1 Introduction
One of the most fundamental facts in representation theory is Schur’s lemma, which implies that if V is a finitedimensional simple module over an associative $\Bbbk $ algebra A, where $\Bbbk $ is an algebraically closed field, then $\operatorname {\mathrm {End}}_A(V) \cong \Bbbk $ . On the other hand, if A is an associative $\Bbbk $ superalgebra, then there are two possibilities: we can have $\operatorname {\mathrm {End}}_A(V) \cong \Bbbk $ or we can have that $\operatorname {\mathrm {End}}_A(V)$ is a twodimensional Clifford superalgebra generated by the parity shift. In the theory of Lie superalgebras, this phenomenon underlies the fact that the general linear Lie algebra $\mathfrak {gl}_n$ has two natural analogues in the super setting: the general linear Lie superalgebra $\mathfrak {gl}_{mn}$ and the isomeric Lie superalgebra $\mathfrak {q}_n$ (following [Reference Nagpal, Sam and SnowdenNSS22], we use the term isomeric instead of the more traditional term queer). The purpose of the current paper is to develop diagrammatic tools for studying the representation theory of the quantum analogue of $\mathfrak {q}_n$ . Our hope is that this is the starting point of the development of isomeric analogues of much of the rich mathematics that has emerged from connections between lowdimensional topology, representation theory and categorification.
Before describing our results, we begin with an overview of the situation for the betterunderstood case of $\mathfrak {gl}_{mn}$ . The finitedimensional complex representation theory of $\mathfrak {gl}_{mn}$ is controlled by the oriented Brauer category
. More precisely,
is a diagrammatic symmetric monoidal category depending on a dimension parameter t. When $t=mn$ , there is a full monoidal functor
to the category of $\mathfrak {gl}_{mn}$ supermodules, sending the generating object of
to the natural supermodule of $\mathfrak {gl}_{mn}$ (see [Reference Comes and WilsonCW12, Section 8.3]) (throughout this Introduction, we work with finitedimensional supermodules). The additive Karoubi envelope (i.e. the idempotent completion of the additive envelope) of
is Deligne’s interpolating category $\mathrm {\underline {Re}p}(\mathrm {GL}_t)$ . Similar statements hold in the orthosymplectic case, where
is replaced by the Brauer category (no longer oriented, due to the fact that the natural supermodule is selfdual) (see [Reference Lehrer and ZhangLZ17, Theorem 5.6]).
Any monoidal category acts on itself via the tensor product. In particular, translation functors, given by tensoring with a given supermodule, are key tools for studying the representation theory of Lie superalgebras. In the case of $\mathfrak {gl}_{mn}$ , this action by tensoring can be enlarged to a monoidal functor
where
is the affine oriented Brauer category of [Reference Brundan, Comes, Nash and ReynoldsBCNR17] and
denotes the monoidal category of endofunctors of a category $\mathcal {C}$ . The category
allows one to study natural transformations between translation functors, provides tools to study cyclotomic Hecke algebras and yields natural elements in the centre of $U(\mathfrak {gl}_{mn})$ . Again, a similar picture exists for the orthosymplectic Lie superalgebras, where
is replaced by the affine Brauer category of [Reference Rui and SongRS19].
Quantum analogues of the above pictures play a particularly important role in connections to link invariants and integrable models in statistical mechanics. The quantum analogue of the oriented Brauer category is the HOMFLYPT skein category
, originally introduced in [Reference TuraevTur89, Section 5.2], where it was called the Hecke category. The affine version
was introduced in [Reference BrundanBru17], and there are monoidal functors
with many of the properties mentioned above for the nonquantum case. We expect that these functors can be generalised to the super setting of $U_q(\mathfrak {gl}_{mn})$ . The generalisation of the first functor should follow from the results [Reference Lehrer, Zhang and ZhangLZZ20], and then the affine case follows from the general affinisation procedure of [Reference Mousaaid and SavageMS21]. Once again, analogues exist in the orthosymplectic case, where the relevant categories are the Kauffman skein category, together with its affine analogue introduced in [Reference Gao, Rui and SongGRS22].
The isomeric analogues of the oriented Brauer category and its affine version are the oriented Brauer–Clifford supercategory
and the degenerate affine oriented Brauer–Clifford supercategory
introduced in [Reference Brundan, Comes and KujawaBCK19]. In analogy with the above, one has monoidal superfunctors
where
denotes the monoidal supercategory of endosuperfunctors of a supercategory $\mathcal {C}$ . The need to move to the setting of supercategories here arises from the super version of Schur’s lemma mentioned earlier. There is an odd endomorphism of the natural representation of $\mathfrak {q}_n$ that corresponds to an odd morphism in
and
(the category of vector superspaces, with parity preserving linear maps, is a monoidal category, with no need to introduce the notion of a monoidal supercategory). Note also the absence of the parameter t that appears for the oriented Brauer category. This is because the natural representation of $\mathfrak {q}_n$ always has superdimension zero.
In the current paper, we develop analogues of the above results for the quantum isomeric superalgebra $U_q(\mathfrak {q}_n)$ . As we will explain below, this case requires several new techniques. We begin by defining the quantum isomeric supercategory
depending on a parameter z in the ground ring. This is a strict monoidal supercategory generated by two objects, $\uparrow $ and $\downarrow $ and morphisms
subject to certain relations (see Definition 2.1). The supercategory
should be viewed as a quantisation of
. In particular,
is isomorphic to
(see Lemma 2.9). From the definition of
, we deduce further relations, showing, in particular, that this supercategory is pivotal. We also prove a basis theorem (Theorem 4.5) showing that the morphism spaces have bases given by tanglelike diagrams, where strands can carry the odd Clifford token
corresponding to the odd endomorphism appearing in the super version of Schur’s lemma. We define, in Theorem 4.3, a monoidal superfunctor
which we call the incarnation superfunctor. This superfunctor is full and asymptotically faithful, in the sense that the induced map on any morphism space in
is an isomorphism for sufficiently large n (Theorem 4.4). This can be viewed as a categorical version of the first fundamental theorem for $U_q(\mathfrak {q}_n)$ invariants.
The endomorphism superalgebras are Hecke–Clifford superalgebras, which appear in quantum Sergeev duality (the quantum isomeric analogue of Schur–Weyl duality). More generally, the endomorphism superalgebras in are isomorphic to the quantum walled Brauer–Clifford superalgebras introduced in [Reference Benkart, Guay, Jung, Kang and WilcoxBGJ+16] (see Corollary 4.9). However, the category contains more information, since it also involves morphism spaces between different objects. Consideration of the entire monoidal category , as opposed to the more traditional approach (e.g. taken in [Reference Benkart, Guay, Jung, Kang and WilcoxBGJ+16]) of treating the endomorphism superalgebras individually, as associative superalgebras, offers significant advantages. In particular, the added structure of cups and caps, arising from the duality between V and $V^*$ , allows us to translate between general morphism spaces and ones of the form . This allows us to recover some of the results of [Reference Benkart, Guay, Jung, Kang and WilcoxBGJ+16] with simplified arguments.
The additive Karoubi envelope of should be viewed as an interpolating category $\mathrm {\underline {Re}p}(U_q(\mathfrak {q}))$ for the quantum isomeric superalgebras. However, since the supercategory does not depend on n, we have an ‘interpolating’ category without a dimension parameter. The same is true for the additive Karoubi envelope $\mathrm {\underline {Re}p}(Q)$ of , which is the isomeric ‘interpolating’ category in the nonquantum setting. Of course, the kernel of the incarnation superfunctor (1.1) does depend on n (see, for example, Theorem 4.4). The semisimplification of , which is the quotient by the tensor ideal of negligible morphisms, is the trivial supercategory with one object, since the identity morphisms of the generating objects $\uparrow $ and $\downarrow $ are negligible. Similar phenomena occur for the periplectic Lie superalgebras. For a discussion of the Deligne interpolating category in that case, we refer the reader to [Reference SerganovaSer14, Section 4.5], [Reference Kujawa and TharpKT17, Section 5], [Reference Coulembier and EhrigCE21, Section 3.1] and [Reference EntovaAizenbud and SerganovaEAS21].
In the second half of the current paper, we define and study the quantum affine isomeric supercategory
. One important difference between the quantum isomeric supercategory and the HOMFLYPT skein category is that the category
is not braided. This corresponds to the fact that $U_q(\mathfrak {q}_n)$ is not a quasitriangular Hopf superalgebra. Diagrammatically, this is manifested in the fact (see Lemma 2.5) that
That is, Clifford tokens slide over crossings but not under them. Since
is not braided, the usual affinisation procedure, which corresponds to considering string diagrams on a cylinder (see [Reference Mousaaid and SavageMS21]) is not appropriate. Instead, we must develop a new approach. To pass from
to the affine version
, we adjoin an odd morphism
satisfying
among other relations (see Definition 6.1). This procedure of odd affinisation (see Remark 7.6) makes apparent a symmetry of
that interchanges
and
and flips all crossings. There does not seem to be any analogous symmetry of the affine HOMFLYPT skein category. The supercategory
is naturally a subsupercategory of
(Proposition 7.7).
We define, in Theorem 8.1, a monoidal superfunctor
which we call the affine action superfunctor. As for the case of the affine HOMFLYPT skein category, the superfunctor (1.2) contains information about supernatural transformations between translation superfunctors acting on $U_q(\mathfrak {q}_n\text {smod})$ . However, in the HOMFLYPT setting, the affine action comes from the braiding in the category. Intuitively, it arises from an action of on corresponding to placing string diagrams representing morphisms of inside the cylinders representing morphisms of . We refer the reader to [Reference Mousaaid and SavageMS21, Section 3] for further details of this interpretation. The fact that is not braided means that we cannot simply apply this general framework, and we must formulate new methods. As a replacement, we develop in Section 5 the concept of a chiral braiding, which is similar to a braiding but is only natural in one argument.
The endomorphism superalgebras are related to the affine Hecke–Clifford superalgebras introduced in [Reference Jones and NazarovJN99], where they are called affine Sergeev algebras (see Section 7). These have played an important role in representation theory and categorification (see, for example, [Reference Brundan and KleshchevBK01]). However, our presentation of these superalgebras is different from the original one appearing in [Reference Jones and NazarovJN99]. There, the affine Hecke–Clifford superalgebra is obtained from the Hecke–Clifford superalgebra by adding a set of pairwisecommuting even elements. In our presentation, we add pairwisesupercommuting odd elements, corresponding to the odd generator appearing on various strands. While the translation between the two presentations is straightforward, the new approach yields a simpler description of the affine Hecke–Clifford superalgebras with an obvious symmetry, corresponding to the symmetry of that interchanges and and flips crossings. The more general endomorphism superalgebras are affine versions of quantum walled Brauer–Clifford superalgebras which do not seem to have appeared in the literature.
As a final application of our approach to the representation theory of the quantum isomeric superalgebra, we use the affine action superfunctor (1.2) to compute an infinite sequence of elements (8.7) in the centre of $U_q(\mathfrak {q}_n)$ . These elements arise from ‘bubbles’ in , which are closed diagrams corresponding to endomorphisms of the unit object. We expect these elements will be useful in a computation of the centre of $U_q(\mathfrak {q}_n)$ , which has yet to appear in the literature. Typically, one uses the Harish–Chandra homomorphism to compute centres. This homomorphism has recently been studied for basic classical Lie superalgebras in [Reference Luo, Wang and YeLWY22], but the quantum isomeric case remains open. It is often not difficult to show that the Harish–Chandra homomorphism is injective. The difficulty lies in showing that its image is as large as expected. By analogy with the $U_q(\mathfrak {gl}_n)$ case, we expect that the central elements (8.7) computed here, together with some obviously central elements, generate the centre of $U_q(\mathfrak {q}_n)$ .
Further directions and open problems
The quantum affine isomeric supercategory should be thought of as an isomeric analogue of the affine HOMFLYPT skein category from [Reference BrundanBru17, Section 4]. The latter is the central charge zero special case of the quantum Heisenberg category of [Reference Brundan, Savage and WebsterBSW20]. A suitable modification of the approach of [Reference Brundan, Savage and WebsterBSW20] should lead to the definition of a quantum isomeric Heisenberg supercategory depending on a central charge $k \in \mathbb {Z}$ . Taking $k=0$ would recover . On the other hand, for nonzero k, this supercategory should act on supercategories of supermodules over cyclotomic Hecke–Clifford superalgebras. Furthermore, we expect that one can adapt the categorical comultiplication technique of [Reference Brundan, Savage and WebsterBSW20] to prove a basis theorem, yielding a proof of Conjecture 6.12 (giving a conjectural basis for each morphism space in ) as a special case.
An even more general quantum Frobenius Heisenberg category was defined in [Reference Brundan, Savage and WebsterBSW22]. This is a monoidal supercategory depending on a central charge $k \in \mathbb {Z}$ and a Frobenius superalgebra A. Taking $A = \Bbbk $ recovers the usual quantum Heisenberg category. It should be possible to define a quantum isomeric Frobenius Heisenberg supercategory, such that specialising $A=\Bbbk $ yields the quantum isomeric Heisenberg category.
The quantum webs of type Q introduced in [Reference Brown, Davidson and KujawaBJK20] should be related to a partial idempotent completion of supercategory . It would be interesting to work out this precise connection, and then use it to define affine versions of quantum webs of type Q, based on the supercategory .
Finally, in [Reference Brundan, Comes and KujawaBCK19], the authors studied cyclotomic quotients of the degenerate affine oriented Brauer–Clifford supercategory. It would be natural to investigate the quantum analogue, namely, cyclotomic quotients of . These could also be thought of as isomeric analogues of the central charge zero case of the cyclotomic quotients considered in [Reference Brundan, Savage and WebsterBSW20, Section 9].
Hidden details
For the interested reader, the tex file of the arXiv version of this paper includes hidden details of some straightforward computations and arguments that are omitted in the pdf file. These details can be displayed by switching the details toggle to true in the tex file and recompiling.
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 $z=qq^{1}$ . 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 paritypreserving 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
for homogeneous $a,a' \in A$ , $b,b' \in B$ . For Asupermodules M and N, we let $\operatorname {\mathrm {Hom}}_A(M,N)$ denote the $\Bbbk $ supermodule of all (i.e. not necessarily paritypreserving) Alinear 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
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 paritypreserving. A superfunctor between supercategories induces a paritypreserving 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:
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:
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
subject to the relations
In the above, we have used left crossings and a right cap defined by
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
It follows that we have left and down analogues of the skein relation (2.8):
We then define the other right crossing so that the right skein relation also holds:
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:
Proof. This follows from Remark 2.2, since all these relations holds in .
We define
It follows that
Lemma 2.4. The following relations hold in for all orientations of the strands:
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 :
Proof. We have
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:
Remark 2.8. In many instances, when we wish to number strands in diagrams, it is most natural to number them from right to left. For instance, we will do so when discussing Jucys–Murphy elements in Section 7. However, at other times, when we want to discuss relationships to superalgebras appearing in the literature, it is useful to number strands from left to right to better match conventions in other papers. The isomorphism $\Omega _\leftrightarrow $ allows us to move back and forth between these two conventions.
It follows from Propositions 2.6 and 2.7 that is strictly pivotal, with duality superfunctor
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).
Lemma 2.9. When $z = 0$ , reversing orientation of strands gives an isomorphism of monoidal supercategories from to the oriented Brauer–Clifford supercategory of [Reference Brundan, Comes and KujawaBCK19, Definition 3.2].
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
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 selfintersections 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.
Proposition 2.11. For any objects $X,Y$ of , the set $B_\bullet (X,Y)$ spans the $\Bbbk $ supermodule over $\Bbbk $ .
Proof. Let X and Y be two objects of . Using (2.20) and (2.21), Clifford tokens can be moved near the termini of strings. Next, using (2.9) and (2.19), we can reduce the number of Clifford tokens to at most one on each string. Then, since all the relations in the HOMFLYPT skein category hold (see Remark 2.2), we have a straightening algorithm to rewrite any diagram representing a morphism $X \to Y$ as a $\Bbbk $ linear combination of the ones in $B_\bullet (X,Y)$ . Here, we also use (2.17) and (2.20) to see that any string diagram with a closed component is equal to zero.
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
satisfying the following relations (for $i,j$ in the allowable range)
We define $\mathrm {BC}_{r,0}(z)$ to be the associative superalgebra generated by even elements $t_1,\dotsc ,t_{r1}$ 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
A straightforward computation shows that we have an isomorphism of superalgebras
We will soon see a diagrammatic interpretation of this isomorphism.
The superalgebra
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
Proof. It is a straightforward computation to verify that the given map is welldefined, that is that it respects the relations in Definition 2.12. Since all elements of $B_\bullet (\uparrow ^{\otimes r} \otimes \downarrow ^{\otimes s}, \uparrow ^{\otimes r} \otimes \downarrow ^{\otimes s})$ can clearly be written as compositions of the given images of the generators of $\mathrm {BC}_{r,s}(z)$ , it follows from Proposition 2.11 that the map is also surjective.
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
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
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
Using this basis, we will identify V with $\Bbbk ^{nn}$ as $\Bbbk $ supermodules and $\operatorname {\mathrm {End}}_\Bbbk (V)$ with $\operatorname {\mathrm {Mat}}_{nn}(\Bbbk )$ as associative superalgebras. Let $E_{ij} \in \operatorname {\mathrm {Mat}}_{nn}(\Bbbk )$ 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 $\mathfrak {gl}_{nn}$ is equal to $\operatorname {\mathrm {End}}_\Bbbk (V)$ as a $\Bbbk $ supermodule, with bracket given by the supercommutator
Let
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 $\mathfrak {gl}_{nn}$ :
The elements
give a $\Bbbk $ basis of $\mathfrak {q}_n$ . The parities of these elements are indicated by their superscripts.
Define
by
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
One can also verify that $\Theta $ satisfies the Yang–Baxter equation:
where
It follows from (3.4) that
and so
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 $\operatorname {\mathrm {Mat}}_{nn}(\Bbbk )$ . In addition, $\Theta $ is invertible with
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
where
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
(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
and antipode S determined by
Note that, viewing L as an element of $\operatorname {\mathrm {Mat}}_{nn}(U_q)$ , 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 finitedimensional $U_q$ supermodules is naturally a rigid monoidal supercategory.
For $\Bbbk $ supermodules U and W, define
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
Consider the opposite comultiplication
Lemma 3.2. We have
Proof. We have
Since
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
and
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
and
The result follows.
Corollary 3.4.

a. We have $u_{aa} u_{kl} = q^{\delta _{a,l}  \delta _{a,k}} u_{kl} u_{aa}$ for all $a \in \{1,2,\dotsc ,n\}$ and $k,l \in \mathtt {I}$ , $k \le l$ .

b. The element $u_{11} u_{22} \dotsm u_{nn}$ lies in the centre of $U_q$ .
Proof.

a. Setting $i=j=a$ in (3.18) gives
$$\begin{align*}q^{\varphi(a,l)} u_{aa} u_{kl} + z \delta_{k \le a < l} u_{al} u_{ka} = q^{\varphi(a,k)} u_{kl} u_{aa} + z \delta_{k < a \le l} u_{al} u_{ka}, \end{align*}$$which implies$$\begin{align*}(q^{\varphi(a,l)}  z \delta_{k<a=l}) u_{aa} u_{kl} = (q^{\varphi(a,k)}  z \delta_{k=a<l}) u_{kl} u_{aa}. \end{align*}$$When $k=l$ , this becomes $u_{aa} u_{kl} = u_{kl} u_{aa}$ , as desired. When $k=l$ , it becomes$$\begin{align*}(q^{\delta_{a,l}}  z \delta_{a,l}) u_{aa} u_{kl} = q^{\delta_{a,l}} u_{kl} u_{aa} \implies u_{aa} u_{kl} = u_{kl} u_{aa}, \end{align*}$$as desired.Now suppose $k \ne l$ . If $a \notin \{k,l\}$ , then
$$\begin{align*}u_{aa} u_{kl} = q^{\varphi(a,k)\varphi(a,l)} u_{kl} u_{aa} = q^{\delta_{a,l}\delta_{a,k}} u_{kl} u_{aa}. \end{align*}$$If $a=k$ , then$$\begin{align*}u_{aa} u_{kl} = (qz) u_{kl} u_{aa} = q^{1} u_{kl} u_{aa}. \end{align*}$$Finally, if $a=l$ , then$$\begin{align*}(qz) u_{aa} u_{kl} = u_{kl} u_{aa} \implies u_{aa} u_{kl} = q u_{kl} u_{aa}. \end{align*}$$ 
b. It follows from (a) that $u_{11} u_{22} \dotsm u_{nn}$ commutes with all $u_{kl}$ , $k \le l$ .
Lemma 3.5. As a unital associative superalgebra, $U_q$ is generated by
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
Thus, it suffices to show that
for $1 \le a \le n1$ and $1 \le b \le n$ . We prove this by induction on a.
First note that, for $1 \le a \le n1$ , taking $i=a$ , $j=a$ , $k=a1$ , $l=a$ in (3.18) gives
Taking $i=a$ , $j=k=a$ and $l=a+1$ in (3.18) gives
Taking $i=a1$ , $j=k=a$ , $l=a+1$ in (3.18) gives
So we have
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 $1 \le c \le n2$ , and that (3.20) holds for $1 \le a, b \le c$ . Then, replacing a by c in (3.23) shows that $u_{c1,c+1} \in \tilde {U}_q$ . Replacing a by $c+1$ in (3.21) and (3.22) then shows that $u_{c2,c+1}, u_{c1,c+2} \in \tilde {U}_q$ . Hence, (3.20) holds for $1 \le a,b \le c+1$ . Thus, by induction, (3.20) holds for $1 \le a,b \le n1$ . Finally, taking $a=n1$ 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.
Proposition 3.6. The square of the antipode of $U_q$ is given by $S^2(u_{ij}) = q^{2j2i} u_{ij}$ , $i,j \in \mathtt {I}$ .
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 $2j2i$ . Thus, the map $u_{ij} \mapsto q^{2j2i} u_{ij}$ 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
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
By Corollary 3.4(a) and (3.17), we have
for all $a \in \{1,2,\dotsc ,n\}$ and $k,l \in \mathtt {I}$ , $k \le l$ . In particular,
for all $a \in \{1,2,\dotsc ,n\}$ and $i,k \in \mathtt {I}$ . Thus,
The proof that $S^2(u_{a1,a}) = q^{2} u_{a1,a}$ is similar.
Next, we have
Finally, (3.24) implies that
Thus
Corollary 3.7. The antipode S is invertible and
It follows from (3.6) and (3.8) that
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
We have the natural evaluation map
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
Then we have the coevaluation map
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 $ .
Lemma 3.8. The map $J \in \operatorname {\mathrm {End}}_\Bbbk (V)$ is an odd isomorphism of $U_q$ supermodules.
Proof. It follows from (3.3) and (3.5) that
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 $z = qq^{1}$ . Recalling the definition (3.4) of $\Theta $ , define
Thus
Therefore, we have
and
Lemma 4.1. The map T is an isomorphism of $U_q$ supermodules.
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
Composing on the left with $\operatorname {\mathrm {flip}}$ , it suffices to show that
This is equivalent to showing that
Since $u_{ij}$ acts on V as $\Theta _{ij}$ , this is equivalent, using (3.12) and (3.16), to
But this is precisely the Yang–Baxter equation (3.6).
Remark 4.2. The map T is a special case of a map $T_{MV}$ to be introduced in (5.6), where $M=V$ . Then Lemma 4.1 will be a special case of Proposition 5.4.
For the computations to follow, it is useful to note that, for $i,j \in \mathtt {I}$ , $i<j$ , we have
Theorem 4.3. For each $n \in \mathbb {N}$ , there exists a unique monoidal superfunctor
, such that
Furthermore,
,
and
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:
the right cup and cap
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 lefthand side is mapped by ${\mathbf {F}}_n$ to the composite
Similarly, the righthand side is mapped by ${\mathbf {F}}_n$ to the composite
Since
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
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 topleft 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 bottomleft horizontal map is given analogously, using
. The right horizontal maps are the usual isomorphisms that hold in any rigid monoidal supercategory. In particular, the topright 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
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
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
We can now improve Proposition 2.11.
Theorem 4.5. For any objects $X,Y$ of , the $\Bbbk $ supermodule is free with basis $B_\bullet (X,Y)$ .
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 $z=qq^{1}$ , 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 $qq^{1}$ . 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 $ .
Remark 4.6. Taking $z=0$ in Theorem 4.5 recovers the basis theorem [Reference Brundan, Comes and KujawaBCK19, Theorem 3.4] for the oriented Brauer–Clifford supercategory (see Lemma 2.9 and Remark 2.10).
Corollary 4.7. Let $X = X_1 \otimes \dotsb \otimes X_r$ and $Y = Y_1 \otimes \dotsb \otimes Y_s$ be objects of for $X_i,Y_j \in \{\uparrow , \downarrow \}$ . Then if the cardinalities of the sets (2.24) are not equal. If they are equal (which implies that $r+s$ is even), then is a free $\Bbbk $ supermodule with even and odd parts each of rank $k!2^{k1}$ , where $k=\frac {r+s}{2}$ is the number of strings in the elements of $B(X,Y)$ .
Proof. This follows immediately from Theorem 4.5. If the sets (2.24) have the same cardinality, then the number of $(X,Y)$ matchings is $k!$ and there are $2^k$ ways of adding Clifford tokens to the strings, half of which yield even string diagrams.
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 $\mathrm {BC}_{r,s}(z)$ .
Corollary 4.9. Let $X = X_1 \otimes \dotsb \otimes X_m$ be an object of for $X_i \in \{\uparrow ,\downarrow \}$ . If r is the number of $i \in \{1,\dotsc ,m\}$ , such that $X_i =\, \uparrow $ , then as associative superalgebras.
Proof. It follows from the third and fourth equalities in (2.7) that is an isomorphism with inverse . Hence, $X \cong \, \uparrow ^{\otimes r} \otimes \downarrow ^{\otimes (mr)}$ , and so the result follows from Corollary 4.8.
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
where
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 :
Proof. We compute
The proofs of the remaining equalities are analogous.
Proposition 5.3. In , we have
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 redblack 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
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 $z = qq^{1}$ . Recall the definition of L from (3.11). For a finitedimensional $U_q$ supermodule M, we let $\rho _M \colon U_q \to \operatorname {\mathrm {End}}_\Bbbk (M)$ denote the corresponding representation. We then define
In particular, we have $L_V = \Theta $ (see (3.26)).
Proposition 5.4. For any $M \in U_q\text {smod}$ , the map
is an isomorphism of $U_q$ supermodules. Furthermore, for all $f \in \operatorname {\mathrm {Hom}}_{U_q}(M,N)$ , we have
Proof. It is clear that $T_{MV}$ is invertible, with
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
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
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
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
It follows that
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,
,
Proof. By Theorem 4.3, to show that ${\mathbf {F}}_n^M$ is welldefined, 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 lefthand side to the map $M \otimes V^{\otimes 2} \to V^{\otimes 2} \otimes M$ given by
On the other hand, ${\mathbf {F}}_n^M$ sends the righthand side to the map
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
Lemma 5.7. For any $U_q$ supermodules M and N, we have
Proof. We have
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
On a morphism
, ${\mathbf {F}}^\bullet _n(f)$ is the natural transformation ${\mathbf {F}}^\bullet _n(X) \to {\mathbf {F}}^\bullet _n(Y)$ whose Mcomponent, for $M \in U_q\text {smod}$ , is
Proof. It follows from (5.13) that the given definition is natural in M.
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
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
It follows that
Lemma 6.3. The following relations hold in :
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
Remark 6.4. Note that, while is isomorphic to the oriented Brauer–Clifford supercategory of [Reference Brundan, Comes and KujawaBCK19, Definition 3.2], as described in Lemma 2.9, the supercategory does not reduce to the definition [Reference Brundan, Comes and KujawaBCK19, Definition 3.2] of the degenerate affine oriented Brauer–Clifford supercategory when $z=0$ . This is analogous to the fact that the degenerate affine Hecke algebra of type A is not simply the $q=1$ specialisation of the affine Hecke algebra of type A.
Define, for $k \in \mathbb {Z}$ ,
Note that both morphisms in (6.6) are of parity $k \pmod 2$ . We then define, for $k \in \mathbb {Z}$ ,
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 $k \in \mathbb {Z}$ :
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 :
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 :
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
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
where $h_0=e_0=1$ . The map $\beta $ sends the lefthand 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 $1 \le r \le k1$ ,
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.
Proposition 6.11. For any objects $X,Y$ of , the set spans the morphism space as a right $\text {Sym}$ supermodule.
Proof. Since this type of argument is standard in categorical representation theory, we only give a sketch of the proof (see also the proof of Proposition 2.11). We have the Reidemeister relations, a skein relation and bubble and zebra sliding relations. These allow diagrams for morphisms in to be transformed in a way similar to the way oriented tangles are simplified in skein categories. Hence, there is a straightening algorithm to rewrite any diagram representing a morphism $X \to Y$ as a linear combination of the ones in .
Conjecture 6.12. For any objects $X,Y$ of , the morphism space is a free right $\text {Sym}$ supermodule with basis .
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
,
In addition,
is the natural transformation $\uparrow \otimes  \to \ \uparrow \otimes $ whose Xcomponent,
, is
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 $t_1,\dotsc ,t_{r1}$ and odd elements $\pi _1,\dotsc ,\pi _r,\varpi _1,\dotsc ,\varpi _r$ ,
satisfying the following relations (for $i,j$ in the allowable range):
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 $t_i^{1} = t_iz$ gives
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 pairwisecommuting invertible even elements $x_1,\dotsc ,x_r$ , subject to the following relations (for $i,j$ in the allowable range):
The isomorphism is given by
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 welldefined homomorphism of superalgebras. It is invertible, with inverse
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
Proof. It is a straightforward computation to verify that the given map is welldefined, that is that it respects the relations in Definition 7.1.
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
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
For $1 \le i \le r$ , define the odd Jucys–Murphy elements
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.
Lemma 7.4. For all $1 \le i \le r$ , we have $J_i = \pi _i J_i^{\text {odd}}$ .
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 $J_{i1} =  \pi _{i1} J_{i1}^{\text {odd}}$ . First note that
Thus, we have
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 ith 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 ith strand from the right.
Remark 7.6. Recall that the ith Jucys–Murphy element in the Iwahori–Hecke algebra of type A is given, in terms of string diagrams, by
where it is the ith 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 .
Proposition 7.7. The superfunctor that is the identity on objects and sends each generating morphism in to the morphism in depicted by the same string diagram is faithful.
Proof. It is straightforward to verify that is left inverse to the superfunctor in the statement of the proposition.
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 $z=qq^{1}$ .
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
Thus, the Mcomponent 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
Proof. The proof is almost identical to that of Proposition 6.13; one merely replaces the thick black strand there (representing the identity morphism $1_X$ ) with a thick red strand.
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
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
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
be the centre of $U_q$ . Evaluation on the identity element of the regular representation defines a canonical superalgebra isomorphism
where $\operatorname {\mathrm {id}}_{\mathcal {C}}$ denotes the identity endosuperfunctor of a supercategory $\mathcal {C}$ . Consider the composite superalgebra homomorphism
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 $k> 0$ ,
where there are a total of $2k$ closed Clifford tokens on alternating sides of the red strand.
For $i,j \in \mathtt {I}$ , define
Note that $y_{ij}$ is of parity $p(i,j)$ . Next, for $i,j \in \mathtt {I}$ , and $m> 0$ , define
These are isomeric analogues of the elements defined in [Reference Brundan, Savage and WebsterBSW20, (5.15)].
Lemma 8.2. We have
where we interpret the righthand side as a natural transformation whose Mcomponent, for a $U_q$ supermodule M with corresponding representation $\rho _M$ , is $\sum _{i,k \in I} \rho _M (y_{ij}) \otimes E_{ij}$ .
Corollary 8.3. We have
Proposition 8.4. For $m>0$ , the image of under (8.2) is
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)).
Acknowledgements
This research was supported by Discovery Grant RGPIN201703854 from the Natural Sciences and Engineering Research Council of Canada. The author would like to thank Jon Brundan and Dimitar Grantcharov for helpful conversations, and the referee for useful comments.
Competing Interests
The author(s) declare none.