\otimes i} \otimes \cap \otimes 1_1^{\otimes n-i}\;:\;n+2\to n \end{equation*}

both for $0\leq i\leq n$ . Let $U_{i,n}=\cup _{i,n-2}\circ \cap _{i,n-2}\;:\;n\to n$ denote the cap–cup morphism for $1\leq i\leq n-2$ , so $U_i^2=\delta U_i$ in $TL$ . Define iterated cups $\Cup _{i,m,n}\;:\;n\to n+2m$ and caps $\Cap _{i,m,n}\;:\;n+2m\to n$ by

\begin{equation*} \Cup _{i,1,n}= \cup _{i,n} \;\; \text{ and } \;\; \Cup _{i,m,n}=\Cup _{i+1,m-1,n+2}\circ \cup _{i,n} \end{equation*}

and

\begin{equation*} \Cap _{i,1,n}= \cup _{i,n} \;\; \text{ and } \;\; \Cap _{i,m,n}=\cap _{i,n}\circ \Cap _{i+1,m-1,n+2}\end{equation*}

both for $0\leq i\leq n$ and $m\geq 1$ . These morphisms in $\mathcal{D}$ all determine morphisms in $TL$ , which we denote by the same symbols, via the canonical functors $\mathcal{D}\hookrightarrow \mathbb{C}\mathcal{D} \to TL$ .

Following [Reference Wenzl27], after [Reference Jones16], we have Jones–Wenzl projections $f_i:i\to i$ in $TL$ , for $0\leq i\leq k+1$ , defined recursively by $f_0=1_0$ , $f_1=1_1$ , and

\begin{equation*}f_i=f_{i-1}\otimes 1+\frac {[i-1]}{[i]} (f_{i-1}\otimes 1)\circ U_{i-1}\circ (f_{i-1}\otimes 1).\end{equation*}

We record some useful facts about the Jones–Wenzl projections. Pictures follow after the proof.

We illustrate the previous proposition graphically, with maps drawn as green boxes. Our pictures should be read going upwards, and we sometimes represent $j$ strands as a single strand from the object $j$ .

Again, we can illustrate these graphically:

As the Jones–Wenzl projections are idempotents, they induce objects $F_i=(i,f_i)$ in $TL$ . Note that $F_0=\unicode{x1D7D9}$ is the unit object.

Let $(F_{k+1})$ denote the ideal of morphisms generated by $F_{k+1}$ .

The following result is standard: see [Reference Chen5, page 47 and Theorem 5.4.5].

Theorem 3.4. $\widetilde {TL}$ is semisimple with non-isomorphic simple objects $F_0$ , $F_1$ , $\ldots$ , $F_k$ , and

\begin{equation*} F_1\otimes F_i\cong F_{i-1}\oplus F_{i+1} \end{equation*}

where $F_\ell =0$ for $\ell \gt k$ .

Recall the definition of maps in the idempotent completion from the start of Section 3.1.

Definition 3.6. Let $i,j\geq 0$ with $i+j\leq n$ . We define maps

\begin{equation*} \iota _{i,j}\;:\;F_{i+j}\to F_i\otimes F_j \;\; \text{ and } \;\; \pi _{i,j}\;:\;F_i\otimes F_j \to F_{i+j}\end{equation*}

using $f_{i+j}$ as follows: $\iota _{i,j}=(f_i\otimes f_j)f_{i+j}f_{i+j}$ and $\pi _{i,j}=f_{i+j}f_{i+j}(f_i\otimes f_j)$ .

We draw these morphisms as follows, with Jones–Wenzl projections drawn as blue boxes:

Note that $\pi _{i,j}\circ \iota _{i,j}=1_{F_{i+j}}$ by the absorption rules. But $\iota _{i,j} \circ \pi _{i,j}\neq 1_{F_i\otimes F_j}$ : for example, when $i=j=1$ , we have $F_1\otimes F_1\cong F_2\oplus F_0$ and $\iota _{1,1} \circ \pi _{1,1}$ projects to the $F_2$ summand.

Lemma 3.7. For maps $g:i\to i$ and $h\;:\;j\to j$ , we have the following equalities of maps in $TL$ :

\begin{equation*} (g\otimes h)\circ \iota _{i,j}=g\otimes h\;:\;F_{i+j}\to F_i\otimes F_j \;\; \text{ and }\;\; \pi _{i,j}\circ (g\otimes h)=g\otimes h\;:\;F_i\otimes F_j\to F_{i+j}.\end{equation*}

Proof. From Definition 3.6, we have

\begin{equation*}(g\otimes h)\circ \iota _{i,j} =(f_i\otimes f_j)(g\otimes h)(f_i\otimes f_j)f_{i+j} =(f_i\otimes f_j)(g\otimes h)f_{i+j} =g\otimes h\;:\;F_{i+j}\to F_i\otimes F_j \end{equation*}

by Proposition 3.2; the equation for $\pi$ is similar.

By Lemma 3.7 with $g$ and $h$ being identity maps, we can represent $\iota$ and $\pi$ as follows:

The following result is well known: see, for example, [Reference Cooper6, Section 5.4.3]. For completeness, we include a proof in our context.

Proof. We can check everything on graded components. First, we check left and right units. As the empty diagram $\unicode{x1D7D9}$ is a strict identity, the left and right unit isomorphisms are just identity maps. But the multiplication maps $f_j\;:\;F_0F_j\to F_j$ and $f_i\;:\;F_iF_0\to F_i$ are identity maps in our idempotent completion. To check associativity, we need commutativity of the following diagram

and this follows from the absorption rules (Proposition 3.2).

3.2. Pivotal and braided structures

As a consequence of Lemma 3.11, we get:

In particular, $\widetilde {TL}$ is a rigid monoidal category. On morphisms, the dual is constructed using $e_i$ and $c_i$ which has the effect of rotating diagrams by $180^\circ$ .

As $\widetilde {TL}$ is a semisimple $\mathbb{C}$ -category (Theorem3.4), it is abelian. We have just seen that $\widetilde {TL}$ is rigid, and by construction the monoidal functor on $\widetilde {TL}$ is bilinear on morphisms. Finally, $\operatorname {End}\nolimits _{\widetilde {TL}}(\unicode{x1D7D9})=\mathbb{C}$ , so $\widetilde {TL}$ is a semisimple tensor category, that is, a fusion category, according to [Reference Etingof, Gelaki, Nikshych and Ostrik8, Definition 4.1.1].

As $\widetilde {TL}$ is semisimple, it follows from Lemma 3.12 that every object is self-dual, and so we get:

The following result will be useful in what follows. A particular case of this result, when $k$ is a multiple of 4, appears in [Reference Morrison, Peters and Snyder19, Lemma2.4].

Proof. Since $F_k \otimes F_k \cong F_0$ , which is simple, we have $\dim _{\mathbb{C}}\operatorname {Hom}\nolimits (F_k \otimes F_k,F_k \otimes F_k)=1$ , thus

\begin{equation*}c_k\circ e_k=\lambda 1_{F_k}\otimes 1_{F_k}\end{equation*}

for some scalar $\lambda$ . Now, we cup and cap the last $k$ strings as follows:

\begin{equation*}(1_{F_k}\otimes e_k )((c_k\circ e_k)\otimes 1_{F_k})(1_{F_k}\otimes c_k) = \lambda (1_{F_k}\otimes e_k )(1_{F_k}\otimes c_k) \end{equation*}

The left-hand side is the identity on $F_k$ , by the duality equations, and by Proposition 3.1 the right-hand side is $(\!-\!1)^k[k+1]1_{F_k}\otimes 1_{F_k}$ . So as $[k+1]=1$ , we have $\lambda =(\!-\!1)^k$ .

We can illustrate Lemmas 3.14 and 3.15 as follows:

We will also need the following structure. Recall from Section 3.1 that we have a fixed square root $t$ of $q$ .

We braid other objects using the axioms for a braiding: for example, $\sigma _{1,2}=(1_1\otimes \sigma _{1,1})\circ (\sigma _{1,1}\otimes 1_1)$ .

We draw $\sigma$ going up the page, with the overcrossing going left to right:

We will make use of the following result: see [Reference Freyd and Yetter10, Corollary 4.6].

Lemma 3.18. We have $f_{a+b}\circ \sigma _{a,b}=t^{ab}f_{a+b}$ and $f_{a+b}\circ \sigma ^{-1}_{a,b}=t^{-ab}f_{a+b}$ as maps $F_a\otimes F_b\to F_{a+b}$ , and $\sigma _{a,b}\circ f_{a+b}=t^{ab}f_{a+b}$ and $\sigma ^{-1}_{a,b}\circ f_{a+b}=t^{-ab}f_{a+b}$ as maps $F_{a+b}\to F_a\otimes F_b$ .

In a braided pivotal category, we have left and right twists $\theta ^\ell _X,\theta ^r_X\;:\;X\to X$ for each object $X$ , defined by:

\begin{equation*} \theta ^\ell _X=(e_X\otimes 1_X)\circ (\iota _{^\vee X}\otimes \sigma _{X,X})\circ (c^r_{X}\otimes 1_X) \;\;\text{ and } \;\; \theta ^r_X=(1_X\otimes e^r_{X})\circ (\sigma _{X,X}\otimes \iota _{^\vee X}^{-1})\circ (1_X\otimes c_X).\end{equation*}

There are explicit descriptions for their inverses and duals: see [Reference Turaev and Virelizier26, Section 3.3.1].

The twists for $\widetilde {TL}$ were studied in [Reference Turaev25, Section XII.6]. We calculate their effect on the Jones–Wenzl idempotents.

Graphically, we have:

Proof. We prove the statement for $(\theta ^\ell _{F_j})^{-1}$ ; the others are similar or follow from [Reference Turaev and Virelizier26, Lemma 3.2]. We use induction on $j$ . The base case follows directly from the definition of the braiding in Proposition 3.16:

\begin{equation*} \big(\theta ^\ell _{F_1}\big)^{-1}=(t^{-1}\delta +t)1_{F_1}=-t^-t^{-3}+t=-t^{-3}.\end{equation*}

Then the inductive step goes as follows:

We remove the twists using Lemma 3.18 to get an additional factor of $t^{-2j}$ , giving overall scalar $(\!-\!1)^{j+1}t^{-j^2-4j-3}=(\!-\!1)^{j+1}t^{-(j+1)^2-2(j+1)}$ .

Turaev showed [Reference Turaev25, Theorem XII.6.6] that $\widetilde {TL}$ is a ribbon category, following the definition in [Reference Turaev25, Section I.1.4]. Note that Lemma 3.19 implies $\theta ^\ell =\theta ^r$ , so we see directly that $\widetilde {TL}$ satisfies the definition in [Reference Turaev and Virelizier26, Section 3.3.2].

3.3. Frobenius structure

We have $k$ fixed for $\widetilde {TL}=\widetilde {TL}_k$ . Throughout this section, we will work with natural numbers $i$ and $j$ such that $i+j=k$ .

Proof. We need to show that the map

\begin{equation*} \varphi \;:\;\Sigma \stackrel {1_\Sigma \otimes c_\Sigma }{\longrightarrow } \Sigma \otimes \Sigma \otimes \Sigma ^\vee \stackrel {m\otimes 1_{\Sigma ^\vee }}{\longrightarrow }\Sigma \otimes \Sigma ^\vee \stackrel {n\otimes 1_{\Sigma ^\vee }}{\longrightarrow } F_k\otimes \Sigma ^\vee \end{equation*}

is invertible. We split the map into components

\begin{equation*} \varphi _{i,j}\;:\;F_i \stackrel {1_{F_i}\otimes c_{j}}{\longrightarrow } F_i\otimes F_j\otimes F_j\stackrel {m\otimes 1_{F_j}}{\longrightarrow }F_{i+j}\otimes F_j\stackrel {n\otimes 1_{F_j}}{\longrightarrow } F_k\otimes F_j \end{equation*}

and notice that the final map $n\otimes 1_{F_j}$ is non-zero iff $i+j=k$ , in which case $n=1_{F_k}\;:\;F_k\to F_k$ . Then, $m=\pi _{i,k-i}$ . So, the non-zero components are

\begin{equation*} \varphi _i:=\varphi _{i,k-i}\;:\;F_i \stackrel {1_{F_i}\otimes c_{k-i}}{\longrightarrow } F_i\otimes F_{k-i}\otimes F_{k-i}\stackrel {\pi _{i,k-i}\otimes 1_{F_{k-i}}}{\longrightarrow }F_k\otimes F_{k-i}. \end{equation*}

We define a map in the opposite direction with components as follows:

\begin{equation*} \frac {(\!-\!1)^{k-i}}{[i+1]}\psi _i:= F_k\otimes F_{k-i}\stackrel {\iota _{i,k-1}\otimes 1_{F_{k-i}}}{\longrightarrow }F_i\otimes F_{k-i}\otimes F_{k-i}\stackrel {1_{F_i}\otimes e_{k-1}}{\longrightarrow }F_i. \end{equation*}

Graphically, we have

Then $\psi _i\circ \varphi _i=1_{F_i}$ by Proposition 3.1(v), using $[j]=[k+2-j]$ and $[1]=1$ .

To compute $\varphi _i\circ \psi _i$ , we use Lemmas 3.14 and 3.15 as follows:

Then using Proposition 3.1(v) again, we get $\varphi _i\circ \psi _i=1_{F_k}\otimes 1_{F_{k-1}}$ . So $\varphi$ and $\psi$ are inverse, and therefore $\varphi$ is invertible.

Recall that dualising a map in $TL$ rotates by $180^\circ$ . So we can construct the $i$ th graded part of the Nakayama morphism $\alpha$ of $\Sigma$ directly from Definition 2.9 as follows:

Proof. Following Definition 2.10, we should calculate the composition

\begin{equation*}A=\unicode{x1D7D9}\otimes A\to V^{\otimes 2}\otimes A\to A\otimes V^{\otimes 2}\to A\otimes \unicode{x1D7D9}=A.\end{equation*}

Let $j=k-i$ . Omitting the scalar $[i+1]^2$ , we get:

where we have used the absorption property and the duality relations and labelled the curves by the number of strands. Now, using more duality relations, this is equal to the first of the following diagrams:

Now, we use the braiding and appeal to Theorem3.17 to show the first diagram above is equal to the second.

From here, we use Lemma 3.14 to obtain the first of the following diagrams and Theorem3.17 again to obtain the second:

Now, we use Lemma 3.18 twice to remove the cossings next to the $F_k$ boxes, which introduces a scalar $t^{2ij}$ . Then we use Proposition 3.1 to remove the $F_k$ boxes, which introduces a scalar $1/[i+1]^2$ , but this cancels with the scalar $[i+1]^2$ we omitted at the start of our calculation. So we have so far shown that $\alpha ^2$ is the following map:

Then by Lemma 3.19, we can remove the twists at a cost of $t^{2i^2+4i}$ , and we get $\alpha _i^2=\lambda _i1_{F_i}$ where

\begin{equation*} \lambda _i=t^{2ij}t^{2i^2+4i} = t^{2i(j+2+i)} = t^{i(2k+4)} = 1^i=1\end{equation*}

so we are done.

3.4. Preprojective algebras

Now let $\mathcal{C}=\widetilde {TL}$ and let $\mathcal{D}=S\!\operatorname {-mod-}\! S$ , where $S=\mathbb{C}\times \cdots \times \mathbb{C}$ ( $k$ copies) is a basic semisimple $\mathbb{C}$ -algebra. Write $e_i=(0,\ldots ,0,1,0,\ldots ,0)$ with $1$ in the $i$ th entry, so $S$ has $\mathbb{C}$ -basis $\{e_1,\ldots ,e_k\}$ . The category $\mathcal{D}$ is monoidal by tensoring over $S$ . We will consider $\mathbb{C}$ -linear monoidal functors $G\;:\;\mathcal{C}\to \mathcal{D}$ . Note that $\mathcal{D}\cong \operatorname {Fun}\nolimits _{\mathbb{C}}(\mathcal{M},\mathcal{M})$ , where $\mathcal{M}=S\operatorname {\!-mod}$ , so monoidal functors $G\;:\;\mathcal{C}\to \mathcal{D}$ correspond to left module categories $\mathcal{C}\times \mathcal{M}\to \mathcal{M}$ : see [Reference Etingof, Gelaki, Nikshych and Ostrik8, Proposition 7.1.3].

As $\widetilde {TL}$ is generated by $F_1$ , the image of a monoidal functor $G\;:\;\mathcal{C}\to \mathcal{D}$ is generated by an $S\text{-} S$ -bimodule $X=G(F_1)$ . The category $\mathcal{D}=S\!\operatorname {-mod-}\! S$ is semisimple and rigid: all simple objects are $1$ -dimensional $\mathbb{C}$ -vector spaces $S_{ij}$ where $e_iS_{ij}e_j\neq 0$ and $S_{ij}^\vee \cong S_{ji}$ . So, $X$ is determined up to isomorphism by a quiver $\overline Q$ with vertices $1,\ldots ,k$ and an arrow $i\to j$ for each summand $S_{ij}$ of $X$ . As $F_1$ is self-dual, we have $X^\vee \cong X$ , so $\overline Q$ has the same number of arrows $i\to j$ as $j\to i$ . Therefore, $X$ is in fact determined by an (unoriented) graph on the vertices $1,\ldots ,k$ .

The fusion rule $F_1\otimes F_i\cong F_{i-1}\oplus F_{i+1}$ (see Theorem3.4) puts further restrictions on the possible graphs that can appear, and it can be shown (see [Reference Ostrik23, Section 6], [Reference Cooper6, Section 3.3.2] and [Reference Goodman, de la Harpe and Jones13, Chapter I]) that the only possibilities are the simply laced Dynkin diagrams of types $A_k$ , $D_k$ , $E_k$ and the tadpole graph $T_k$ .

We now restrict to the $ADE$ cases, excluding the tadpole graphs. Let $Q$ be a quiver whose underlying graph $\Delta$ , where we replace each arrow with an unoriented edge, is an $ADE$ Dynkin diagram. For each arrow $a:i\to j$ , we add an arrow $a^*\;:\;j\to i$ to obtain a new quiver $\overline Q$ , as above. Following [Reference Dlab and Ringel7, Reference Gelfand and Ponomarev12], the preprojective algebra $\Pi$ of $\Delta$ is the quotient of the path algebra $\mathbb{C} \overline Q$ by the relations $\sum _{a\in Q_1}aa^*-a^*a$ ; this is independent of the orientation of $Q$ . The algebra $\Pi$ can also be constructed directly from the Auslander–Reiten theory of $\mathbb{C} Q$ , as in [Reference Baer, Geigle and Lenzing2].

Recall the algebra object $\Sigma$ in $\widetilde {TL}$ from Definition 3.8. The following result seems to have been known in folklore; a careful statement and proof can be found in [Reference Cooper6, Proposition 5.6.13].

As a consequence of Theorem3.23 together with Lemmas 2.12 and 3.21 and Theorem3.22, we get the following: