2 The quantum isomeric supercategory

Throughout the paper, we work over a commutative ring $\Bbbk $ , whose characteristic is not equal to two, and we fix an element $z \in \Bbbk $ . Statements about abstract categories will typically be at this level of generality. When making statements involving supermodules over the quantum isomeric superalgebra, we will specialise to $\Bbbk = \mathbb {C}(q)$ and $z=q-q^{-1}$ . We let $\mathbb {N}$ denote the set of nonnegative integers.

All vector spaces, algebras, categories and functors will be assumed to be linear over $\Bbbk $ unless otherwise specified. Almost everything in the paper will be enriched over the category of vector superspaces with parity-preserving morphisms. We write $\bar {v}$ for the parity of a homogeneous vector v in a vector superspace. When we write formulae involving parities, we assume the elements in question are homogeneous; we then extend by linearity.

For associative superalgebras A and B, multiplication in the superalgebra $A \otimes B$ is defined by

(2.1) $$ \begin{align} (a' \otimes b) (a \otimes b') = (-1)^{\bar a \bar b} a'a \otimes bb' \end{align} $$

for homogeneous $a,a' \in A$ , $b,b' \in B$ . For A-supermodules M and N, we let $\operatorname {\mathrm {Hom}}_A(M,N)$ denote the $\Bbbk $ -supermodule of all (i.e. not necessarily parity-preserving) A-linear maps from M to N. The opposite superalgebra $A^{\mathrm {op}}$ is a copy $\{a^{\mathrm {op}} : a \in A\}$ of the vector superspace A with multiplication defined from

(2.2) $$ \begin{align} a^{\mathrm{op}} b^{\mathrm{op}} := (-1)^{\bar a \bar b} (ba)^{\mathrm{op}}. \end{align} $$

A superalgebra homomorphism $A \to B^{\mathrm {op}}$ is equivalent to an antihomomorphism of superalgebras $A \to B$ . When viewing it in this way, we will often omit the superscript ‘ $\mathrm {op}$ ’ on elements of B.

Throughout this paper, we will work with strict monoidal supercategories, in the sense of [Reference Brundan and EllisBE17]. We summarise here a few crucial properties that play an important role in the present paper. A supercategory means a category enriched in . Thus, its morphism spaces are vector superspaces and composition is parity-preserving. A superfunctor between supercategories induces a parity-preserving linear map between morphism superspaces. For superfunctors $F,G \colon \mathcal {A} \to \mathcal {B}$ , a supernatural transformation $\alpha \colon F \Rightarrow G$ of parity $r\in \mathbb {Z}/2$ is the data of morphisms $\alpha _X\in \operatorname {\mathrm {Hom}}_{\mathcal {B}}(FX, GX)$ of parity r, for each $X \in \mathcal {A}$ , such that $Gf \circ \alpha _X = (-1)^{r \bar f}\alpha _Y\circ Ff$ for each homogeneous $f \in \operatorname {\mathrm {Hom}}_{\mathcal {A}}(X, Y)$ . Note when r is odd that $\alpha $ is not a natural transformation in the usual sense due to the sign. A supernatural transformation $\alpha \colon F \Rightarrow G$ is of the form $\alpha = \alpha _0 + \alpha _1$ , with each $\alpha _r$ being a supernatural transformation of parity r.

In a strict monoidal supercategory, morphisms satisfy the super interchange law:

(2.3) $$ \begin{align} (f' \otimes g) \circ (f \otimes g') = (-1)^{\bar f \bar g} (f' \circ f) \otimes (g \circ g'). \end{align} $$

We denote the unit object by and the identity morphism of an object X by $1_X$ . We will use the usual calculus of string diagrams, representing the horizontal composition $f \otimes g$ (respectively, vertical composition $f \circ g$ ) of morphisms f and g diagrammatically by drawing f to the left of g (respectively, drawing f above g). Care is needed with horizontal levels in such diagrams due to the signs arising from the super interchange law:

(2.4)

If $\mathcal {A}$ is a supercategory, the category of superfunctors $\mathcal {A} \to \mathcal {A}$ and supernatural transformations is a strict monoidal supercategory. The notation $\mathcal {A}^{\mathrm {op}}$ denotes the opposite supercategory and, if $\mathcal {A}$ is also monoidal, $\mathcal {A}^{\mathrm {rev}}$ denotes the reverse monoidal supercategory (changing the order of the tensor product); these are defined as for categories but with appropriate signs.

Definition 2.1. We define the quantum isomeric supercategory to be the strict monoidal supercategory generated by objects $\uparrow $ and $\downarrow $ and morphisms

(2.5)

(2.6)

subject to the relations

(2.7)

(2.8)

(2.9)

(2.10)

In the above, we have used left crossings and a right cap defined by

(2.11)

The parity of is odd, and all the other generating morphisms are even. We refer to as a Clifford token (later, we will refer to this as a closed Clifford token) (see Definition 6.1).

In addition to the left crossing and right cap defined in (2.11), we define

(2.12)

It follows that we have left and down analogues of the skein relation (2.8):

(2.13)

We then define the other right crossing so that the right skein relation also holds:

(2.14)

We call , , and positive crossings, and we call , , and negative crossings.

Remark 2.2. Given $z,t \in \Bbbk ^\times $ , the HOMFLYPT skein category

is the quotient of the category of framed oriented tangles by the Conway skein relation (2.8) and the relations

This category was first introduced in [Reference TuraevTur89, Section 5.2], where it was called the Hecke category (not to be confused with the more modern use of this term, which is related to the category of Soergel bimodules). We borrow the notation

, which comes oriented skein, from [Reference BrundanBru17]. It follows from [Reference BrundanBru17, Theorem 1.1], which gives a presentation of

, that all of the relations in

hold in

. More precisely, reflecting diagrams in the vertical axis and flipping crossings (i.e. interchanging positive and negative crossings), we see that (2.7), (2.8), (2.10) and the last equality in (2.9) correspond to the relations given in [Reference BrundanBru17, Theorem 1.1] with $t=1$ . Thus, by that result, all relations in

hold in

after reflecting in the vertical axis and flipping crossings. But

is invariant under this transformation, and so all its relations hold in

. In fact,

is the strict monoidal supercategory obtained from

by adjoining the Clifford token, subject to the relations (2.9) involving the Clifford token. Note that the condition $t=1$ is essentially forced by the skein relation and the last relation in (2.9), since

Hence, $t = \pm 1$ . If $t=-1$ , we can rescale the crossings by $-1$ and replace z by $-z$ to reduce to the case $t=1$ . This explains why the category

depends on only one parameter $z \in \Bbbk $ .

Lemma 2.3. The following relations hold in for all orientations of the strands:

(2.15)

(2.16)

(2.17)

We define

(2.18)

It follows that

(2.19)

Lemma 2.4. The following relations hold in for all orientations of the strands:

(2.20)

Proof. Composing the second relation in (2.9) on the top and bottom with , we see that the first two relations in (2.20) hold when both strands are oriented up. Attaching a left cup to the bottom of (2.18) and using (2.10), we see that the third relation in (2.20) holds for the strand oriented to the left. Similarly, attaching a left cap to the top of (2.18), we see that the fourth relation in (2.20) also holds for the strand oriented to the left. Then, using the definitions (2.11) and (2.12) of the left and down crossings, we see that the first two relations in (2.20) hold for the strands oriented to the left or oriented down. Next, taking the second relation in (2.20) for the strands oriented to the left, and composing on the top and bottom with , we see that the first relation in (2.20) holds for the strands oriented to the right. Similarly, taking the first relation in (2.20) for the strands oriented to the left, and composing on the top and bottom with , we see that the second relation in (2.20) holds for the strands oriented to the right.

So we have now proved the first two relations in (2.20) for all orientations of the strands, and the third and fourth relations for the strands oriented to the left. Next we compute

So the last equality in (2.20) holds for both orientations of the strand. We also have

An analogous argument shows that the fourth relation in (2.20) holds for the strands oriented to the right.

It follows from (2.20) that Clifford tokens slide over all crossings. However, they do not slide under crossings. In fact, we have the following result.

Lemma 2.5. The following relations hold in :

(2.21)

Proof. We have

(2.22)

The proof of the second relation is analogous.

We now describe several symmetries of the category

. First note that we have an isomorphism of monoidal supercategories

that is the identity objects and, on morphisms, multiplies all crossings by $-1$ .

Proposition 2.6. There is a unique isomorphism of monoidal supercategories

determined on objects by $\uparrow \ \mapsto \ \downarrow $ , $\downarrow \ \mapsto \ \uparrow $ and sending

The superfunctor $\Omega _{\updownarrow }$ acts on the other crossings, cups, caps and Clifford tokens as follows:

Proposition 2.7. There is a unique isomorphism of monoidal supercategories

determined on objects by $\uparrow \ \mapsto \ \uparrow $ , $\downarrow \ \mapsto \ \downarrow $ and sending

The superfunctor $\Omega _\leftrightarrow $ acts on the other crossings, cups, caps and Clifford tokens as follows:

It follows from Propositions 2.6 and 2.7 that is strictly pivotal, with duality superfunctor

(2.23)

defined by rotating diagrams through and multiplying by $(-1)^{\binom {y}{2}}$ , where y is the number of Clifford tokens in the diagram. Intuitively, this means that morphisms are invariant under isotopy fixing the endpoints, multiplying by the appropriate sign when odd elements change height. Thus, for example, we have rightward, leftward and downward versions of the relations (2.21).

Proof. When $z=0$ , (2.8) implies that

It is then straightforward to verify that the relations of Definition 2.1, without the last relation in (2.9), become the relations in [Reference Brundan, Comes and KujawaBCK19, Definition 3.2] with the orientations of strands reversed. The last relation in (2.9) also holds in the oriented Brauer–Clifford supercategory by [Reference Brundan, Comes and KujawaBCK19, (3.16)].

Remark 2.10. The reason we need to reverse orientation in Lemma 2.9 is that [Reference Brundan, Comes and KujawaBCK19, Definition 3.2] includes the relation

which matches the sign in (2.19) but not in the first relation in (2.9). If $\sqrt {-1} \in \Bbbk $ , then we have an automorphism of

that reverses orientation of strands and multiplies Clifford tokens by $\sqrt {-1}$ . In this case, there is an isomorphism from

to the oriented Brauer–Clifford category that multiplies Clifford tokens by $\sqrt {-1}$ , with no need to reverse orientation.

Let $X = X_1 \otimes \dotsb \otimes X_r$ and $Y = Y_1 \otimes \dotsb \otimes Y_s$ be objects of for $X_i,Y_j \in \{\uparrow , \downarrow \}$ . An $(X,Y)$ -matching is a bijection between the sets

(2.24) $$ \begin{align} \{i : X_i =\, \uparrow\} \sqcup \{j : Y_j =\, \downarrow\} \quad \text{and} \quad \{i : X_i =\, \downarrow\} \sqcup \{j : Y_j =\, \uparrow\}. \end{align} $$

A positive reduced lift of an $(X,Y)$ -matching is a string diagram representing a morphism $X \to Y$ , such that

• • the endpoints of each string are points that correspond under the given matching;

• • there are no Clifford tokens on any string and no closed strings (i.e. strings with no endpoints);

• • there are no self-intersections of strings and no two strings cross each other more than once;

• • all crossings are positive.

It follows from (2.16) that any two positive reduced lifts of a given $(X,Y)$ -matching are equal as morphisms in .

For each $(X,Y)$ , fix a set $B(X,Y)$ consisting of a choice of positive reduced lift for each $(X,Y)$ -matching. Then let $B_\bullet (X,Y)$ denote the set of all morphisms that can be obtained from elements of $B(X,Y)$ by adding at most one (and possibly zero) Clifford token near the terminus of each string. We require that all Clifford tokens occurring on strands whose terminus is at the top of the diagram to be at the same height; similarly, we require that all Clifford tokens occurring on strands whose terminus is at the bottom of the diagram to be at the same height, and below those Clifford tokens on strands whose terminus is at the top of the diagram.

We will prove later, in Theorem 4.5, that the sets $B_\bullet (X,Y)$ are actually bases of the morphism spaces.

Definition 2.12 [Reference Benkart, Guay, Jung, Kang and WilcoxBGJ+16, Definition 3.4].

For $r,s \in \mathbb {Z}_{> 0}$ and $z \in \Bbbk $ , the quantum walled Brauer–Clifford superalgebra $\mathrm {BC}_{r,s}(z)$ is the associative superalgebra generated by

$$\begin{align*}\text{even elements } t_1,\dotsc,t_{r-1}, t_1^*,\dotsc,t_{s-1}^*, e \text{ and odd elements } \pi_1,\dotsc,\pi_r,\pi_1^*,\dotsc \pi_s^* \end{align*}$$

satisfying the following relations (for $i,j$ in the allowable range)

$$ \begin{align*} t_i^2 &= z t_i + 1, & (t_i^*)^2 &= z t_i^* + 1, \\ t_i t_{i+1} t_i &= t_{i+1} t_i t_{i+1}, & t_i^* t_{i+1}^* t_i^* &= t_{i+1}^* t_i^* t_{i+1}^*, \\ t_i t_j &= t_j t_i \text{ for } |i-j|> 1, & t_i^* t_j^* &= t_j^* t_i^* \text{ for } |i-j| > 1, \\ \pi_i^2 = -1,&\ \pi_i \pi_j = - \pi_j \pi_i \text{ for } i \ne j, & (\pi_i^*)^2 = 1,&\ \pi_i^* \pi_j^* = - \pi_j^* \pi_i^* \text{ for } i \ne j, \\ t_i \pi_i &= \pi_{i+1} t_i, & t_i^* \pi_i^* &= \pi_{i+1}^* t_i^*, \\ t_i \pi_j &= \pi_j t_i \text{ for } j \ne i,i+1, & t_i^* \pi_j^* &= \pi_j^* t_i^* \text{ for } j \ne i,i+1, \\ e^2 = 0,\ et_{r-1}e = e,&\ et_j = t_j e \text{ for } j \ne r-1, & e t_1^* e = e,&\ e t_j^* = t_j^* e \text{ for } j \ne 1, \\ t_i \pi_j^* = \pi_j^* t_i,&\ \pi_r e = \pi_1^* e,& t_i^* \pi_j = \pi_j t_i^*,&\ e \pi_r = e \pi_1^*,\\ \pi_j e &= e\pi_j \text{ for } j \ne r,& \pi_j^* e &= e \pi_j^* \text{ for } j \ne 1, \\ e t_{r-1}^{-1} t_1^* e t_1^* t_{r-1}^{-1} &= t_{r-1}^{-1} t_1^* e t_1^* t_{r-1}^{-1} e,& e \pi_r e &= 0. \end{align*} $$

We define $\mathrm {BC}_{r,0}(z)$ to be the associative superalgebra generated by even elements $t_1,\dotsc ,t_{r-1}$ and odd elements $\pi _1,\dots ,\pi _r$ subject to the above relations involving only these elements. We define $\mathrm {BC}_{0,s}(z)$ similarly. Finally, we define $\mathrm {BC}_{0,0}(z) = \Bbbk $ .

The relations in the first line in Definition 2.12 imply that $t_i$ and $t_i^*$ are invertible, with $t_i^{-1} = t_i - z$ and $(t_i^*)^{-1} = t_i^* - z$ . Then, multiplying both sides of the relation $t_i \pi _i = \pi _{i+1} t_i$ on the left and right by $t_i^{-1}$ gives the relation

(2.25) $$ \begin{align} \pi_i t_i = t_i \pi_{i+1} + z(\pi_i-\pi_{i+1}). \end{align} $$

A straightforward computation shows that we have an isomorphism of superalgebras

(2.26) $$ \begin{align} \mathrm{BC}_{r,s}(z) \xrightarrow{\cong} \mathrm{BC}_{s,r}(z)^{\mathrm{op}},\qquad t_i \mapsto t^*_{r-i},\ t_i^* \mapsto t_{s-i},\ \pi_i \mapsto \pi_{r+1-i},\ \pi_i^* \mapsto \pi^*_{s+1-i},\ e \mapsto e. \end{align} $$

We will soon see a diagrammatic interpretation of this isomorphism.

The superalgebra

$$\begin{align*}\mathrm{HC}_r(z) := \mathrm{BC}_{r,0}(z) \end{align*}$$

is the Hecke–Clifford superalgebra, which first appeared in [Reference OlshanskiOls92, Definition 5.1]. It follows from (2.26) that we have an isomorphism of superalgebras $\mathrm {BC}_{0,s}(z) \cong \mathrm {HC}_s(z)^{\mathrm {op}}$ .

Proposition 2.13. For $r,s \in \mathbb {N}$ , we have a surjective homomorphism of associative superalgebras

given by

We will show in Corollary 4.8 that the homomorphism of Proposition 2.13 is actually an isomorphism.

3 The quantum isomeric superalgebra

In this section, we recall the definition of the quantum isomeric superalgebra and prove some results about it that will be used in the sequel (recall, as mentioned in the Introduction, that this superalgebra is traditionally called the quantum queer superalgebra). Throughout this section, we work over the field $\Bbbk = \mathbb {C}(q)$ and we set $z := q - q^{-1}$ . To simplify the expressions to follow, we first introduce some notation and conventions. Fix an index set

$$\begin{align*}\mathtt{I} := \{1,2,\dotsc,n,-1,-2,\dotsc,-n\}. \end{align*}$$

We will use $a,b,c,d$ to denote elements of $\{1,2,\dotsc ,n\}$ and $i,j,k,l$ to denote elements of $\mathtt {I}$ . For $i,j \in \mathtt {I}$ , we define

(3.1) $$ \begin{align} p(i) &:= \begin{cases} 0 & \text{if } i> 0, \\ 1 & \text{if } i < 0, \end{cases} & p(i,j) &:= p(i)+p(j), \end{align} $$

(3.2) $$ \begin{align} \operatorname{\mathrm{sgn}}(i) &:= (-1)^{p(i)} = 1-2p(i),& \varphi(i,j) &:= \delta_{|i|,|j|} \operatorname{\mathrm{sgn}}(j). \end{align} $$

If C is some condition, we define $\delta _C = 1$ if the condition is satisfied, and $\delta _C = 0$ otherwise. Then, for $i,j \in \mathtt {I}$ , $\delta _{ij} := \delta _{i=j}$ is the usual Kronecker delta.

Let V denote the $\Bbbk $ -supermodule with basis $v_i$ , $i \in \mathtt {I}$ , where the parity of $v_i$ is given by

$$\begin{align*}\overline{v_i} = p(i). \end{align*}$$

Using this basis, we will identify V with $\Bbbk ^{n|n}$ as $\Bbbk $ -supermodules and $\operatorname {\mathrm {End}}_\Bbbk (V)$ with $\operatorname {\mathrm {Mat}}_{n|n}(\Bbbk )$ as associative superalgebras. Let $E_{ij} \in \operatorname {\mathrm {Mat}}_{n|n}(\Bbbk )$ denote the matrix with a $1$ in the $(i,j)$ -position and a $0$ in all other positions. Then the parity of $E_{ij}$ is $p(i,j)$ . The general linear Lie superalgebra $\mathfrak {gl}_{n|n}$ is equal to $\operatorname {\mathrm {End}}_\Bbbk (V)$ as a $\Bbbk $ -supermodule, with bracket given by the supercommutator

$$\begin{align*}[X,Y] = XY - (-1)^{\bar{X} \bar{Y}} YX. \end{align*}$$

Let

$$\begin{align*}J := \sum_{i \in \mathtt{I}} (-1)^{p(i)} E_{-i,i} = \begin{pmatrix} 0 & -I_n \\ I_n & 0 \end{pmatrix} \in \operatorname{\mathrm{Mat}}_{n|n}(\Bbbk), \end{align*}$$

where $I_n$ is the $n \times n$ identity matrix. Multiplication by J is an odd linear automorphism of V, and $J^2 = -1$ . The isomeric Lie superalgebra $\mathfrak {q}_n$ is the Lie superalgebra equal to the centraliser of J in $\mathfrak {gl}_{n|n}$ :

$$\begin{align*}\mathfrak{q}_n := \left\{ X \in \mathfrak{gl}_{n|n} : JX = (-1)^{\bar{X}} X J \right\}. \end{align*}$$

The elements

$$\begin{align*}e^0_{ab} := E_{ab} + E_{-a,-b},\qquad e^1_{ab} := E_{a,-b} + E_{-a,b},\qquad a,b \in \{1,2,\dotsc,n\}, \end{align*}$$

give a $\Bbbk $ -basis of $\mathfrak {q}_n$ . The parities of these elements are indicated by their superscripts.

Define

(3.3) $$ \begin{align} \Theta = \sum_{i,j \in \mathtt{I}} \Theta_{ij} \otimes E_{ij} \in \operatorname{\mathrm{End}}_\Bbbk(V)^{\otimes 2} = \operatorname{\mathrm{Mat}}_{n|n}(\Bbbk)^{\otimes 2}, \end{align} $$

by

(3.4) $$ \begin{align} \begin{aligned} \Theta &= \sum_{a,j} q^{\delta_{a,j} - \delta_{a,-j}} e^0_{aa} \otimes E_{jj} + z \sum_{a<b} e^0_{ba} \otimes E_{ab} - z \sum_{a>b} e^0_{ba} \otimes E_{-a,-b} - z \sum_{a,b} e^1_{ba} \otimes E_{-a,b} \\ &= \sum_{i,j} q^{\varphi(i,j)} E_{ii} \otimes E_{jj} + z \sum_{i<j} (-1)^{p(i)} (E_{ji} + E_{-j,-i}) \otimes E_{ij}. \end{aligned} \end{align} $$

The definition of $\Theta $ first appeared in [Reference OlshanskiOls92, Section 4], where it is denoted S. We use the notation $\Theta $ to reserve the notation S for the antipode, which will play an important role in the current paper. It follows immediately from the definition that

(3.5) $$ \begin{align} \Theta(J \otimes 1) = (J \otimes 1)\Theta. \end{align} $$

One can also verify that $\Theta $ satisfies the Yang–Baxter equation:

(3.6) $$ \begin{align} \Theta^{12} \Theta^{13} \Theta^{23} = \Theta^{23} \Theta^{13} \Theta^{12}, \end{align} $$

where

$$\begin{align*}\Theta^{12} = \Theta \otimes 1,\qquad \Theta^{23} = 1 \otimes \Theta,\qquad \Theta^{13} = \sum_{i,j \in \mathtt{I}} \Theta_{ij} \otimes 1 \otimes E_{ij}. \end{align*}$$

It follows from (3.4) that

(3.7) $$ \begin{align} \Theta_{ii} = \sum_a q^{\delta_{a,i} - \delta_{a,-i}} e^0_{aa} \qquad \text{for all } i \in \mathtt{I}, \end{align} $$

and so

(3.8) $$ \begin{align} \Theta_{ii} \Theta_{-i,-i} = 1 = \Theta_{-i,-i} \Theta_{i,i} \qquad \text{for all } i \in \mathtt{I}. \end{align} $$

When $q=1$ , we have $\Theta = 1 \otimes 1$ , and so $\Theta _{ij} = \delta _{ij} 1_V$ .

Note that all the second tensor factors appearing in (3.4) are upper triangular elements of $\operatorname {\mathrm {Mat}}_{n|n}(\Bbbk )$ . In addition, $\Theta $ is invertible with

(3.9) $$ \begin{align} \Theta^{-1} &= \sum_{a,j} q^{\delta_{a,-j} - \delta_{a,j}} e^0_{aa} \otimes E_{jj} - z \sum_{a<b} e^0_{ba} \otimes E_{ab} + z \sum_{a>b} e^0_{ba} \otimes E_{-a,-b} + z \sum_{a,b} e^1_{ba} \otimes E_{-a,b} \nonumber\\ &= \sum_{i,j} q^{-\varphi(i,j)} E_{ii} \otimes E_{jj} - z \sum_{i<j} (-1)^{p(i)} (E_{ji} + E_{-j,-i}) \otimes E_{ij}. \end{align} $$

Note that $\Theta ^{-1}$ is obtained from $\Theta $ by replacing q by $q^{-1}$ .

Definition 3.1. The quantum isomeric superalgebra $U_q = U_q(\mathfrak {q}_n)$ is the unital associative superalgebra over $\Bbbk $ generated by elements $u_{ij}$ , $i,j \in \mathtt {I}$ , $i \le j$ , subject to the relations

(3.10) $$ \begin{align} u_{ii} u_{-i,-i} = 1 = u_{-i,-i} u_{ii},\qquad L^{12} L^{13} \Theta^{23} = \Theta^{23} L^{13} L^{12}, \end{align} $$

where

(3.11) $$ \begin{align} L := \sum_{\substack{i,j \in \mathtt{I} \\ i \le j}} u_{ij} \otimes E_{ij},\qquad L^{12} = L \otimes 1,\qquad L^{13} = \sum_{\substack{i,j \in \mathtt{I} \\ i \le j}} u_{ij} \otimes 1 \otimes E_{ij}, \end{align} $$

and the last equality in (3.10) takes place in $U_q \otimes \operatorname {\mathrm {End}}_\Bbbk (V)^{\otimes 2}$ . The parity of $u_{ij}$ is $p(i,j)$ .

The quantum isomeric Lie superalgebra was first defined in [Reference OlshanskiOls92, Definition 4.2]. It is a Hopf superalgebra with comultiplication determined by

(3.12) $$ \begin{align} \Delta(L) :=& \sum_{\substack{i,j \in \mathtt{I} \\ i \le j}} \Delta(u_{ij}) \otimes E_{ij} = L^{13} L^{23}, \quad \text{or, more explicitly,} \end{align} $$

(3.13) $$ \begin{align} \Delta(u_{ij}) =& \sum_{\substack{k \in \mathtt{I} \\ i \le k \le j}} (-1)^{p(i,k)p(k,j)} u_{ik} \otimes u_{kj} = \sum_{\substack{k \in \mathtt{I} \\ i \le k \le j}} u_{ik} \otimes u_{kj} \end{align} $$

(where the final equality holds since, for $i \le k \le j$ , we must have $p(k)=p(i)$ or $p(k)=p(j)$ ), counit determined by

(3.14) $$ \begin{align} \varepsilon(L) := \sum_{\substack{i,j \in \mathtt{I} \\ i \le j}} \varepsilon(u_{ij}) E_{ij} = 1 \quad \text{or, more explicitly,} \quad \varepsilon(u_{ij}) = \delta_{ij}, \end{align} $$

and antipode S determined by

(3.15) $$ \begin{align} \sum_{\substack{i,j \in \mathtt{I} \\ i \le j}} S(u_{ij}) \otimes E_{ij} = L^{-1}. \end{align} $$

Note that, viewing L as an element of $\operatorname {\mathrm {Mat}}_{n|n}(U_q)$ , it follows from its definition and (3.10) that it is triangular with invertible diagonal entries. Thus, L is indeed invertible. Since $U_q$ is a Hopf superalgebra, the supercategory $U_q\text {-smod}$ of finite-dimensional $U_q$ -supermodules is naturally a rigid monoidal supercategory.

For $\Bbbk $ -supermodules U and W, define

$$\begin{align*}\operatorname{\mathrm{flip}}_{U,W} \colon U \otimes W \to W \otimes U,\quad \operatorname{\mathrm{flip}}_{U,W}(u \otimes w) = (-1)^{\bar{u} \bar{w}} w \otimes u. \end{align*}$$

When U and W are clear from the context, we will sometimes write $\operatorname {\mathrm {flip}}$ instead of $\operatorname {\mathrm {flip}}_{U,W}$ . Note that

$$\begin{align*}\operatorname{\mathrm{flip}}_{V,V} = \sum_{i,j} (-1)^{p(j)} E_{ij} \otimes E_{ji}. \end{align*}$$

Consider the opposite comultiplication

$$\begin{align*}\Delta^{\mathrm{op}} = \operatorname{\mathrm{flip}} \circ \Delta. \end{align*}$$

Lemma 3.2. We have

(3.16) $$ \begin{align} \Delta^{\mathrm{op}}(L) := \sum_{\substack{i,j \in \mathtt{I} \\ i \le j}} \Delta^{\mathrm{op}}(u_{ij}) \otimes E_{ij} = L^{23} L^{13}. \end{align} $$

Proof. We have

$$\begin{align*}\Delta^{\mathrm{op}}(u_{ij}) = \operatorname{\mathrm{flip}} \sum_{\substack{k \in \mathtt{I} \\ i \le k \le j}} (-1)^{p(i,k)p(k,j)} u_{ik} \otimes u_{kj} = \sum_{\substack{k \in \mathtt{I} \\ i \le k \le j}} u_{kj} \otimes u_{ik}. \end{align*}$$

Since

$$\begin{align*}L^{23} L^{13} = \left( \sum_{i \le k} 1 \otimes u_{ik} \otimes E_{ik} \right) \left( \sum_{l \le j} u_{lj} \otimes 1 \otimes E_{lj} \right) = \sum_{\substack{k \in \mathtt{I} \\ i \le k \le j}} u_{kj} \otimes u_{ik} \otimes E_{ij}, \end{align*}$$

the result follows.

The following result is stated in [Reference OlshanskiOls92, Theorem 6.1] without proof.

Proposition 3.3. The quantum isomeric superalgebra $U_q$ is isomorphic is the unital associative superalgebra over $\Bbbk $ generated by the elements $u_{ij}$ , $i,j \in \mathtt {I}$ , $i \le j$ , subject to the relations

(3.17) $$ \begin{align} u_{ii} u_{-i,-i} = 1 = u_{-i,-i} u_{ii}, \qquad i \in \mathtt{I}, \end{align} $$

and

(3.18) $$ \begin{align} &(-1)^{p(i,j)p(k,l)} q^{\varphi(j,l)} u_{ij} u_{kl} + z \delta_{i \le l} \delta_{k \le j < l} \theta(i,j,k) u_{il} u_{kj} + z \delta_{i \le -l < j \le -k} \theta(-i,-j,k) u_{i,-l} u_{k,-j} \nonumber\\ &\quad= q^{\varphi(i,k)} u_{kl} u_{ij} + z \delta_{k < i \le l} \delta_{k \le j} \theta(i,j,k) u_{il} u_{kj} + z \delta_{-l \le i < -k \le j} \theta(-i,-j,k) u_{-i,l} u_{-k,j}, \end{align} $$

for all $i,j,k,l \in \mathtt {I}$ , $i \le j$ , $k \le l$ , where $\theta (i,j,k) = (-1)^{p(i)p(j) + p(j)p(k) + p(i)p(k)}$ .

Proof. It suffices to prove that the relations (3.18) are equivalent to the second relation in (3.10). Direct computation shows that

$$ \begin{align*} &L^{12} L^{13} \Theta^{23} = \sum_{i \le j,\, k \le l} (-1)^{p(i,j)p(k,l)} q^{\varphi(j,l)} u_{ij} u_{kl} \otimes E_{ij} \otimes E_{kl} \\ &+ z \sum_{i \le l,\, k \le j < l} \theta(i,j,k) u_{il} u_{kj} \otimes E_{ij} \otimes E_{kl} + z \sum_{i \le -l < j \le -k} \theta(-i,-j,k) u_{i,-l} u_{k,-j} \otimes E_{ij} \otimes E_{kl} \end{align*} $$

and

$$ \begin{align*} &\Theta^{23} L^{13} L^{12} = \sum_{i \le j,\, k \le l} q^{\varphi(i,k)} u_{kl} u_{ij} \otimes E_{ij} \otimes E_{kl} \\ &+ z \sum_{k < i \le l,\, k \le j} \theta(i,j,k) u_{il} u_{kj} \otimes E_{ij} \otimes E_{kl} + z \sum_{-l \le i < -k \le j} \theta(-i,-j,k) u_{-i,l} u_{-k,j} \otimes E_{ij} \otimes E_{kl}. \end{align*} $$

The result follows.

Lemma 3.5. As a unital associative superalgebra, $U_q$ is generated by

(3.19) $$ \begin{align} u_{a,a+1},\quad u_{-a-1,-a},\quad u_{ii},\quad u_{-1,1},\qquad 1 \le a \le n-1,\quad i \in \mathtt{I}. \end{align} $$

Proof. Let $\tilde {U}_q$ be the unital associative subsuperalgebra of $U_q$ generated by the elements (3.19). It is shown in [Reference Grantcharov, Jung, Kang and KimGJKK10, Theorem 2.1] that $U_q$ is generated by

$$\begin{align*}u_{a,a+1},\quad u_{-a-1,-a},\quad u_{ii},\quad u_{-a-1,a},\quad u_{-a,a+1},\quad u_{-b,b},\qquad\! 1 \le a \le n-1,\ 1 \le b \le n,\ i \in \mathtt{I}. \end{align*}$$

Thus, it suffices to show that

(3.20) $$ \begin{align} u_{-a-1,a},\ u_{-a,a+1},\ u_{-b,b} \in \tilde{U}_q \end{align} $$

for $1 \le a \le n-1$ and $1 \le b \le n$ . We prove this by induction on a.

First note that, for $1 \le a \le n-1$ , taking $i=-a$ , $j=a$ , $k=-a-1$ , $l=-a$ in (3.18) gives

$$\begin{align*}q^{-1} u_{-a,a} u_{-a-1,-a} = u_{-a-1,-a} u_{-a,a} - z u_{-a,-a} u_{-a-1,a}. \end{align*}$$

Taking $i=-a$ , $j=k=a$ and $l=a+1$ in (3.18) gives

$$\begin{align*}u_{-a,a} u_{a,a+1} + z u_{-a,a+1} u_{aa} = q u_{a,a+1} u_{-a,a}. \end{align*}$$

Taking $i=-a-1$ , $j=k=-a$ , $l=a+1$ in (3.18) gives

$$\begin{align*}u_{-a-1,-a} u_{-a,a+1} - z u_{-a-1,a+1} u_{-a,-a} + z u_{-a-1,-a-1} u_{-a, a} = u_{-a,a+1} u_{-a-1,-a}. \end{align*}$$

So we have

(3.21) $$ \begin{align} u_{-a-1,a} &= z^{-1} u_{aa} u_{-a-1,-a} u_{-a,a} - q^{-1}z^{-1} u_{aa} u_{-a,a} u_{-a-1,-a}, \end{align} $$

(3.22) $$ \begin{align} u_{-a,a+1} &= qz^{-1} u_{a,a+1} u_{-a,a} u_{-a,-a} - z^{-1} u_{-a,a} u_{a,a+1} u_{-a,-a}, \end{align} $$

(3.23) $$ \begin{align} u_{-a-1,a+1} &= z^{-1} u_{-a-1,-a} u_{-a,a+1} u_{aa} + u_{-a-1,-a-1} u_{-a,a} u_{aa} - z^{-1} u_{-a,a+1} u_{-a-1,-a} u_{aa}. \end{align} $$

Taking $a=1$ in (3.21) and (3.22) shows that $u_{-2,1}, u_{-1,2} \in \tilde {U}_q$ . Thus, (3.20) holds for $a=b=1$ . Now suppose that $1 \le c \le n-2$ , and that (3.20) holds for $1 \le a, b \le c$ . Then, replacing a by c in (3.23) shows that $u_{-c-1,c+1} \in \tilde {U}_q$ . Replacing a by $c+1$ in (3.21) and (3.22) then shows that $u_{-c-2,c+1}, u_{-c-1,c+2} \in \tilde {U}_q$ . Hence, (3.20) holds for $1 \le a,b \le c+1$ . Thus, by induction, (3.20) holds for $1 \le a,b \le n-1$ . Finally, taking $a=n-1$ in (3.23) shows that $u_{-n,n} \in U_q$ .

It will be useful for future arguments to compute the square of the antipode.

Proof. It follows from the defining relations that $U_q$ is a $\mathbb {Z}$ -graded Hopf superalgebra, where we define the degree of $u_{ij}$ to be $2|j|-2|i|$ . Thus, the map $u_{ij} \mapsto q^{2|j|-2|i|} u_{ij}$ is a homomorphism of superalgebras. Since the antipode is an antihomomorphism of superalgebras, its square is a homomorphism of superalgebras. Thus, by Lemma 3.5, it suffices to prove that

$$\begin{align*}S^2(u_{a,a+1}) = q^2 u_{a,a+1},\quad\kern-3pt S^2(u_{-a-1,-a}) = q^{-2} u_{-a-1,-a},\quad\kern-3pt S^2(u_{ii}) = u_{ii},\quad\kern-3pt S^2(u_{-1,1}) = u_{-1,1}, \end{align*}$$

for $1 \le a \le n$ , $i \in \mathtt {I}$ .

Using the definition (3.15) of the antipode, which involves inverting an upper triangular matrix, we see that

$$ \begin{gather*} S(u_{a,a+1}) = - u_{-a,-a} u_{a,a+1} u_{-a-1,-a-1},\qquad S(u_{-a-1,-a}) = - u_{a+1,a+1} u_{-a-1,-a} u_{aa}, \\ S(u_{ii}) = u_{ii}^{-1} = u_{-i,-i},\qquad S(u_{-1,1}) = - u_{11} u_{-1,1} u_{-1,-1}. \end{gather*} $$

By Corollary 3.4(a) and (3.17), we have

(3.24) $$ \begin{align} u_{aa} u_{kl} = q^{\delta_{a,|l|} - \delta_{a,|k|}} u_{kl}u_{aa} \qquad \text{and} \qquad u_{-a,-a} u_{kl} = q^{\delta_{a,|k|} - \delta_{a,|l|}} u_{kl} u_{-a,-a}, \end{align} $$

for all $a \in \{1,2,\dotsc ,n\}$ and $k,l \in \mathtt {I}$ , $k \le l$ . In particular,

$$\begin{align*}u_{-a,-a} u_{a,a+1} = q u_{a,a+1} u_{-a,-a},\quad u_{a+1,a+1} u_{a,a+1} = q u_{a,a+1} u_{a+1,a+1},\quad u_{ii} u_{kk} = u_{kk} u_{ii}, \end{align*}$$

for all $a \in \{1,2,\dotsc ,n\}$ and $i,k \in \mathtt {I}$ . Thus,

$$\begin{align*}S^2(u_{a,a+1}) &= - S(u_{-a-1,-a-1}) S(u_{a,a+1}) S(u_{-a,-a}) \\&\quad= u_{a+1,a+1} u_{-a,-a} u_{a,a+1} u_{-a-1,-a-1} u_{aa} = q^2 u_{a,a+1}. \end{align*}$$

The proof that $S^2(u_{-a-1,-a}) = q^{-2} u_{-a-1,-a}$ is similar.

Next, we have

$$\begin{align*}S^2(u_{ii}) = S(u_{-i,-i}) = u_{ii}. \end{align*}$$

Finally, (3.24) implies that

$$\begin{align*}u_{11} u_{-1,1} = u_{-1,1} u_{11} \qquad \text{and} \qquad u_{-1,-1} u_{-1,1} = u_{-1,1} u_{-1,-1}. \end{align*}$$

Thus

$$\begin{align*}S^2(u_{-1,1}) = -S \left( u_{11} u_{-1,1} u_{-1,-1} \right) = u_{11}^2 u_{-1,1} u_{-1,-1}^2 = u_{-1,1}.\\[-35pt] \end{align*}$$

Corollary 3.7. The antipode S is invertible and

(3.25) $$ \begin{align} S^{-1}(u_{ij}) = q^{2|i|-2|j|} S(u_{ij}) \qquad i,j \in \mathtt{I}. \end{align} $$

It follows from (3.6) and (3.8) that

(3.26) $$ \begin{align} \rho \colon U_q \to \operatorname{\mathrm{End}}_\Bbbk(V),\qquad u_{ij} \mapsto \Theta_{ij},\qquad i,j \in \mathtt{I}, \end{align} $$

defines a representation of $U_q$ on V. The $U_q$ -supermodule structure on the dual space $V^* := \operatorname {\mathrm {Hom}}_\Bbbk (V,\Bbbk )$ is given by

$$\begin{align*}(xf)(v) = (-1)^{\bar{x}\bar{f}} f(S(x)v),\qquad x \in U_q,\ f \in V^*,\ v \in V. \end{align*}$$

We have the natural evaluation map

(3.27) $$ \begin{align} \operatorname{\mathrm{ev}} \colon V^* \otimes V \to \Bbbk,\qquad f \otimes v \mapsto f(v). \end{align} $$

Let $v_i^*$ , $i \in \mathtt {I}$ , be the basis of $V^*$ dual to the basis $v_i$ , $i \in \mathtt {I}$ , of V, so that

$$\begin{align*}v_i^*(v_j) = \delta_{ij},\qquad i,j \in \mathtt{I}. \end{align*}$$

Then we have the coevaluation map

(3.28) $$ \begin{align} \operatorname{\mathrm{coev}} \colon \Bbbk \to V \otimes V^*,\qquad 1 \mapsto \sum_{i \in \mathtt{I}} v_i \otimes v_i^*. \end{align} $$

It is a straightforward exercise, using only the properties of Hopf superalgebras, to verify that $\operatorname {\mathrm {ev}}$ and $\operatorname {\mathrm {coev}}$ are both homomorphisms of $U_q$ -supermodules, where $\Bbbk $ is the trivial $U_q$ -supermodule, with action given by the counit $\varepsilon $ .

Proof. It follows from (3.3) and (3.5) that

$$\begin{align*}\Theta_{ij} J = (-1)^{p(i,j)} J \Theta_{ij} \qquad \text{for all } i,j \in \mathtt{I}. \end{align*}$$

Since $u_{ij}$ acts on V as $\Theta _{ij}$ , it follows that J is an odd endomorphism of $U_q$ -supermodules. Since $J^2 = -1$ , it is an isomorphism.

4 The incarnation superfunctor

In this section, we prove some of our main results. We describe a full monoidal superfunctor from to the category of $U_q$ -supermodules, give explicit bases for the morphism spaces in and identify the endomorphism superalgebras of with walled Brauer–Clifford superalgebras.

Until further notice later in this section, we assume that $\Bbbk = \mathbb {C}(q)$ and $z = q-q^{-1}$ . Recalling the definition (3.4) of $\Theta $ , define

(4.1) $$ \begin{align} T := \operatorname{\mathrm{flip}} \circ \Theta, \qquad \text{so that} \qquad T^{-1} = \Theta^{-1} \circ \operatorname{\mathrm{flip}}. \end{align} $$

Thus

$$ \begin{align*} T &= \sum_{i,j} (-1)^{p(i)} q^{\varphi(i,j)} E_{ji} \otimes E_{ij} + z \sum_{i<j} E_{ii} \otimes E_{jj} - z \sum_{i<j} (-1)^{p(i,j)} E_{i,-i} \otimes E_{-j,j}, \\ T^{-1} &= \sum_{i,j} (-1)^{p(i)} q^{-\varphi(j,i)} E_{ji} \otimes E_{ij} - z \sum_{i>j} E_{ii} \otimes E_{jj} - z \sum_{i<j} (-1)^{p(i,j)} E_{i,-i} \otimes E_{-j,j}. \end{align*} $$

Therefore, we have

$$ \begin{align*} T(v_i \otimes v_j) &= (-1)^{p(i)p(j)} q^{\varphi(i,j)} v_j \otimes v_i + z \delta_{i<j} v_i \otimes v_j + z \delta_{i+j>0} (-1)^{p(j)} v_{-i} \otimes v_{-j}, \\ T^{-1}(v_i \otimes v_j) &= (-1)^{p(i)p(j)} q^{-\varphi(j,i)} v_j \otimes v_i - z \delta_{i > j} v_i \otimes v_j + z \delta_{i+j>0} (-1)^{p(j)} v_{-i} \otimes v_{-j}, \end{align*} $$

and

(4.2) $$ \begin{align} T - T^{-1} = z 1_{V \otimes V}. \end{align} $$

Proof. Since it is invertible, it remains to show that it is a homomorphism of $U_q$ -supermodules. To do this, it suffices to show that, as operators on $V \otimes V$ , we have an equality

$$\begin{align*}T \Delta(u_{ij}) = \Delta(u_{ij}) T \qquad \text{for all } i,j \in \mathtt{I}. \end{align*}$$

Composing on the left with $\operatorname {\mathrm {flip}}$ , it suffices to show that

$$\begin{align*}\Theta \Delta(u_{ij}) = \Delta^{\mathrm{op}}(u_{ij}) \Theta \qquad \text{for all } i,j \in \mathtt{I}. \end{align*}$$

This is equivalent to showing that

$$\begin{align*}\sum_{i,j \in \mathtt{I}} \Theta \Delta(u_{ij}) \otimes E_{ij} = \sum_{i,j \in \mathtt{I}} \Delta^{\mathrm{op}}(u_{ij}) \Theta \otimes E_{ij}. \end{align*}$$

Since $u_{ij}$ acts on V as $\Theta _{ij}$ , this is equivalent, using (3.12) and (3.16), to

$$\begin{align*}\Theta^{12} \Theta^{13} \Theta^{23} = \Theta^{23} \Theta^{13} \Theta^{12}. \end{align*}$$

But this is precisely the Yang–Baxter equation (3.6).

For the computations to follow, it is useful to note that, for $i,j \in \mathtt {I}$ , $i<j$ , we have

(4.3) $$ \begin{align} z \sum_{\substack{k \in \mathtt{I} \\ i < k < j}} (-1)^{p(k)} q^{2|k|} &= q^{2|j|-\operatorname{\mathrm{sgn}}(j)} - q^{2|i|+\operatorname{\mathrm{sgn}}(i)}, \end{align} $$

(4.4) $$ \begin{align} z \sum_{\substack{k \in \mathtt{I} \\ i < k < j}} (-1)^{p(k)} q^{-2|k|} &= q^{-2|i|-\operatorname{\mathrm{sgn}}(i)} - q^{-2|j|+\operatorname{\mathrm{sgn}}(j)}. \end{align} $$

Theorem 4.3. For each $n \in \mathbb {N}$ , there exists a unique monoidal superfunctor

, such that

Furthermore,

,

and

(4.5)

We call ${\mathbf {F}}_n$ the incarnation superfunctor. Before giving the proof of Theorem 4.3, we compute, using the definitions (2.11) to (2.13), the images under ${\mathbf {F}}$ of the leftward and downward crossings:

(4.6)

(4.7)

(4.8)

(4.9)

the right cup and cap

(4.10)

and the positive right crossing

(see Remark 5.6 for another description of the images under ${\mathbf {F}}_n$ of the various crossings).

Proof of Theorem 4.3

We first show existence, taking , and as in (4.5). We must show that ${\mathbf {F}}_n$ respects the relations in Definition 2.1.

The first two relations in (2.7) are clear. To verify the third relation in (2.7), we compute

and

and, for $i \ne \pm j$ ,

Thus

So ${\mathbf {F}}_n$ respects the third relation in (2.7). Since $V \otimes V^*$ is finite dimensional, it follows that we also have

Hence, ${\mathbf {F}}_n$ also respects the fourth relation in (2.7).

Next we verify the braid relation (the last relation in (2.7)). The left-hand side is mapped by ${\mathbf {F}}_n$ to the composite

$$\begin{align*}(T \otimes 1_V) (1_V \circ T) (T \otimes 1_V) = \operatorname{\mathrm{flip}}^{12} \Theta^{12} \operatorname{\mathrm{flip}}^{23} \Theta^{23} \operatorname{\mathrm{flip}}^{12} \Theta^{12} = \operatorname{\mathrm{flip}}^{12} \operatorname{\mathrm{flip}}^{23} \operatorname{\mathrm{flip}}^{12} \Theta^{23} \Theta^{13} \Theta^{12}. \end{align*}$$

Similarly, the right-hand side is mapped by ${\mathbf {F}}_n$ to the composite

$$\begin{align*}(1_V \otimes T) (T \otimes 1_V) (1_V \otimes T) = \operatorname{\mathrm{flip}}^{23} \Theta^{23} \operatorname{\mathrm{flip}}^{12} \Theta^{12} \operatorname{\mathrm{flip}}^{23} \Theta^{23} = \operatorname{\mathrm{flip}}^{23} \operatorname{\mathrm{flip}}^{12} \operatorname{\mathrm{flip}}^{23} \Theta^{12} \Theta^{13} \Theta^{23}. \end{align*}$$

Since

$$\begin{align*}\operatorname{\mathrm{flip}}^{12} \operatorname{\mathrm{flip}}^{23} \operatorname{\mathrm{flip}}^{12} = \operatorname{\mathrm{flip}}^{23} \operatorname{\mathrm{flip}}^{12} \operatorname{\mathrm{flip}}^{23} \colon u \otimes v \otimes w \mapsto (-1)^{\bar{u}\bar{v} + \bar{u}\bar{w} + \bar{v}\bar{w}} w \otimes v \otimes u, \end{align*}$$

and $\Theta ^{23} \Theta ^{13} \Theta ^{12} = \Theta ^{12} \Theta ^{13} \Theta ^{23}$ by (3.6), we see that ${\mathbf {F}}_n$ respects the braid relation.

Since

the superfunctor ${\mathbf {F}}_n$ respects the skein relation (2.8). We also have

and so ${\mathbf {F}}_n$ respects the first relation in (2.9). Next, we compute

Thus, ${\mathbf {F}}_n$ preserves the second relation in (2.9). For the third equality in (2.9), we compute

For the last equality in (2.9), we compute

Finally, the relations (2.10) are straightforward to verify.

It remains to prove uniqueness. Suppose ${\mathbf {F}}_n$ is a monoidal superfunctor as described in the first sentence of the statement of the theorem. Then

and

are uniquely determined by the fact that they must be inverse to

and

, respectively. Next, suppose that

Then, for all $k \in \mathtt {I}$ ,

It follows that $a_{ij} = \delta _{ij}$ for all $i,j \in \mathtt {I}$ , and so

.

To simplify notation, we will start writing objects of as sequences of $\uparrow $ ’s and $\downarrow $ ’s, omitting the $\otimes $ symbol. For such an object X, we define $V^X := {\mathbf {F}}_n(X)$ , and we let $\# X$ denote the length of the sequence.

Theorem 4.4. The superfunctor ${\mathbf {F}}_n$ is full for all $n \in \mathbb {N}$ . Furthermore, the induced map

(4.11)

is an isomorphism when $\# X + \# Y \le 2n$ .

Proof. Our proof is similar to that of [Reference Brundan, Comes and KujawaBCK19, Theorem 4.1], which treats the case $z=0$ . We need to show that, for all objects X and Y in , the map (4.11) is surjective, and that it is also injective when $\# X + \# Y \le 2n$ . Suppose that X (respectively, Y) is a tensor product of $r_X$ (respectively, $r_Y$ ) copies of $\uparrow $ and $s_X$ (respectively, $s_Y$ ) copies of $\downarrow $ . Consider the following commutative diagram:

The top-left horizontal map is given by composing on the top and bottom of diagrams with

to move $\uparrow $ ’s on the top to the left and $\uparrow $ ’s on the bottom to the right. The bottom-left horizontal map is given analogously, using

. The right horizontal maps are the usual isomorphisms that hold in any rigid monoidal supercategory. In particular, the top-right horizontal map is the $\mathbb {C}(q)$ -linear isomorphism given on diagrams by

with inverse

where the rectangle denotes some diagram.

Since all the horizontal maps are isomorphisms, it suffices to show that the rightmost vertical map has the desired properties. Thus, we must show that the map

(4.12)

is surjective for all $r,s \in \mathbb {N}$ , and that it is injective when $r+s \le 2n$ . We first consider the case where $r \ne s$ . Let $x = u_{11} u_{22} \dotsm u_{nn}$ be the central element of Corollary 3.4(b). Since $u_{ii}$ acts on V by $\Theta _{ii}$ , it follows from (3.7) that x acts on V as multiplication by q. By (3.13), we have $\Delta (x) = x \otimes x$ . Thus, x acts on $V^{\otimes r}$ as multiplication by $q^r$ . Since x is central, this implies that $\operatorname {\mathrm {Hom}}_{U_q}(V^{\otimes r}, V^{\otimes s}) = 0$ . Since we also have in this case, by Proposition 2.11, the map (4.12) is an isomorphism when $r \ne s$ .

Now suppose $r=s$ , and consider the composite

(4.13)

where $\varphi $ is the surjective homomorphism of Proposition 2.13 with $s=0$ . This composite is precisely the map of [Reference OlshanskiOls92, Theorem 5.2]. Surjectivity is asserted, without proof, in [Reference OlshanskiOls92, Theorem 5.3]. For the more precise statement, with proof, that this map is also an isomorphism when $\# X + \# Y = 2r \le 2n$ , see [Reference Benkart, Guay, Jung, Kang and WilcoxBGJ+16, Theorem 3.28]. It follows that is always surjective, and that it is an isomorphism when $r \le n$ , as desired.

Note that $\# X + \# Y$ is twice the number of strands in any string diagram representing a morphism in from X to Y. Thus, Theorem 4.4 asserts that ${\mathbf {F}}_n$ induces an isomorphism on morphism spaces whenever the number of strands is less than or equal to n.

We now loosen our assumption on the ground field. For the remainder of this section

$$\begin{align*}\Bbbk \ \,is\, an\, arbitrary\, commutative\, ring\, of\, characteristic\, not\, equal\, to\, two,\, and\,\, z \in \Bbbk. \end{align*}$$

We can now improve Proposition 2.11.

Proof. In light of Proposition 2.11, it remains to prove that the elements of $B_\bullet (X,Y)$ are linearly independent. We first prove this when $\Bbbk = \mathbb {C}(z)$ . Consider the superalgebra homomorphisms (4.13). By [Reference Benkart, Guay, Jung, Kang and WilcoxBGJ+16, Theorem 3.28], the composite ${\mathbf {F}}_n \circ \varphi $ , which is the map denoted $\rho _{n,q}^{r,0}$ there, is an isomorphism for $n \ge r$ . Since the map $\varphi $ is independent of n, it follows that $\varphi $ is injective, and hence an isomorphism. Thus,

where the last equality is [Reference Jones and NazarovJN99, Proposition 2.1] (the statement in [Reference Jones and NazarovJN99, Proposition 2.1] is over the field $\mathbb {C}(q)$ , with $z=q-q^{-1}$ , but the proof is the same over $\mathbb {C}(z)$ ). Now suppose that X (respectively, Y) is a tensor product of $r_X$ (respectively, $r_Y$ ) copies of $\uparrow $ and $s_X$ (respectively, $s_Y$ ) copies of $\downarrow $ . As in the proof of Theorem 4.4, we have a linear isomorphism

Thus

This dimension is equal to the number of elements of $B_\bullet (X,Y)$ . Indeed, there are $k! (X,Y)$ -matchings and $2^k$ ways of adding Clifford tokens to the strings in a positive reduced lift. It follows that $B_\bullet (X,Y)$ is a basis for

. This completes the proof of Theorem 4.5 for $\Bbbk = \mathbb {C}(z)$ .

To complete the proof over more general base rings, note that $\mathbb {C}(q)$ is a free $\mathbb {Z}[z]$ -module, with z acting as $q-q^{-1}$ . Thus, any linear dependence relation over $\mathbb {Z}[z]$ yields a linear dependence relation over $\mathbb {C}(q)$ after extending scalars. Therefore, it follows from the above that the elements of $B_\bullet (\uparrow ^r, \uparrow ^r)$ are a basis over $\mathbb {Z}[z]$ and hence, by extension of scalars, over any commutative ring $\Bbbk $ of characteristic not equal to two and $z \in \Bbbk $ .

Corollary 4.8. The homomorphism of Proposition 2.13 is an isomorphism of associative superalgebras

Proof. By Proposition 2.13, the map is surjective. When $\Bbbk = \mathbb {C}(q)$ , one can then conclude that it is an isomorphism by comparing dimensions. Indeed, by Corollary 4.7 and [Reference Benkart, Guay, Jung, Kang and WilcoxBGJ+16, Corollary 3.25], we have

More generally, one can argue as in Step 1 of the proof of [Reference Jung and KangJK14, Theorem 5.1] to show that $\mathrm {BC}_{r,s}(z)$ has a spanning set that maps to the basis $B_\bullet (\uparrow ^r \downarrow ^s,\uparrow ^r \downarrow ^s)$ of

. It follows that this spanning set is linearly independent, hence a basis of $\mathrm {BC}_{r,s}(z)$ .

As a special case of Corollary 4.9, we have an isomorphism of associative superalgebras . This recovers [Reference Benkart, Guay, Jung, Kang and WilcoxBGJ+16, Theorem 4.19], which describes the walled Brauer–Clifford superalgebras in terms of bead tangle superalgebras. When converting string diagrams representing endomorphisms in to the bead tangle diagrams of [Reference Benkart, Guay, Jung, Kang and WilcoxBGJ+16, Section 4], one should forget the orientations of strings, and then rotate diagrams by . This transformation is needed since the convention in [Reference Benkart, Guay, Jung, Kang and WilcoxBGJ+16, Section 4] for composing diagrams is the opposite of ours.

Remark 4.10. The full monoidal subsupercategory of $U_q\text {-smod}$ generated by V and $V^*$ is not semisimple. Indeed, it follows from Theorem 4.4 that, for $n \ge 2$ , $\operatorname {\mathrm {End}}_{U_q}(V \otimes V^*)$ is isomorphic to

, which, by Theorem 4.5, has basis

By the last equalities in (2.17) and (2.20), the span of the last four diagrams above is a nilpotent ideal. Thus, $\operatorname {\mathrm {End}}_{U_q}(V \otimes V^*)$ is not semisimple. Note, however, that the full monoidal subsupercategory of $U_q\text {-smod}$ generated by V is semisimple (see [Reference Grantcharov, Jung, Kang and KimGJKK10, Theorem 6.5]).

5 The chiral braiding

This section is the start of the second part of the current paper. Our goal is to define and study an affine version of the quantum isomeric supercategory. For braided monoidal supercategories, there is a general affinisation procedure (see [Reference Mousaaid and SavageMS21]). However, the supercategory is not braided since the Clifford dots do not slide through crossings both ways. This corresponds, under the incarnation superfunctor, to the fact that $U_q$ is not a quasitriangular Hopf superalgebra. In this section, we discuss a chiral braiding, which is like a braiding but only natural in one argument. We begin this section with the assumption that $\Bbbk $ is an arbitrary commutative ring of characteristic not equal to two, and $z \in \Bbbk $ .

Definition 5.1. Let

be the strict monoidal supercategory obtained from

by adjoining an additional generating object

and two additional even morphisms

subject to the relations

(5.1)

where

(5.2)

Note that we do not have morphisms corresponding to a red strand passing under a black strand. We also do not have red cups or caps.

Lemma 5.2. The following relations hold in :

(5.3)

Proof. We compute

The proofs of the remaining equalities are analogous.

Proposition 5.3. In , we have

(5.4)

where f is any string diagram in not containing Clifford tokens.

Proof. First note that the second equality in (5.4) follows from the first after composing on the top and bottom with the appropriate red-black crossings and using the first four relations in (5.1). Therefore, we prove only the first equality. It suffices to prove it for f equal to each of the generating morphisms

,

,

,

,

. Since

it is also enough to show it holds for

.

For , the first equality in (5.4) follows from the last relation in (5.1). Composing both sides of the last relation in (5.1) on the top with and on the bottom with shows that the first equality in (5.4) also holds with .

To prove that the first equality in (5.4) holds with and , we must show that

(5.5)

The first relation in (5.5) follows from the first relation in (5.3) after composing on the bottom with and using the third relation in (5.1). Similarly, the second relation in (5.5) follows from the second relation in (5.3) after composing on the top with and using the fourth relation in (5.1). The proofs for and are analogous, using the last two equalities in (5.3).

In the remainder of this section, we will be discussing connections to $U_q\text {-smod}$ . Thus, we now begin supposing that $\Bbbk = \mathbb {C}(q)$ and $z = q-q^{-1}$ . Recall the definition of L from (3.11). For a finite-dimensional $U_q$ -supermodule M, we let $\rho _M \colon U_q \to \operatorname {\mathrm {End}}_\Bbbk (M)$ denote the corresponding representation. We then define

$$\begin{align*}L_M = (\rho_M \otimes 1_V)(L) = \sum_{\substack{i, j \in \mathtt{I} \\ i \le j}} \rho_M(u_{ij}) \otimes E_{ij} \in \operatorname{\mathrm{End}}_\Bbbk(M) \otimes \operatorname{\mathrm{End}}_\Bbbk(V). \end{align*}$$

In particular, we have $L_V = \Theta $ (see (3.26)).

Proposition 5.4. For any $M \in U_q\text {-smod}$ , the map

(5.6) $$ \begin{align} T_{MV} := \operatorname{\mathrm{flip}} \circ L_M \colon M \otimes V \to V \otimes M \end{align} $$

is an isomorphism of $U_q$ -supermodules. Furthermore, for all $f \in \operatorname {\mathrm {Hom}}_{U_q}(M,N)$ , we have

(5.7) $$ \begin{align} T_{NV} \circ (f \otimes 1_V) = (1_V \otimes f) \circ T_{MV}. \end{align} $$

Proof. It is clear that $T_{MV}$ is invertible, with

(5.8) $$ \begin{align} T_{MV}^{-1} = (\rho_M \otimes 1_V)(L^{-1}) \circ \operatorname{\mathrm{flip}} \overset{(3.15)}{=} \left( \sum_{i \le j} \rho_M(S(u_{ij})) \otimes E_{ij} \right) \circ \operatorname{\mathrm{flip}}. \end{align} $$

To show that $T_{MV}$ is a homomorphism of $U_q$ -supermodules, it suffices to prove that it commutes with the action of $u_{kl}$ , $k,l \in \mathtt {I}$ , $k \le l$ . By (3.12), it is enough to show that

(5.9) $$ \begin{align} \operatorname{\mathrm{flip}}^{12} L_M^{12} L_M^{13} L_V^{23} = L_V^{13} L_M^{23} \operatorname{\mathrm{flip}}^{12} L_M^{12} \quad \text{as maps } M \otimes V \otimes V \to V \otimes M \otimes V. \end{align} $$

Composing both sides of (5.9) on the left with the invertible map $\operatorname {\mathrm {flip}}^{12}$ , and using the fact that $L_V = \Theta $ , we see that (5.9) is equivalent to

(5.10) $$ \begin{align} L_M^{12} L_M^{13} \Theta^{23} = \Theta^{23} L_M^{13} L_M^{12} \quad \text{as maps } M \otimes V \otimes V \to M \otimes V \otimes V, \end{align} $$

which follows from the last equality in (3.10).

It remains to prove (5.7). For $f \in \operatorname {\mathrm {Hom}}_{U_q}(M,N)$ , $m \in M$ and $v \in V$ , we have

$$ \begin{align*} T_{NV} \circ (f \otimes 1_V)(m \otimes v) &= \operatorname{\mathrm{flip}} \circ \left( \sum_{i \le j} (-1)^{(\bar{f}+\bar{m})p(i,j)} u_{ij} f(m) \otimes E_{ij} v \right) \\ &= \operatorname{\mathrm{flip}} \circ \left( \sum_{i \le j} (-1)^{\bar{m}p(i,j)} f(u_{ij} m) \otimes E_{ij} v \right) \\ &= \operatorname{\mathrm{flip}} \circ (f \otimes 1_V) \circ \left( \sum_{i \le j} u_{ij} \otimes E_{ij} \right) (m \otimes v) \\ &= (1_V \otimes f) \circ T_{MV} (m \otimes v).\\[-35pt] \end{align*} $$

Note that $T_{VV} = T$ (see (4.1)), and so Proposition 5.4 is a generalisation of Lemma 4.1. Now, for $i,j \in \mathtt {I}$ , define

$$\begin{align*}E_{ij}^* \colon V^* \to V^*, \qquad f \mapsto (-1)^{p(i,j)\bar{f}} f \circ E_{ij}. \end{align*}$$

It follows that

(5.11) $$ \begin{align} E_{ij}^* v_k^* = \delta_{ik} (-1)^{p(i)+p(i)p(j)} v_j^* \qquad \text{and} \qquad E_{ij}^* E_{kl}^* = \delta_{il} (-1)^{p(i,j)p(k,l)} E_{kj}^*, \end{align} $$

for all $i,j,k,l \in \mathtt {I}$ .

Theorem 5.5. For each $U_q$ -supermodule M, the superfunctor ${\mathbf {F}}_n$ of Theorem 4.3 extends to a unique monoidal superfunctor

, such that

Furthermore,

,

(5.12)

Proof. By Theorem 4.3, to show that ${\mathbf {F}}_n^M$ is well-defined, it suffices to show that ${\mathbf {F}}_n^M$ respects the relations (5.1). First of all, uniqueness of the inverse implies that , and then the first two relations in (5.1) are satisfied.

Next we show, using (5.2), that the equalities (5.12) must hold. For $m \in M$ and $k \in \mathtt {I}$ , we have

Next, we compute

Now, for the third relation in (5.1), we compute

The proof of the fourth relation in (5.1) is analogous.

For the last relation in (5.1), we compute that ${\mathbf {F}}_n^M$ sends the left-hand side to the map $M \otimes V^{\otimes 2} \to V^{\otimes 2} \otimes M$ given by

$$\begin{align*}T_{VV}^{12} T_{MV}^{23} T_{MV}^{12} = \operatorname{\mathrm{flip}}^{12} \Theta^{12} \operatorname{\mathrm{flip}}^{23} L^{23} \operatorname{\mathrm{flip}}^{12} L^{12} = \operatorname{\mathrm{flip}}^{12} \operatorname{\mathrm{flip}}^{23} \operatorname{\mathrm{flip}}^{12} \Theta^{23} L^{13} L^{12}. \end{align*}$$

On the other hand, ${\mathbf {F}}_n^M$ sends the right-hand side to the map

$$\begin{align*}T_{MV}^{23} T_{MV}^{12} T_{VV}^{23} = \operatorname{\mathrm{flip}}^{23} L^{23} \operatorname{\mathrm{flip}}^{12} L^{12} \operatorname{\mathrm{flip}}^{23} \Theta^{23} = \operatorname{\mathrm{flip}}^{23} \operatorname{\mathrm{flip}}^{12} \operatorname{\mathrm{flip}}^{23} L^{12} L^{13} \Theta^{23}. \end{align*}$$

Then the last relation in (5.1) follows from (3.10) and the fact that $\operatorname {\mathrm {flip}}^{12} \operatorname {\mathrm {flip}}^{23} \operatorname {\mathrm {flip}}^{12} = \operatorname {\mathrm {flip}}^{23} \operatorname {\mathrm {flip}}^{12} \operatorname {\mathrm {flip}}^{23}$ .

Remark 5.6. There are natural superfunctors

sending

to $\uparrow $ and $\downarrow $ , respectively. It is then straightforward to verify that following diagrams commute:

In particular, we have

For $M \in U_q\text {-smod}$ , we will denote the image of a string diagram in

under ${\mathbf {F}}_n^M$ by labeling the red strands by M. Thus, for example

Then (5.7) is equivalent to

(5.13)

Lemma 5.7. For any $U_q$ -supermodules M and N, we have

(5.14)

Proof. We have

$$ \begin{align*} &T_{M \otimes N,V} \overset{(3.12)}{=} \operatorname{\mathrm{flip}}_{M \otimes N,V} \circ L^{13} L^{23} = (\operatorname{\mathrm{flip}}_{MV} \otimes 1_N) \circ (1_M \otimes \operatorname{\mathrm{flip}}_{NV}) \circ L^{13} L^{23} \\ &= (\operatorname{\mathrm{flip}}_{MV} \otimes 1_N) \circ L^{12} \circ (1_M \otimes \operatorname{\mathrm{flip}}_{NV}) \circ L^{23} = (T_M \otimes 1_N) \circ (1_M \otimes T_N).\\[-35pt] \end{align*} $$

The relations (5.1), (5.4), (5.13) and (5.14) show that , , and , together with the crossings in , almost endow with the structure of a braided monoidal category. However, we do not truly have a braiding since, for example, we do not have a morphism corresponding to a red strand passing under a black strand. Furthermore, closed Clifford tokens do not pass under crossings. In general, we can define a crossing for any sequence of strands in passing over any sequence of strands in . All morphisms in pass over such crossings, but only some morphisms in pass under them. In other words, the crossings are only natural in one argument. Because of this asymmetry, we refer to this structure as a chiral braiding.

We now restrict our attention to diagrams with a single red strand. Let denote the full subsupercategory of on objects that are tensor products of $\uparrow $ , $\downarrow $ and , with exactly one occurrence of . Thus, objects of are of the form for . Note that is not a monoidal supercategory.

Theorem 5.8. There is a unique superfunctor

defined as follows: On an object

, ${\mathbf {F}}^\bullet _n(X)$ is the superfunctor

$$ \begin{align*} {\mathbf{F}}^{\bullet}_n(X) \colon U_q\text{-smod} \to U_q\text{-smod},\quad M \mapsto {\mathbf{F}}_n^M(X). \end{align*} $$

On a morphism

, ${\mathbf {F}}^\bullet _n(f)$ is the natural transformation ${\mathbf {F}}^\bullet _n(X) \to {\mathbf {F}}^\bullet _n(Y)$ whose M-component, for $M \in U_q\text {-smod}$ , is

$$ \begin{align*} {\mathbf{F}}^{\bullet}_n(f)_M = {\mathbf{F}}_n^M(f). \end{align*} $$

6 The quantum affine isomeric supercategory

In this section, we introduce an affine version of the quantum isomeric supercategory and examine some of its properties. Throughout this section, $\Bbbk $ is an arbitrary commutative ring of characteristic not equal to two, and $z \in \Bbbk $ .

Definition 6.1. The quantum affine isomeric supercategory

is the strict monoidal supercategory obtained from

by adjoining an additional odd morphism

subject to the relations

(6.1)

We refer to

as an open Clifford token. To emphasise the difference, we will henceforth refer to

as a closed Clifford token.

It is important to note that we do not impose a relation for sliding open Clifford tokens past closed ones. It follows immediately from the defining relations that we have the following symmetry of .

Lemma 6.2. There is a unique isomorphism of monoidal supercategories

determined on objects by $\uparrow \ \mapsto \ \uparrow $ , $\downarrow \ \mapsto \ \downarrow $ and sending

On arbitrary diagrams, the isomorphism acts by interchanging open and closed Clifford tokens and flipping crossings.

Define

(6.2)

It follows that

(6.3)

Lemma 6.3. The following relations hold in :

(6.4)

(6.5)

where, in (6.4), the relations hold for all orientations of the strands.

It follows from the above discussion that the isomorphisms $\Omega _-$ , $\Omega _\updownarrow $ and $\Omega _\leftrightarrow $ defined in Section 2 extend to isomorphisms of monoidal supercategories

These are defined as in Section 2 for the generators of

and, on the open Clifford token, are defined by

Furthermore,

is strictly pivotal, with duality superfunctor

Define, for $k \in \mathbb {Z}$ ,

(6.6)

Note that both morphisms in (6.6) are of parity $k \pmod 2$ . We then define, for $k \in \mathbb {Z}$ ,

(6.7)

We refer to the decorations as zebras. We have coloured them and their labels mahogany to help distinguish these labels from coefficients in linear combinations of diagrams. The morphism should be thought of as a quantum analogue of the even morphism of [Reference Brundan, Comes and KujawaBCK19].

Recall our convention

That is, when zebras appear at the same height, the entire zebra on the left should be considered as above the entire zebra on the right. Note that composition of zebras is a bit subtle, since the labels do not add in general. We have a homomorphism of superalgebras

Conjecture 6.12 below would imply that this map is injective.

Lemma 6.5. The following relations hold in for all $k \in \mathbb {Z}$ :

(6.8)

(6.9)

where, in (6.8), the relations hold for both orientations of the strands.

Proof. It follows from (2.10), (2.20), (6.4) and (6.7) that

Then relations (6.8) follow from (2.20) and (6.4).

For $k \ge 0$ , we have

The case $k<0$ , as well as the proof of the second equality in (6.9), are analogous.

For the third equality in (6.9), it suffices to consider the case $k>0$ . In this case, we have

The proof of the last equality in (6.9) is similar.

Lemma 6.6. The following relations hold in :

(6.10)

Proof. For the first relation, we have

The proof of the second relation is analogous.

Corollary 6.7. For all $k \in \mathbb {Z}_{>0}$ , the following relations hold in :

(6.11)

(6.12)

Proof. We prove (6.11) by induction on k. The case $k=1$ is (6.10). Then, for $k \ge 2$ , we have

where we used the induction hypothesis in the first equality. Relation (6.12) then follows by composing (6.11) on the bottom with

and on the top with

.

Lemma 6.8. For all $k> 0$ , we have

(6.13)

Proof. We have

Let $\text {Sym}$ denote the $\Bbbk $ -algebra of symmetric functions over $\Bbbk $ . For $r \ge 0$ , let $e_r$ and $h_r$ denote the degree r elementary and complete homogeneous symmetric functions, respectively, with the convention that $e_0 = h_0 = 1$ .

Proposition 6.9. We have a homomorphism of rings

Proof. The $\Bbbk $ -algebra $\text {Sym}$ is generated by $e_r,h_r$ , $r>0$ , modulo the identities

$$\begin{align*}\sum_{r=0}^k (-1)^r e_{k-r} h_r = 0,\quad k> 0, \end{align*}$$

where $h_0=e_0=1$ . The map $\beta $ sends the left-hand side of this identity to

which is equal to zero by (6.13).

Proposition 6.11 below implies that the map $\beta $ is surjective, while Conjecture 6.12 would imply it is an isomorphism. We next deduce a bubble slide relation.

Lemma 6.10. For all $k \ge 0$ , we have

Proof. The case $k=0$ follows immediately from (2.7). Thus, we suppose $k>0$ . We first compute

Next, note that, for $1 \le r \le k-1$ ,

Similarly, for $1 \le r \le k$ ,

where the last sum is zero when $r=k$ . We also have

Therefore,

For any two objects

, the morphism space

is a right $\text {Sym}$ -supermodule with action given by

As in Section 2, for each $(X,Y)$ , fix a set $B(X,Y)$ consisting of a choice of positive reduced lift for each $(X,Y)$ -matching. Then let

denote the set of all morphisms that can be obtained from the elements of $B(X,Y)$ by adding a zebra, labelled by some integer (possibly zero) near the terminus of each string. We require that all zebras occurring on strands whose terminus is at the top of the diagram to be at the same height; similarly, we require that all zebras occurring on strands whose terminus is at the bottom of the diagram to be at the same height, and below those zebras on strands whose terminus is at the top of the diagram.

As noted in the Introduction, we expect that Conjecture 6.12 could be proved using the categorical comultiplication technique of [Reference Brundan, Savage and WebsterBSW20], after introducing the more general quantum isomeric Heisenberg supercategory.

Proposition 6.13. There is a unique monoidal superfunctor

defined as follows. On objects

and morphisms

,

$$\begin{align*}\mathbf{C}(X) = X \otimes - \qquad \text{and} \qquad \mathbf{C}(f) = f \otimes -. \end{align*}$$

In addition,

is the natural transformation $\uparrow \otimes - \to \ \uparrow \otimes -$ whose X-component,

, is

(6.14)

where the thick strand labelled X is the identity morphism $1_X$ of X.

Proof. Naturality of $\mathbf {C}(f)$ is clear for . For , it follows from the fact that the generating morphisms (2.5) and (2.6) slide over crossings.

All the relations appearing in Definition 2.1 are clearly respected by $\mathbf {C}$ . It remains to verify the relations (6.1). The first relation is straightforward. For the second relation, we compute (dropping the label X on the thick strand)

Finally, for the last relation in (6.1), we compute

The superfunctor $\mathbf {C}$ , which we will call the collapsing superfunctor, should be viewed as an odd analogue of the one appearing in [Reference Mousaaid and SavageMS21, Theorem 3.2], which describes actions of the affinisation of a braided monoidal category. In that setting, the analogue of the open Clifford token is the affine dot, which acts as

See Remark 7.6 for additional discussion.

7 Affine endomorphism superalgebras

In this section, we describe the relationship between the endomorphism superalgebras in the quantum affine isomeric supercategory and affine Hecke–Clifford superalgebras. We also use the collapsing superfunctor of Proposition 6.13 to explain how the Jucys–Murphy elements in the Hecke–Clifford superalgebra arise naturally in this context. Throughout this section, $\Bbbk $ is an arbitrary commutative ring of characteristic not equal to two, and $z \in \Bbbk $ .

Definition 7.1. For $r \in \mathbb {Z}_{>0}$ and $z \in \Bbbk $ , let $\mathrm {AHC}_r(z)$ denote the associative superalgebra generated by

even elements $t_1,\dotsc ,t_{r-1}$ and odd elements $\pi _1,\dotsc ,\pi _r,\varpi _1,\dotsc ,\varpi _r$ ,

satisfying the following relations (for $i,j$ in the allowable range):

(7.1) $$ \begin{align} t_i^2 &= z t_i + 1, \end{align} $$

(7.2) $$ \begin{align} t_i t_{i+1} t_i &= t_{i+1} t_i t_{i+1},& t_i t_j &= t_j t_i,& |i-j|> 1, \end{align} $$

(7.3) $$ \begin{align} \pi_i^2 &= -1,& \pi_i \pi_j &= - \pi_j \pi_i,& i \ne j, \end{align} $$

(7.4) $$ \begin{align} \varpi_i^2 &= -1,& \varpi_i \varpi_j &= - \varpi_j \varpi_i,& i \ne j, \end{align} $$

(7.5) $$ \begin{align} t_i \pi_i &= \pi_{i+1} t_i,& t_i \pi_j &= \pi_j t_i,& j \ne i,i+1, \end{align} $$

(7.6) $$ \begin{align} t_i \varpi_{i+1} &= \varpi_i t_i,& t_i \varpi_j &= \varpi_j t_i,& j \ne i,i+1. \end{align} $$

Equivalently, $\mathrm {AHC}_r(z)$ is the associative superalgebra generated by $\mathrm {HC}_r(z)$ , together with odd elements $\varpi _1,\dotsc ,\varpi _r$ , subject to the relations (7.4) and (7.6).

Multiplying both sides of the first relation in (7.6) on the left and right by $t_i^{-1} = t_i-z$ gives

(7.7) $$ \begin{align} t_i \varpi_i = \varpi_{i+1} t_i + z(\varpi_i - \varpi_{i+1}). \end{align} $$

The next result shows that $\mathrm {AHC}_r(z)$ is isomorphic to the affine Hecke–Clifford superalgebra, which was first introduced in [Reference Jones and NazarovJN99, Section 3], where it was called the affine Sergeev algebra.

Lemma 7.2. For $r \in \mathbb {Z}_{>0}$ and $z \in \Bbbk $ , $\mathrm {AHC}_r(z)$ is isomorphic to the associative superalgebra $\mathrm {AHC}_r'(z)$ generated by $\mathrm {HC}_r(z)$ , together with pairwise-commuting invertible even elements $x_1,\dotsc ,x_r$ , subject to the following relations (for $i,j$ in the allowable range):

$$ \begin{align*} t_i x_i &= x_{i+1} t_i - z (x_{i+1} - \pi_i \pi_{i+1} x_i), \\ t_i x_{i+1} &= x_i t_i + z(1 + \pi_i \pi_{i+1}) x_{i+1}, \\ t_i x_j &= x_j t_i,& j \ne i,i+1, \\ \pi_i x_i &= x_i^{-1} \pi_i, \\ x_i x_j &= x_j x_i,& j \ne i. \end{align*} $$

The isomorphism is given by

(7.8) $$ \begin{align} \mathrm{AHC}^{\prime}_r(z) \xrightarrow{\cong} \mathrm{AHC}_r(z), \qquad t_i \mapsto t_i,\quad \pi_i \mapsto \pi_i,\quad x_i \mapsto \pi_i \varpi_i. \end{align} $$

Proof. It is a straightforward exercise to verify that (7.8) respects the defining relations of $\mathrm {AHC}^{\prime }_r(z)$ . Thus, the map (7.8) is a well-defined homomorphism of superalgebras. It is invertible, with inverse

$$\begin{align*}\mathrm{AHC}_r(z) \to \mathrm{AHC}^{\prime}_r(z),\qquad t_i \mapsto t_i,\quad \pi_i \mapsto \pi_i,\quad \varpi_i \mapsto - \pi_i x_i.\\[-35pt] \end{align*}$$

In light of Lemma 7.2, we will simply refer to $\mathrm {AHC}_r(z)$ as the affine Hecke–Clifford superalgebra.

Proposition 7.3. For $r \in \mathbb {N}$ , we have a homomorphism of associative superalgebras

given by

Note that, under the homomorphism of Proposition 7.3, we have

The difference between the new presentation of the affine Hecke–Clifford superalgebra given in Definition 7.1 and the one in Lemma 7.2 that has appeared previously in the literature is that the former presentation involves the odd generators $\varpi _i$ , whereas the latter involves the even generator $x_i = \pi _i \varphi _i$ . We prefer the presentation of Definition 7.1 since the relations are simpler and a natural symmetry of $\mathrm {AHC}_r(z)$ becomes apparent. In particular, we have an automorphism of $\mathrm {AHC}_r(z)$ given by

(7.9) $$ \begin{align} \pi_i \mapsto \varphi_i,\qquad \varpi_i \mapsto \pi_i,\qquad t_j \mapsto - t_j^{-1},\qquad 1 \le i \le r,\ 1 \le j \le r-1. \end{align} $$

For the remainder of this section, we reverse our numbering convention for strands in diagrams (see Remark 2.8). The composite of the map of Proposition 7.3 with the automorphism of

induced by the superfunctor $\Omega _\leftrightarrow $ yields a homomorphism of associative superalgebras

given by

This is also equal to the automorphism (7.9) followed by the map of Proposition 7.3. We have

The Jucys–Murphy elements $J_1,\dotsc ,J_r \in \mathrm {HC}_r(z)$ were defined recursively in [Reference Jones and NazarovJN99, (3.10)] by

$$\begin{align*}J_i := \begin{cases} 1 & \text{for } i=1, \\ (t_{i-1} - z \pi_{i-1} \pi_i) J_{i-1} t_{i-1} & \text{for } i=2,\dotsc,n. \end{cases} \end{align*}$$

For $1 \le i \le r$ , define the odd Jucys–Murphy elements

(7.10) $$ \begin{align} J_i^{\text{odd}} = t_{i-1}^{-1} \dotsm t_2^{-1} t_1^{-1} \pi_1 t_1 t_2 \dotsm t_{i-1} \in \mathrm{HC}_r(z), \end{align} $$

where, by convention, we have $J_1^{\text {odd}} = \pi _1$ . The following result gives a direct (i.e. nonrecursive) expression for the even Jucys–Murphy elements.

Proof. We prove the result by induction on i. Since $- \pi _1 J_1^{\text {odd}} = -\pi _1^2 = 1 = J_1$ , the result holds for $i=1$ . Now suppose that $i>1$ , and that $J_{i-1} = - \pi _{i-1} J_{i-1}^{\text {odd}}$ . First note that

$$\begin{align*}\pi_i t_{i-1}^{-1} = \pi_i (t_{i-1}-z) \overset{(7.5)}{=} t_{i-1} \pi_{i-1} - z \pi_i. \end{align*}$$

Thus, we have

$$\begin{align*}- \pi_i J_i^{\text{odd}} \kern-1.3pt{=}\kern-1.3pt - \pi_i t_{i-1}^{-1} J_{i-1}^{\text{odd}} t_{i-1} \kern-1.3pt{=}\kern-1.3pt - (t_{i-1} \pi_{i-1} - z \pi_i) J_{i-1}^{\text{odd}} t_{i-1} \kern-1.3pt{=}\kern-1.3pt (t_{i-1} - z \pi_{i-1} \pi_i) J_{i-1} t_{i-1} \kern-1.3pt{=}\kern-1.3pt J_i.\\[-13pt] \end{align*}$$

Evaluation on the unit object

yields a superfunctor

Note that this is not a monoidal superfunctor. Recall the collapsing superfunctor $\mathbf {C}$ of Proposition 6.13.

Proposition 7.5. For $1 \le i \le r$ , we have

Proof. We have

where it is the i-th strand from the right that passes under other strands. The proof of the second equality in the statement of the proposition follows after adding a closed Clifford token to the top of the i-th strand from the right.

Remark 7.6. Recall that the i-th Jucys–Murphy element in the Iwahori–Hecke algebra of type A is given, in terms of string diagrams, by

where it is the i-th strand from the right that loops around other strands. See [Reference Mousaaid and SavageMS21, Section 6] for a discussion of Jucys–Murphy elements in a more general setting, related to the affinisation of braided monoidal categories. The above discussion suggests there may be a general notion of odd affinisation, where the above diagram is replaced by the one appearing in the proof of Proposition 7.5.

The next result shows that we can naturally view as a subcategory of .

8 The affine action superfunctor

In this final section, we define an action of on the category of $U_q$ -supermodules. We then use this action to define a sequence of elements in the centre of $U_q$ . Throughout this section, we assume that $\Bbbk = \mathbb {C}(q)$ and $z=q-q^{-1}$ .

The image of

under the superfunctor ${\mathbf {F}}^\bullet _n$ of Theorem 5.8 is the superfunctor

of tensoring on the left with V. Define the natural transformation

(8.1)

Thus, the M-component of K, for $M \in U_q\text {-smod}$ , is the $U_q$ -supermodule homomorphism

It is straightforward to verify that $K^2=-\operatorname {\mathrm {id}}$ , where $\operatorname {\mathrm {id}}$ is the identity natural transformation.

Theorem 8.1. There is a unique monoidal superfunctor

such that

We call the superfunctor $\widehat {{\mathbf {F}}}_n$ the affine action superfunctor. It endows $U_q\text {-smod}$ with the structure of an -supermodule category. Note that

$$\begin{align*}\widehat{{\mathbf{F}}}_n(\uparrow) = V \otimes - \qquad \text{and} \qquad \widehat{{\mathbf{F}}}_n(\downarrow) = V^* \otimes - \end{align*}$$

are the translation endosuperfunctors of $U_q\text {-smod}$ given by tensoring on the left with V and $V^*$ , respectively. Thus, combining Proposition 7.3 and Theorem 8.1, we have a homomorphism of associative superalgebras

$$\begin{align*}\mathrm{AHC}_r(z) \to \operatorname{\mathrm{End}}_{U_q}(V^{\otimes r} \otimes M) \end{align*}$$

for any $U_q$ -supermodule M and $r,s \in \mathbb {N}$ . This is a quantum analogue of [Reference Hill, Kujawa and SussanHKS11, Theorem 7.4.1].

Let

$$\begin{align*}Z_q := \{x \in U_q : xy = (-1)^{\bar{x}\bar{y}} yx \text{ for all } y \in U_q\} \end{align*}$$

be the centre of $U_q$ . Evaluation on the identity element of the regular representation defines a canonical superalgebra isomorphism

$$\begin{align*}\operatorname{\mathrm{End}}(\operatorname{\mathrm{id}}_{U_q\text{-smod}}) \xrightarrow{\cong} Z_q, \end{align*}$$

where $\operatorname {\mathrm {id}}_{\mathcal {C}}$ denotes the identity endosuperfunctor of a supercategory $\mathcal {C}$ . Consider the composite superalgebra homomorphism

(8.2)

Our goal is now to compute the image of this homomorphism. By Proposition 6.11, it suffices to compute the image of the zebra bubbles , $k> 0$ .

We begin with a simplifying computation. Using (5.4) and the relations in , we have, for $k> 0$ ,

(8.3)

where there are a total of $2k$ closed Clifford tokens on alternating sides of the red strand.

For $i,j \in \mathtt {I}$ , define

(8.4) $$ \begin{align} y_{ij} := - \sum_{k = \max(i,j)}^n (-1)^{p(i,k)p(j,k)} S(u_{ik}) u_{-k,-j} \in U_q. \end{align} $$

Note that $y_{ij}$ is of parity $p(i,j)$ . Next, for $i,j \in \mathtt {I}$ , and $m> 0$ , define

$$\begin{align*}y_{ij}^{(m)} := (-1)^{p(i)p(j)} \sum_{i=i_0,i_1,\dotsc,i_{m-1},i_m=j} (-1)^{\sum_{k=1}^{m-1} p(i_k) + \sum_{k=0}^{m-1} p(i_k)p(i_{k+1})} y_{i_0,i_1} \dotsm y_{i_{m-1},i_m}. \end{align*}$$

These are isomeric analogues of the elements defined in [Reference Brundan, Savage and WebsterBSW20, (5.15)].

Lemma 8.2. We have

(8.5)

where we interpret the right-hand side as a natural transformation whose M-component, for a $U_q$ -supermodule M with corresponding representation $\rho _M$ , is $\sum _{i,k \in I} \rho _M (y_{ij}) \otimes E_{ij}$ .

Proof. We have

Replacing l by $-j$ yields (8.5).

Corollary 8.3. We have

(8.6)

Proposition 8.4. For $m>0$ , the image of under (8.2) is

(8.7) $$ \begin{align} \sum_{i \in \mathtt{I}} (-1)^{p(i)} q^{2|i|-2n-1} y_{ii}^{(m)}. \end{align} $$

Proof. It suffices to compute the action of

on the element $1$ of the regular representation. Using (8.3), this is given by

Proposition 8.4 is an isomeric analogue of [Reference Brundan, Savage and WebsterBSW20, (5.29)], giving the image of the analogous diagrams for the affine HOMFLYPT skein category, which is the $U_q(\mathfrak {gl_n})$ -analogue of . On the other hand, Proposition 8.4 can also be viewed as a quantum analogue of [Reference Brundan, Comes and KujawaBCK19, Theorem 4.5], which treats the degenerate (i.e. nonquantum) case. In particular, the elements (8.7) are quantum analogues of central elements in $U(\mathfrak {q}_n)$ introduced by Sergeev in [Reference SergeevSer83] (see [Reference Brundan, Comes and KujawaBCK19, Proposition 4.6]). In the degenerate case, these elements generate the centre of $U(\mathfrak {q}_n)$ (see [Reference Nazarov and SergeevNS06, Proposition 1.1]). It seems likely that the elements (8.7) do not quite generate the centre of $U_q(\mathfrak {q}_n)$ , by analogy with the case of $U_q(\mathfrak {gl}_n)$ , where one needs to add one additional generator (see [Reference Brundan, Savage and WebsterBSW20, Corollary 5.11] and Corollary 3.4(b)).