THE QUANTUM ISOMERIC SUPERCATEGORY

Abstract We introduce the quantum isomeric supercategory and the quantum affine isomeric supercategory. These diagrammatically defined supercategories, which can be viewed as isomeric analogues of the HOMFLYPT skein category and its affinisation, provide powerful categorical tools for studying the representation theory of the quantum isomeric superalgebras (commonly known as quantum queer superalgebras).


Introduction
One of the most fundamental facts in representation theory is Schur's lemma, which implies that if V is a finite-dimensional simple module over an associative k-algebra A, where k is an algebraically closed field, then End A (V ) ∼ = k.On the other hand, if A is an associative k-super algebra, then there are two possibilities: we can have End A (V ) ∼ = k or we can have that 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 gl n has two natural analogues in the super setting: the general linear Lie superalgebra gl m|n and the isomeric Lie superalgebra q n .(Following [NSS22], 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 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 low-dimensional topology, representation theory, and categorification.
Before describing our results, we begin with an overview of the situation for the better-understood case of gl m|n .The finite-dimensional complex representation theory of gl m|n is controlled by the oriented Brauer category OB(m − n).More precisely, OB(t) is a diagrammatic symmetric monoidal category depending on a dimension parameter t.When t = m − n, there is a full monoidal functor OB(m − n) → gl m|n -smod to the category of gl m|n -supermodules, sending the generating object of OB(m − n) to the natural supermodule of gl m|n ; see [CW12,§8.3].(Throughout this introduction we work with finitedimensional supermodules.)The additive Karoubi envelope (i.e. the idempotent completion of the additive envelope) of OB(t) is Deligne's interpolating category Rep(GL t ).Similar statements hold in the orthosymplectic case, where OB(t) is replaced by the Brauer category (no longer oriented, due to the fact that the natural supermodule is self-dual); see [LZ17,Th. 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 gl m|n , this action by tensoring can be enlarged to a monoidal functor AOB(m − n) → End (gl m|n -smod), where AOB(t) is the affine oriented Brauer category of [BCNR17] and End (C) denotes the monoidal category of endofunctors of a category C. The category AOB(t) allows one to study natural transformations between translation functors, provides tools to study cyclotomic Hecke algebras, and yields natural elements in the center of U (gl m|n ).Again, a similar picture exists for the orthosymplectic Lie superalgebras, where AOB(t) is replaced by the affine Brauer category of [RS19].
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 OS(z, t), originally introduced in [Tur89, §5.2],where it was called the Hecke category.The affine version AOS(z, t) was introduced in [Bru17], and there are monoidal functors OS(q − q −1 , q n ) → U q (gl n )-mod, AOS(q − q −1 , q n ) → End (U q (gl n )-mod), with many of the properties mentioned above for the non-quantum case.We expect that these functors can be generalized to the super setting of U q (gl m|n ).The generalization of the first functor should follow from the results [LZZ20], and then the affine case follows from the general affinization procedure of [MS21].Once again, analogues exist in the orthosymplectic case, where the relevant categories are the Kauffman skein category, together with its affine analogue introduced in [GRS22].
The isomeric analogues of the oriented Brauer category and its affine version are the oriented Brauer-Clifford supercategory OBC and the degenerate affine oriented Brauer-Clifford supercategory AOBC introduced in [BCK19].In analogy with the above, one has monoidal superfunctors OBC → q n -smod, AOBC → SEnd (q n -smod), where SEnd (C) denotes the monoidal supercategory of endosuperfunctors of a supercategory C.
The need to move to the setting of super categories here arises from the super version of Schur's lemma mentioned earlier.There is an odd endomorphism of the natural representation of q n that corresponds to an odd morphism in OBC and AOBC .(The category of vector superspaces, with parity preserving linear maps, is a monoidal category, with no need to introduce the notion of a monoidal super category.)Note also the absence of the parameter t that appears for the oriented Brauer category.This is because the natural representation of q n always has superdimension zero.
In the current paper we develop analogues of the above results for the quantum isomeric superalgebra U q (q n ).As we will explain below, this case requires several new techniques.We begin by defining the quantum isomeric supercategory Q (z) depending on a parameter z in the ground ring.This is a strict monoidal supercategory generated by two objects, ↑ and ↓, and morphisms subject to certain relations; see Definition 2.1.The supercategory Q (z) should be viewed as a quantization of OBC .In particular, Q (0) is isomorphic to OBC ; see Lemma 2.9.From the definition of Q (z), 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 tangle-like 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 (1.1) Q (q − q −1 ) → U q (q n )-smod, 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 Q (q − q −1 ) 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 (q n )-invariants.
The endomorphism superalgebras End Q (z) (↑ ⊗r ) are Hecke-Clifford superalgebras, which appear in quantum Sergeev duality (the quantum isomeric analogue of Schur-Weyl duality).More generally, the endomorphism superalgebras in Q (z) are isomorphic to the quantum walled Brauer-Clifford superalgebras introduced in [BGJ + 16]; see Corollary 4.9.However, the category Q (z) contains more information, since it also involves morphism spaces between different objects.Consideration of the entire monoidal category Q (z), as opposed to the more traditional approach (e.g.taken in [BGJ + 16]) of treating the endomorphisms 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 End Q (z) (↑ ⊗r ).This allows us to recover some of the results of [BGJ + 16] with simplified arguments.
The additive Karoubi envelope of Q (z) should be viewed as an interpolating category Rep(U q (q)) for the quantum isomeric superalgebras.However, since the supercategory Q (z) does not depend on n, we have an "interpolating" category without a dimension parameter.The same is true for the additive Karoubi envelope Rep(Q) of OBC , which is the isomeric "interpolating" category in the non-quantum setting.Of course, the kernel of the incarnation superfunctor (1.1) does depend on n; see, for example, Theorem 4.4.The semisimplification of Q (z), 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 ↑ and ↓ 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 [Ser14, §4.5], [KT17, §5], [CE21, §3.1], and [EAS21].
In the second half of the current paper, we define and study the quantum affine isomeric supercategory AQ (z).One important difference between the quantum isomeric supercategory and the HOMFLYPT skein category is that the category Q (z) is not braided.This corresponds to the fact that U q (q n ) is not a quasi-triangular 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 Q (z) is not braided, the usual affinization procedure, which corresponds to considering string diagrams on a cylinder (see [MS21]) is not appropriate.Instead, we must develop a new approach.To pass from Q (z) to the affine version AQ (z), we adjoin an odd morphism : ↑ → ↑ satisfying = , among other relations; see Definition 6.1.This procedure of odd affinization (see Remark 7.6) makes apparent a symmetry of AQ (z) that interchanges and and flips all crossings.There does not seem to be any analogous symmetry of the affine HOMFLYPT skein category.The supercategory 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 (q n -smod).However, in the HOMFLYPT setting, the affine action comes from the braiding in the category.Intuitively, it arises from an action of AOS(z, t) on OS(z, t) corresponding to placing string diagrams representing morphisms of OS(z, t) inside the cylinders representing morphisms of AOS(z, t).We refer the reader to [MS21, §3] for further details of this interpretation.The fact that Q (z) 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 End AQ (z) (↑ ⊗r ) are related to the affine Hecke-Clifford superalgebras introduced in [JN99], where they are called affine Sergeev algebras; see Section 7.These have played an important role in representation theory and categorification; see, for example, [BK01].However, our presentation of these superalgebras is different from the original one appearing in [JN99].There, the affine Hecke-Clifford superalgebra is obtained from the Hecke-Clifford superalgebra by adding a set of pairwise-commuting 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 AQ (z) that interchanges and and flips crossings.The more general endomorphism superalgebras End AQ (z) (↑ ⊗r ⊗ ↓ ⊗s ) 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 center of U q (q n ).These elements arise from "bubbles" in AQ (z), which are closed diagrams corresponding to endomorphisms of the unit object.We expect these elements will be useful in a computation of the center of U q (q n ), which has yet to appear in the literature.Typically one uses the Harish-Chandra homomorphism to compute centers.This homomorphism has recently been studied for basic classical Lie superalgebras in [LWY22], 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 (gl n ) case, we expect that the central elements (8.7) computed here, together with some obviously central elements, generate the center of U q (q n ).
Further directions and open problems.The quantum affine isomeric supercategory AQ (z) should be thought of as an isomeric analogue of the affine HOMFLYPT skein category from [Bru17,§4].The latter is the central charge zero special case of the quantum Heisenberg category of [BSW20].A suitable modification of the approach of [BSW20] should lead to the definition of a quantum isomeric Heisenberg supercategory depending on a central charge k ∈ Z. Taking k = 0 would recover AQ (z).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 [BSW20] to prove a basis theorem, yielding a proof of Conjecture 6.12 (giving a conjectural basis for each morphism space in AQ (z)) as a special case.
An even more general quantum Frobenius Heisenberg category was defined in [BSW22].This is a monoidal supercategory depending on a central charge k ∈ Z and a Frobenius superalgebra A. Taking A = k recovers the usual quantum Heisenberg category.It should be possible to define a quantum isomeric Frobenius Heisenberg supercategory such that specializing A = k yields the quantum isomeric Heisenberg category.
The quantum webs of type Q introduced in [BJK20] should be related to a partial idempotent completion of supercategory Q (z).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 AQ (z).
Finally, in [BCK19], 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 AQ (z).These could also be thought of as isomeric analogues of the central charge zero case of the cyclotomic quotients considered in [BSW20,§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.
Acknowledgements.This research was supported by Discovery Grant RGPIN-2017-03854 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.

The quantum isomeric supercategory
Throughout the paper we work over a commutative ring k whose characteristic is not equal to two, and we fix an element z ∈ k.Statements about abstract categories will typically be at this level of generality.When making statements involving supermodules over the quantum isomeric superalgebra, we will specialize to k = C(q) and z = q − q −1 .We let N denote the set of nonnegative integers.
All vector spaces, algebras, categories, and functors will be assumed to be linear over k unless otherwise specified.Almost everything in the paper will be enriched over the category SVec of vector superspaces with parity-preserving morphisms.We write 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 ⊗ B is defined by for homogeneous a, a ′ ∈ A, b, b ′ ∈ B. For A-supermodules M and N , we let Hom A (M, N ) denote the k-supermodule of all (i.e.not necessarily parity-preserving) A-linear maps from M to N .The opposite superalgebra A op is a copy {a op : a ∈ A} of the vector superspace A with multiplication defined from (2.2) a op b op := (−1) āb (ba) op .
A superalgebra homomorphism A → B op is equivalent to an antihomomorphism of superalgebras A → B. When viewing it in this way, we will often omit the superscript 'op' on elements of B.
Throughout this paper we will work with strict monoidal supercategories, in the sense of [BE17].We summarize here a few crucial properties that play an important role in the present paper.A supercategory means a category enriched in SVec.Thus, its morphism spaces are vector superspaces and composition is parity-preserving.A superfunctor between supercategories induces a paritypreserving linear map between morphism superspaces.For superfunctors F, G : A → B, a supernatural transformation α : Note when r is odd that α is not a natural transformation in the usual sense due to the sign.A supernatural transformation α : F ⇒ G is of the form α = α 0 + α 1 , with each α 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 ½ 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 ⊗ g (resp.vertical composition f • g) of morphisms f and g diagrammatically by drawing f to the left of g (resp. 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 A is a supercategory, the category SEnd (A) of superfunctors A → A and supernatural transformations is a strict monoidal supercategory.The notation A op denotes the opposite supercategory and, if A is also monoidal, A 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 Q (z) to be the strict monoidal supercategory generated by objects ↑ and ↓ and morphisms subject to the relations 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: ( This category was first introduced in [Tur89, §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 OS(z, t), which comes oriented skein, from [Bru17].It follows from [Bru17, Th. 1.1], which gives a presentation of OS(z, t), that all of the relations in OS(z, 1) hold in Q (z).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 [Bru17, Th. 1.1] with t = 1.Thus, by that result, all relations in OS(z, 1) hold in Q (z) after reflecting in the vertical axis and flipping crossings.But OS(z, 1) is invariant under this transformation, and so all its relations hold in Q (z).In fact, Q (z) is the strict monoidal supercategory obtained from OS(z, 1) 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 t = (2.8) Hence t = ±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 Q (z) depends on only one parameter z ∈ k.
Lemma 2.3.The following relations hold in Q (z) for all orientations of the strands: Proof.This follows from Remark 2.2, since all these relations holds in OS(z, 1).
It follows that Lemma 2.4.The following relations hold in Q (z) 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 (2.12) = = (2.11)
So the last equality in (2.20) holds for both orientations of the strand.We also have (2.12)
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 Q (z): Proof.We have The proof of the second relation is analogous.
We now describe several symmetries of the category Q (z).First note that we have an isomorphism of monoidal supercategories that is the identity objects and, on morphisms, multiplies all crossings by −1.
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 Ω ↔ allows us to move back and forth between these two conventions.
It follows from Propositions 2.6 and 2.7 that Q (z) is strictly pivotal, with duality superfunctor (2.23) defined by rotating diagrams through 180°and multiplying by (−1) ( 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 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 [BCK19, Def.3.2] with the orientations of strands reversed.The last relation in (2.9) also holds in the oriented Brauer-Clifford supercategory by [BCK19,(3.16)].
Remark 2.10.The reason we need to reverse orientation in Lemma 2.9 is that [BCK19, Def.3.2] includes the relation = , which matches the sign in (2.19), but not in the first relation in (2.9).If √ −1 ∈ k, then we have an automorphism of Q (z) that reverses orientation of strands and multiplies Clifford tokens by √ −1.In this case, there is an isomorphism from Q (0) to the oriented Brauer-Clifford category that multiplies Clifford tokens by √ −1, with no need to reverse orientation.
• 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 Q (z).
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 • (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 Q (z), the set B • (X, Y ) spans the k-supermodule Proof.Let X and Y be two objects of Q (z).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 → Y as a k-linear combination of the ones in B • (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 • (X, Y ) are actually bases of the morphism spaces.
The relations in the first line in Definition 2.12 imply that t i and t * i are invertible, with Then, multiplying both sides of the relation t i π i = π i+1 t i on the left and right by t −1 i gives the relation (2.25) A straightforward computation shows that we have an isomorphism of superalgebras We will soon see a diagrammatic interpretation of this isomorphism.
Proposition 2.13.For r, s ∈ N, we have a surjective homomorphism of associative superalgebras given by Proof.It is a straightforward computation to verify that the given map is well-defined, i.e. that it respects the relations in Definition 2.12.Since all elements of B • (↑ ⊗r ⊗ ↓ ⊗s , ↑ ⊗r ⊗ ↓ ⊗s ) can clearly be written as compositions of the given images of the generators of 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.

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 k = 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, . . ., n} and i, j, k, l to denote elements of I.For i, j ∈ I, we define If C is some condition, we define δ C = 1 if the condition is satisfied, and δ C = 0 otherwise.Then, for i, j ∈ I, δ ij := δ i=j is the usual Kronecker delta.
Let V denote the k-supermodule with basis v i , i ∈ I, where the parity of v i is given by Using this basis, we will identify V with k n|n as k-supermodules, and End k (V ) with Mat n|n (k) as associative superalgebras.Let E ij ∈ Mat n|n (k) 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 gl n|n is equal to End k (V ) as a k-supermodule, with bracket given by the supercommutator where I n is the n × n identity matrix.Multiplication by J is an odd linear automorphism of V , and J 2 = −1.The isomeric Lie superalgebra q n is the Lie superalgebra equal to the centralizer of J in gl n|n : The elements give a k-basis of q n .The parities of these elements are indicated by their superscripts.Define The definition of Θ first appeared in [Ols92,§4], where it is denoted S. We use the notation Θ 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) Θ(J ⊗ 1) = (J ⊗ 1)Θ.
One can also verify that Θ satisfies the Yang-Baxter equation: where and so for all i ∈ I.
When q = 1, we have Θ = 1 ⊗ 1, and so Note that all the second tensor factors appearing in (3.4) are upper triangular elements of Mat n|n (k).In addition, Θ is invertible with (3.9) Note that Θ −1 is obtained from Θ by replacing q by q −1 .
Definition 3.1.The quantum isomeric superalgebra U q = U q (q n ) is the unital associative superalgebra over k generated by elements u ij , i, j ∈ I, i ≤ j, subject to the relations and the last equality in (3.10) takes place in U q ⊗ End k (V ) ⊗2 .The parity of u ij is p(i, j).
The quantum isomeric Lie superalgebra was first defined in [Ols92, Def.4.2].It is a Hopf superalgebra with comultiplication determined by (where the final equality holds since, for i ≤ k ≤ j, we must have p and antipode S determined by Note that, viewing L as an element of 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 -smod of finite-dimensional U q -supermodules is naturally a rigid monoidal supercategory.For k-supermodules U and W , define When U and W are clear from the context, we will sometimes write flip instead of flip U,W .Note that flip V,V = i,j Consider the opposite comultiplication Lemma 3.2.We have Proof.We have The following result is stated in [Ols92, Th. 6.1] without proof.
Proposition 3.3.The quantum isomeric superalgebra U q is isomorphic is the unital associative superalgebra over k generated by the elements u ij , i, j ∈ I, i ≤ j, subject to the relations
Proof.It suffices to prove that the relations (3.18) are equivalent to the second relation in (3.10).
(a) Setting i = j = a in (3.18) gives q ϕ(a,l) u aa u kl + zδ k≤a<l u al u ka = q ϕ(a,k) u kl u aa + zδ k<a≤l u al u ka , which implies (q ϕ(a,l) − zδ k<a=l )u aa u kl = (q ϕ(a,k) − zδ k=a<l )u kl u aa .
When k = l, this becomes u aa u kl = u kl u aa , as desired.When k = −l, it becomes (q δ a,l − zδ a,l )u aa u kl = q −δ a,l u kl u aa =⇒ u aa u kl = u kl u aa , as desired.Now suppose |k| = |l|.If a / ∈ {k, l}, then u aa u kl = q ϕ(a,k)−ϕ(a,l) u kl u aa = q δ a,|l| −δ a,|k| u kl u aa .
If a = k, then u aa u kl = (q − z)u kl u aa = q −1 u kl u aa .Finally, if a = l, then (q − z)u aa u kl = u kl u aa =⇒ u aa u kl = qu kl u aa .
(b) It follows from (a) that u 11 u 22 • • • u nn commutes with all u kl , k ≤ l.
Lemma 3.5.As a unital associative superalgebra, U q is generated by Proof.Let Ũq be the unital associative sub-superalgebra of U q generated by the elements (3.19).It is shown in [GJKK10, Th. 2.1] that U q is generated by Thus it suffices to show that We prove this by induction on a.
So we have 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 2|j|−2|i| u ij , i, j ∈ I.
Proof.It follows from the defining relations that U q is a Z-graded Hopf superalgebra, where we define the degree of u ij to be 2|j| − 2|i|.Thus the map u ij → 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 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 (3.24)u aa u kl = q δ a,|l| −δ a,|k| u kl u aa and u −a,−a u kl = q δ a,|k| −δ a,|l| u kl u −a,−a , for all a ∈ {1, 2, . . ., n} and k, l ∈ I, k ≤ l.In particular, u −a,−a u a,a+1 = qu a,a+1 u −a,−a , u a+1,a+1 u a,a+1 = qu a,a+1 u a+1,a+1 , u ii u kk = u kk u ii , for all a ∈ {1, 2, . . ., n} and i, k ∈ I. Thus, Finally, (3.24) implies that 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 * := Hom k (V, k) is given by We have the natural evaluation map Then we have the coevaluation map It is a straightforward exercise, using only the properties of Hopf superalgebras, to verify that ev and coev are both homomorphisms of U q -supermodules, where k is the trivial U q -supermodule, with action given by the counit ε.
Lemma 3.8.The map J ∈ End k (V ) is an odd isomorphism of U q -supermodules.
Since u ij acts on V as Θ ij , it follows that J is an odd endomorphism of U q -supermodules.Since J 2 = −1, it is an isomorphism.

The incarnation superfunctor
In this section we prove some of our main results.We describe a full monoidal superfunctor from Q (z) to the category of U q -supermodules, give explicit bases for the morphism spaces in Q (z), and identify the endomorphism superalgebras of Q (z) with walled Brauer-Clifford superalgebras.
Until further notice later in this section, we assume that k = C(q) and z = q − q −1 .Recalling the definition (3.4) of Θ, define (4.1) Therefore, we have 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 ⊗ V , we have an equality Composing on the left with flip, it suffices to show that This is equivalent to showing that i,j∈I Since u ij acts on V as Θ 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 M V 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 ∈ I, i < j, we have z k∈I i<k<j (−1) p(k) q 2|k| = q 2|j|−sgn(j) − q 2|i|+sgn(i) , (4.3) z k∈I i<k<j (−1) p(k) q −2|k| = q −2|i|−sgn(i) − q −2|j|+sgn(j) .(4.4) Theorem 4.3.For each n ∈ N, there exists a unique monoidal superfunctor We call 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 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 F n of the various crossings.) Proof of Theorem 4.3.We first show existence, taking F n ( ) = T −1 , F n ( ) = coev, and F n ( ) as in (4.5).We must show that 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 So F n respects the third relation in (2.7).Since V ⊗ V * is finite dimensional, it follows that we also have Hence 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 F n to the composite Similarly, the right-hand side is mapped by F n to the composite and Θ 23 Θ 13 Θ 12 = Θ 12 Θ 13 Θ 23 by (3.6), we see that F n respects the braid relation.Since the superfunctor F n respects the skein relation (2.8).We also have and so F n respects the first relation in (2.9).Next, we compute Thus 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 F n is a monoidal superfunctor as described in the first sentence of the statement of the theorem.Then F n ( ) and F n ( ) are uniquely determined by the fact that they must be inverse to F n ( ) and F n ( ), respectively.Next, suppose that Then, for all k ∈ I, It follows that a ij = δ ij for all i, j ∈ I, and so F n ( ) = coev.
To simplify notation, we will start writing objects of Q (z) as sequences of ↑'s and ↓'s, omitting the ⊗ symbol.For such an object X, we define V X := F n (X), and we let #X denote the length of the sequence.
Theorem 4.4.The superfunctor F n is full for all n ∈ N. Furthermore, the induced map Proof.Our proof is similar to that of [BCK19, Th 4.1], which treats the case z = 0. We need to show that, for all objects X and Y in Q (z), the map (4.11) is surjective, and that it is also injective when #X + #Y ≤ 2n.Suppose that X (respectively, Y ) is a tensor product of r X (respectively, r Y ) copies of ↑ and s X (respectively, s Y ) copies of ↓.Consider the following commutative diagram: The top-left horizontal map is given by composing on the top and bottom of diagrams with to move ↑'s on the top to the left and ↑'s on the bottom to the right.The bottom-left horizontal map is given analogously, using F n ( ).The right horizontal maps are the usual isomorphisms that hold in any rigid monoidal supercategory.In particular, the top-right horizontal map is the C(q)-linear isomorphism given on diagrams by 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 ∈ N, and that it is injective when r + s ≤ 2n.We first consider the case where r = s.Let x = u 11 u 22 • • • u nn be the central element of Corollary 3.4(b).Since u ii acts on V by Θ ii , it follows from (3.7) that x acts on V as multiplication by q.By (3.13), we have ∆(x) = x ⊗ x.Thus x acts on V ⊗r as multiplication by q r .Since x is central, this implies that Hom Uq (V ⊗r , V ⊗s ) = 0. Since we also have Hom Q (z) (↑ r , ↑ s ) = 0 in this case, by Proposition 2.11, the map (4.12) is an isomorphism when r = s.Now suppose r = s, and consider the composite where ϕ is the surjective homomorphism of Proposition 2.13 with s = 0.This composite is precisely the map of [Ols92, Th. 5.2].Surjectivity is asserted, without proof, in [Ols92, Th. 5.3].For the more precise statement, with proof, that this map is also an isomorphism when ) is always surjective, and that it is an isomorphism when r ≤ n, as desired.
Note that #X+#Y is twice the number of strands in any string diagram representing a morphism in Q (z) from X to Y .Thus, Theorem 4.4 asserts that 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 k is an arbitrary commutative ring of characteristic not equal to two, and z ∈ k.
We can now improve Proposition 2.11.
Theorem 4.5.For any objects X, Y of Q (z), the k-supermodule Hom Q (z) (X, Y ) is free with basis Proof.In light of Proposition 2.11, it remains to prove that the elements of B • (X, Y ) are linearly independent.We first prove this when k = C(z).Consider the superalgebra homomorphisms (4.13).By [BGJ + 16, Th. 3.28], the composite F n • ϕ, which is the map denoted ρ r,0 n,q there, is an isomorphism for n ≥ r.Since the map ϕ is independent of n, it follows that ϕ is injective, and hence an isomorphism.Thus, where the last equality is [JN99, Prop.2.1].(The statement in [JN99, Prop.2.1] is over the field C(q), with z = q − q −1 , but the proof is the same over C(z).)Now suppose that X (respectively, Y ) is a tensor product of r X (respectively, r Y ) copies of ↑ and s X (respectively, s Y ) copies of ↓.As in the proof of Theorem 4.4, we have a linear isomorphism This dimension is equal to the number of elements of B • (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 • (X, Y ) is a basis for End Q (z) (↑ r ).This completes the proof of Theorem 4.5 for k = C(z).
To complete the proof over more general base rings, note that C(q) is a free Z[z]-module, with z acting as q − q −1 .Thus, any linear dependence relation over Z[z] yields a linear dependence relation over C(q) after extending scalars.Therefore, it follows from the above that the elements of B • (↑ r , ↑ r ) are a basis over Z[z] and hence, by extension of scalars, over any commutative ring k of characteristic not equal to two and z ∈ k.
Then Hom Q (z) (X, Y ) = 0 if the cardinalities of the sets (2.24) are not equal.If they are equal (which implies that r + s is even), then Hom Q (z) (X, Y ) is a free k-supermodule with even and odd parts each of rank k!2 k−1 , where k = 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 BC r,s (z) Proof.By Proposition 2.13, the map is surjective.When k = C(q), one can then conclude that it is an isomorphism by comparing dimensions.Indeed, by Corollary 4.7 and [BGJ + 16, Cor.3.25], we have dim C(q) BC r,s (z) = (r + s)!2 r+s = dim C(q) End Q (z) (↑ r ↓ s ).

More generally, one can argue as in
Step 1 of the proof of [JK14, Th. 5.1] to show that BC r,s (z) has a spanning set that maps to the basis B It follows that this spanning set is linearly independent, hence a basis of BC r,s (z).
As a special case of Corollary 4.9, we have an isomorphism of associative superalgebras BC r,s (z) ∼ = End Q (z) (↓ s ↑ r ).This recovers [BGJ + 16, Th. 4.19], which describes the walled Brauer-Clifford superalgebras in terms of bead tangle superalgebras.When converting string diagrams representing endomorphisms in Q (z) to the bead tangle diagrams of [BGJ + 16, §4], one should forget the ori- entations of strings, and then rotate diagrams by 180°.This transformation is needed since the convention in [BGJ + 16, §4] for composing diagrams is the opposite of ours.
Remark 4.10.The full monoidal sub-supercategory of U q -smod generated by V and V * is not semisimple.Indeed, it follows from Theorem 4.4 that, for n ≥ 2, End Uq (V ⊗ V * ) is isomorphic to End Q (z) (↑↓), 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, End Uq (V ⊗ V * ) is not semisimple.Note, however, that the full monoidal subsupercategory of U q -smod generated by V is semisimple; see [GJKK10, Th. 6.5].

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 affinization procedure; see [MS21].However, the supercategory Q (z) 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 k is an arbitrary commutative ring of characteristic not equal to two, and z ∈ k.
Definition 5.1.Let W (z) be the strict monoidal supercategory obtained from Q (z) by adjoining an additional generating object and two additional even morphisms 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 W (z): Proof.We compute The proofs of the remaining equalities are analogous.
Proposition 5.3.In W (z), we have where f is any string diagram in Q (z) not containing Clifford tokens.
Proof.First note that the second equality in 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 , , , it is also enough to show it holds for f ∈ { , , , , , }.For f = , 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 f = .To prove that the first equality in (5.4) holds with f = and f = , we must show that (5.5) = and = .
The first relation in (5.5) follows from the first relation in (5.3) after composing on the bottom with and using 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 f = and f = are analogous, using the last two equalities in (5.3).
In the remainder of this section we will be discussing connections to U q -smod.Thus, we now begin supposing that k = 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 ρ M : U q → End k (M ) denote the corresponding representation.We then define In particular, we have L V = Θ; see (3.26).
Proposition 5.4.For any M ∈ U q -smod, the map is an isomorphism of U q -supermodules.Furthermore, for all f ∈ Hom Uq (M, N ), we have To show that T M V is a homomorphism of U q -supermodules, it suffices to prove that it commutes with the action of u kl , k, l ∈ I, k ≤ l.By (3.12), it is enough to show that (5.9) Composing both sides of (5.9) on the left with the invertible map flip 12 , and using the fact that L V = Θ, we see that (5.9) is equivalent to (5.10) which follows from the last equality in (3.10).It remains to prove (5.7).For f ∈ Hom Uq (M, N ), m ∈ M , and v ∈ V , we have Note that T V V = T (see (4.1)), and so Proposition 5.4 is a generalization of Lemma 4.1.Now, for i, j ∈ I, define

It follows that
(5.11) Theorem 5.5.For each U q -supermodule M , the superfunctor F n of Theorem 4.3 extends to a unique monoidal superfunctor (5.12) Proof.By Theorem 4.3, to show that F M n is well defined, it suffices to show that F M n respects the relations (5.1).First of all, uniqueness of the inverse implies that F M n ( ) = T −1 M V , 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 ∈ M and k ∈ 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 F M n sends the left-hand side to the map M ⊗V ⊗2 → V ⊗2 ⊗ M given by Remark 5.6.There are natural superfunctors F ↑ , F ↓ : W (z) → Q (z) sending to ↑ and ↓, respec- tively.It is then straightforward to very that following diagrams commute: In particular, we have For M ∈ U q -smod, we will denote the image of a string diagram in W (z) under F M n 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 The relations (5.1), (5.4), (5.13), and (5.14) show that , , , and , together with the crossings in Q (z), almost endow W (z) 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 W (z) passing over any sequence of strands in Q (z).All morphisms in W (z) pass over such crossings, but only some morphisms in Q (z) 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 W 1 (z) denote the full sub-supercategory of W (z) on objects that are tensor products of ↑, ↓, and , with exactly one occurrence of .Thus, objects of W 1 (z) are of the form X ⊗ ⊗ Y for X, Y ∈ Q (z).Note that W 1 (z) is not a monoidal supercategory.
Theorem 5.8.There is a unique superfunctor Proof.It follows from (5.13) that the given definition is natural in M .

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, k is an arbitrary commutative ring of characteristic not equal to two, and z ∈ k.Definition 6.1.The quantum affine isomeric supercategory AQ (z) is the strict monoidal supercategory obtained from Q (z) by adjoining an additional odd morphism We refer to as an open Clifford token.To emphasize 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 AQ (z).Lemma 6.2.There is a unique isomorphism of monoidal supercategories AQ (z) → AQ (−z) determined on objects by ↑ → ↑, ↓ → ↓, and sending On arbitrary diagrams, the isomorphism acts by interchanging open and closed Clifford tokens and flipping crossings.
Lemma 6.3.The following relations hold in AQ (z): where, in (6.4), the relations hold for all orientations of the strands.
It follows from the above discussion that the isomorphisms Ω − , Ω , and Ω ↔ defined in Section 2 extend to isomorphisms of monoidal supercategories These are defined as in Section 2 for the generators of Q (z) and, on the open Clifford token, are defined by Furthermore, AQ (z) is strictly pivotal, with duality superfunctor Remark 6.4.Note that, while Q (0) is isomorphic to the oriented Brauer-Clifford supercategory of [BCK19, Def.3.2], as described in Lemma 2.9, the supercategory AQ (z) does not reduce to the definition [BCK19, Def.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 specialization of the affine Hecke algebra of type A.
Define, for k ∈ Z, Note that both morphisms in (6.6) are of parity k (mod 2).We then define, for k ∈ Z, (6.7) We refer to the decorations as zebras.We have colored 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 [BCK19].
Recall our convention k l = k l .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 AQ (z) for all k ∈ Z: where, in (6.8), the relations hold for both orientations of the strands.
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 AQ (z): Proof.For the first relation, we have (2.9) The proof of the second relation is analogous.
Corollary 6.7.For all k ∈ Z >0 , the following relations hold in AQ (z): Proof.We prove (6.11) by induction on k.The case k = 1 is (6.10).Then, for k ≥ 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 −2k and on the top with −2k .
Lemma 6.8.For all k > 0, we have Let Sym denote the k-algebra of symmetric functions over k.For r ≥ 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 k-algebra Sym is generated by e r , h r , r > 0, modulo the identities where h 0 = e 0 = 1.The map β 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 β is surjective, while Conjecture 6.12 would imply it is an isomorphism.We next deduce a bubble slide relation.Proof.The case k = 0 follows immediately from (2.7).Thus, we suppose k > 0. We first compute where the last sum is zero when r = k.We also have For any two objects X, Y ∈ AQ (z), the morphism space Hom AQ (z) (X, Y ) is a right Symsupermodule 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 B (X, Y ) 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 AQ (z), the set B (X, Y ) spans the morphism space Hom AQ (z) (X, Y ) as a right Sym-supermodule.
Proof.Since this type of argument is standard in categorical representation theory, we only give a sketch of the proof.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 AQ (z) 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 → Y as a linear combination of the ones in B (X, Y ).
Conjecture 6.12.For any objects X, Y of AQ (z), the morphism space Hom AQ (z) (X, Y ) is a free right Sym-supermodule with basis B (X, Y ).
As noted in the introduction, we expect that Conjecture 6.12 could be proved using the categorical comultiplication technique of [BSW20], after introducing the more general quantum isomeric Heisenberg supercategory.Proposition 6.13.There is a unique monoidal superfunctor C : AQ (z) → SEnd (Q (z)) defined as follows.On objects X ∈ AQ (z) and morphisms f ∈ { , , , , , }, where the thick strand labelled X is the identity morphism 1 X of X.
All the relations appearing in Definition 2.1 are clearly respected by 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 C, which we will call the collapsing superfunctor, should viewed as an odd analogue of the one appearing in [MS21, Th. 3.2], which describes actions of the affinization 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.

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, k is an arbitrary commutative ring of characteristic not equal to two, and z ∈ k.Definition 7.1.For r ∈ Z >0 and z ∈ k, let AHC r (z) denote the associative superalgebra generated by even elements t 1 , . . ., t r−1 and odd elements π 1 , . . ., π r , ̟ 1 , . . ., ̟ r , satisfying the following relations (for i, j in the allowable range): Equivalently, AHC r (z) is the associative superalgebra generated by HC r (z), together with odd elements ̟ 1 , . . ., ̟ 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 −1 i = t i − z gives (7.7) The next result shows that AHC r (z) is isomorphic to the affine Hecke-Clifford superalgebra, which was first introduced in [JN99, §3], where it was called the affine Sergeev algebra.
Lemma 7.2.For r ∈ Z >0 and z ∈ k, AHC r (z) is isomorphic to the associative superalgebra AHC ′ r (z) generated by HC r (z), together with pairwise-commuting invertible even elements x 1 , . . ., 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 AHC ′ r (z).Thus the map (7.8) is well-defined homomorphism of superalgebras.It is invertible, with inverse In light of Lemma 7.2, we will simply refer to AHC r (z) as the affine Hecke-Clifford superalgebra.
Proposition 7.3.For r ∈ N, we have a homomorphism of associative superalgebras given by Proof.It is a straightforward computation to verify that the given map is well-defined, i.e. that it respects the relations in Definition 7.1.
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 ̟ i , whereas the latter involves the even generator x i = π i ̟ i .We prefer the presentation of Definition 7.1 since the relations are simpler and a natural symmetry of AHC r (z) becomes apparent.In particular, we have an automorphism of AHC r (z) given by (7.9) 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 End AQ (z) (↑ ⊗r ) induced by the superfunctor Ω ↔ yields a homomorphism of associative superalgebras ı r : AHC r (z) → End AQ (z) (↑ ⊗r ) given by t i → − ↑ ⊗(r−i−1) ⊗ ⊗ ↑ ⊗(i−1) , 1 ≤ i ≤ r − 1, This is also equal to the automorphism (7.9) followed by the map of Proposition 7.3.We have ı r (x i ) = ↑ ⊗(r−i) ⊗ ⊗ ↑ ⊗(i−1) , 1 ≤ i ≤ r.
For 1 ≤ i ≤ r, define the odd Jucys-Murphy elements (7.10) , where, by convention, we have J odd 1 = π 1 .The following result gives a direct (i.e.non-recursive) expression for the even Jucys-Murphy elements.
Lemma 7.4.For all 1 ≤ i ≤ r, we have J i = −π i J odd i .
Proof.We prove the result by induction on i.Since −π 1 J odd 1 = −π 2 1 = 1 = J 1 , the result holds for i = 1.Now suppose that i > 1 and that J i−1 = −π i−1 J odd i−1 .First note that Thus, we have Evaluation on the unit object ½ yields a superfunctor Note that this is not a monoidal superfunctor.Recall the collapsing superfunctor C of Proposition 6.13.
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 [MS21, §6] for a discussion of Jucys-Murphy elements in a more general setting, related to the affinization of braided monoidal categories.The above discussion suggests there may be a general notion of odd affinization, 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 Q (z) as a subcategory of AQ (z).
Proposition 7.7.The superfunctor Q (z) → AQ (z) that is the identity on objects and sends each generating morphism in Q (z) to the morphism in AQ (z) depicted by the same string diagram is faithful.
Proof.It is straightforward to verify that Ev ½ •C is left inverse to the superfunctor in the statement of the proposition.

The affine action superfunctor
In this final section, we define an action of AQ (z) on the category of U q -supermodules.We then use this action to define a sequence of elements in the center of U q .Throughout this section we assume that k = C(q) and z = q − q −1 .The image of ↑ under the superfunctor F • n of Theorem 5.8 is the superfunctor F • n (↑ ) = V ⊗ − : U q -smod → U q -smod of tensoring on the left with V .Define the natural transformation (8.1) Thus the M -component of K, for M ∈ U q -smod, is the U q -supermodule homomorphism It is straightforward to verify that K 2 = − id, where id is the identity natural transformation.
Theorem 8.1.There is a unique monoidal superfunctor F n : AQ (z) → SEnd (U q -smod), 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 F n the affine action superfunctor.It endows U q -smod with the structure of an AQ (z)-supermodule category.Note that are the translation endosuperfunctors of U q -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 AHC r (z) → End Uq (V ⊗r ⊗ M ) for any U q -supermodule M and r, s ∈ N.This is a quantum analogue of [HKS11, Th. 7.4.1].Let Z q := {x ∈ U q : xy = (−1) xȳ yx for all y ∈ U q } be the center of U q .Evaluation on the identity element of the regular representation defines a canonical superalgebra isomorphism End(id Uq-smod ) where id C denotes the identity endosuperfunctor of a supercategory C. Consider the composite superalgebra homomorphism (8.2) End AQ (z) (½) 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 2k , k > 0.
We begin with a simplifying computation.Using (5.4) and the relations in Q (z) we have, for (−1) p(i,k)p(j,k) S(u ik )u −k,−j ∈ U q .
Note that y ij is of parity p(i, j).Next, for i, j ∈ I, and m > 0, define These are isomeric analogues of the elements defined in [BSW20, (5.15)].
Lemma 8.2.We have where we interpret the right-hand side as a natural transformation whose M -component, for a U qsupermodule M with corresponding representation ρ M , is i,k∈I ρ M (y ij ) ⊗ E ij .
Proof.We have Replacing l by −j yields (8.5).