Affine oriented Frobenius Brauer categories
aa r X i v : . [ m a t h . R T ] J a n AFFINE ORIENTED FROBENIUS BRAUER CATEGORIES
ALEXANDRA MCSWEEN AND ALISTAIR SAVAGE
Abstract.
To any Frobenius superalgebra A we associate an oriented Frobenius Brauer category and an affine oriented Frobenius Brauer categeory . We define natural actions of these categories oncategories of supermodules for general linear Lie superalgebras gl m | n ( A ) with entries in A . Theseactions generalize those on module categories for general linear Lie superalgebras and queer Liesuperalgebras, which correspond to the cases where A is the ground field and the two-dimensionalClifford algebra, respectively. Introduction
The oriented Brauer category OB is the free linear rigid symmetric monoidal category generatedby a single object ↑ . This universal property immediately implies the existence of a monoidal functor F : OB → mod- gl n from OB to the category of right modules for the general linear Lie algebra gl n . (One can also workwith left modules, but right modules turn out to be easier for the current paper.) This functor sendsthe generating object ↑ and its dual ↓ to the defining gl n -module V and its dual V ∗ , respectively.For r ≥ , the endomorphism algebra End OB ( ↑ ⊗ r ) is the group algebra of the symmetric group on r letters, and the algebra homomorphism End OB ( ↑ ⊗ r ) → End gl n ( V ⊗ r ) induced by F is the classical one appearing in Schur–Weyl duality. More generally, End OB ( ↑ ⊗ r ⊗ ↓ ⊗ s ) are walled Brauer algebras and the induced algebra homomorphisms End OB ( ↑ ⊗ r ⊗ ↓ ⊗ s ) → End gl n ( V ⊗ r ⊗ ( V ∗ ) ⊗ s ) were originally defined and studied by Turaev [Tur89] and Koike [Koi89].The rank n of gl n appears as a parameter in OB . In fact, the definition of OB makes sensefor any value of this parameter, i.e. it need not be a positive integer. This observation leads tothe definition of Deligne’s interpolating category for the general linear Lie groups [Del07]; thisinterpolating category is the additive Karoubi envelope of OB .The functor F yields an action of OB on mod- gl n . More precisely, for a category C , let End ( C ) denote the corresponding strict monoidal category of endofunctors and natural transformations.Then we have a functor OB → End ( mod- gl n ) , X F ( X ) ⊗ − , f F ( f ) ⊗ − , for objects X and morphisms f in OB . In [BCNR17], this was extended to an action of the affineoriented Brauer category AOB on mod- gl n . The category AOB is obtained from OB by adjoiningan additional endomorphism of the generating object ↑ , subject to certain natural relations. Thisadditional endomorphism acts by a natural transformation of the functor V ⊗ − arising from multi-plication by a certain canonical element of gl n ⊗ gl n . Restricting to endomorphism spaces recoversactions of affine walled Brauer algebras studied in [RS15, Sar14]. Mathematics Subject Classification.
Key words and phrases.
Monoidal category, supercategory, Lie superalgebra, Brauer algebra.
ALEXANDRA MCSWEEN AND ALISTAIR SAVAGE
In fact, much of the above picture can be generalized, replacing gl n by the general linear Lie super algebra gl m | n . Remarkably, one does not need to modify OB or AOB at all. Here the corre-sponding Schur–Weyl duality was established by Sergeev [Ser84a] and Berele–Regev [BR87], whilethe action of the walled Brauer algebras was described in [BS12, Th. 7.8]. The analogue of thefunctor F above is described in [ES16, Th. 4.16]. The extension of the action to AOB does notseem to have appeared in the literature, but is certainly expected by experts. For example, it ismentioned in the introduction to [BCNR17].The affine oriented Brauer category
AOB is a special case of more general category. The
Heisen-berg category at central charge − was first introduced by Khovanov [Kho14] as a tool to study therepresentation theory of the symmetric group. In [MS18], it was generalized to arbitrary negativecentral charge, which corresponds to replacing the symmetric group by more general degeneratecyclotomic Hecke algebras of type A . In [Bru18], Brundan gave a simplified presentation of theHeisenberg category Heis k at arbitrary central charge k . When k = 0 , the Heisenberg category isprecisely the affine oriented Brauer category.The Heisenberg category has been further generalized in [RS17, Sav19, BSW20a] to the Frobe-nius Heisenberg category
Heis k ( A ) depending on a Frobenius superalgebra A . When A = k , thisconstruction recovers the Heisenberg category. When k = 0 , the category Heis k ( A ) acts naturallyon categories of modules over the cyclotomic wreath product algebras defined in [Sav20]. However,actions in the case k = 0 have not yet been studied. In some sense, central charge zero yields thesimplest and most interesting case. For example, Heis k ( A ) is symmetric monoidal if and only if k = 0 .The goal of the current paper is to fill in this gap in the literature. In analogy with the A = k case, we call the Frobenius Heisenberg category at central charge zero the affine oriented FrobeniusBrauer category AOB ( A ) . It contains a natural Frobenius algebra analogue of the oriented Brauercategory, which we call the oriented Frobenius Brauer category OB ( A ) . We define, in Theorems 5.1and 5.2, natural functors OB ( A ) → smod- gl m | n ( A ) , AOB ( A ) → End ( smod- gl m | n ( A )) , where smod- gl m | n ( A ) denotes the monoidal supercategory of right supermodules for the generallinear Lie superalgebra with entries in the Frobenius superalgebra A .When A = k , we recover the functors described above for gl m | n = gl m | n ( k ) . On the otherhand, if A = Cl is the two-dimensional Clifford superalgebra, then gl m | n (Cl) is isomorphic to thequeer Lie superalgebra q ( m + n ) , and our functors recover those defined in [BCK19]. As in the A = k case, these functors extend Schur–Weyl duality results for queer Lie superalgebras [Ser84b],actions of walled Brauer–Clifford superalgebras [JK14], and actions of affine walled Brauer–Cliffordsuperalgebras [BCK19, GRSS19]. In fact, the Clifford superalgebra is the main example of interestwhere the Frobenius superalgebra is not symmetric. Since this case has already been studied in theaforementioned papers, we assume throughout the current document that A is symmetric, as thissimplifies the exposition. We have indicated in Remark 4.6 the modification that needs to be madeto handle the more general case.The results of the current paper extend the powerful category theoretic tools that have been usedto study the representation theory of general linear Lie superalgebras and queer Lie superalgebrasto the setting of general linear Lie superalgebras over Frobenius superalgebras. For example, inProposition 5.3, we see that these functors yield central elements in the universal enveloping algebrageneralizing the known generators of this center in the A = k and A = Cl cases. When A = k [ x ] / ( x l ) ,then gl m | n ( A ) is a truncated current superalgebra (also called a Takiff algebra when m = 0 or n = 0 ).In this case, Brauer category type methods do not seem to have appeared in the literature before. FFINE ORIENTED FROBENIUS BRAUER CATEGORIES 3
The results of the current paper lead naturally to several avenues of further research. We concludethis introduction by listing a few such directions here.(a)
Interpolating categories . The idempotent completion of OB ( A ) is a natural candidate foran interpolating category for smod- gl m | n ( A ) , which could potentially be used to generalizework of Deligne and others in the case n = 0 , A = k .(b) Frobenius Schur algebras
One should be able to define Schur algebras depending on a Frob-nenius superalgebra A such that, when A = k , one recovers the usual Schur algebras.(c) Cyclotomic quotients . In the cases A = k and A = Cl , cyclotomic quotients of AOB ( A ) havebeen studied in [BCNR17, BCK19]. We expect that many of these results can be extendedto the setting of general Frobenius superalgebras.(d) Quantum analogues . The quantum Frobenius Heisenberg categories , introduced in [BSW20b],are natural quantum analogues of Frobenius Heisenberg categories. The special case of cen-tral charge zero yields a natural quantum affine oriented Frobenius Brauer category . When A = k , this is the affine HOMFLY-PT skein category. Then quantum affine oriented Frobe-nius Brauer categories should act on as-yet-to-be-defined quantum enveloping algebras of gl m | n ( A ) and yield Frobenius analogues of the HOMFLY-PT link invariant. Acknowledgements.
This research was supported by Discovery Grant RGPIN-2017-03854 fromthe Natural Sciences and Engineering Research Council of Canada (NSERC).2.
Monoidal supercategories
Throughout the paper, we fix a ground field k . All vector spaces, algebras, categories, and func-tors will be assumed to be linear over k unless otherwise specified. Unadorned tensor products denotetensor products over k . Most things in the article will be enriched over the category SVec of vectorsuperspaces , that is, Z / Z -graded vector spaces V = V ¯0 ⊕ V ¯1 with parity-preserving morphisms.Writing ¯ v ∈ Z / Z for the parity of a homogeneous vector v ∈ V , the category SVec is a symmetricmonoidal category with symmetric braiding V ⊗ W → W ⊗ V defined by v ⊗ w ( − ¯ v ¯ w w ⊗ v forhomogeneous v, w , and extended by linearity.For superalgebras A = A ¯0 ⊕ A ¯1 and B = B ¯0 ⊕ B ¯1 , multiplication in the superalgebra A ⊗ B isdefined by(2.1) ( a ′ ⊗ b )( a ⊗ b ′ ) = ( − ¯ a ¯ b a ′ a ⊗ bb ′ for homogeneous a, a ′ ∈ A , b, b ′ ∈ B . The center Z ( A ) is the subalgebra generated by all homoge-neous a ∈ A such that(2.2) ab = ( − ¯ a ¯ b ba for all homogeneous b ∈ A . The cocenter C ( A ) is the quotient of A by the subspace spanned by ab − ( − ¯ a ¯ b ba for all homogeneous a, b ∈ A . Note that, a priori, C ( A ) is only a vector superspace,and not a superalgebra.Throughout this document, we will be working with strict monoidal supercategories in the sense of[BE17]. A supercategory is a category enriched in SVec . Thus, its morphism spaces are superspacesand composition is parity preserving. A superfunctor between supercategories induces a parity-preserving linear map between morphism superspaces. For superfunctors
F, G : A → B , a supernat-ural transformation α : F ⇒ G of parity r ∈ Z / Z is the data of morphisms α X ∈ Hom B ( F X, GX ) r for each X ∈ A such that Gf ◦ α X = ( − r ¯ f α Y ◦ F f for each homogeneous f ∈ Hom A ( X, Y ) . A supernatural transformation α : F ⇒ G is α = α ¯0 + α ¯1 with each α r being a supernatural transfor-mation of parity r . ALEXANDRA MCSWEEN AND ALISTAIR SAVAGE
In a strict monoidal supercategory, the super interchange law is(2.3) ( f ′ ⊗ g ) ◦ ( f ⊗ g ′ ) = ( − ¯ f ¯ g ( f ′ ◦ f ) ⊗ ( g ◦ g ′ ) . We denote the unit object by and the identity morphism of an object X by X . We will usethe usual calculus of string diagrams, representing the horizontal composition f ⊗ g (resp. verticalcomposition 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 arisingfrom the super interchange law:(2.4) f g = f g = ( − ¯ f ¯ g f g . We refer the reader to [Sav21] for a brief overview of string diagrams suited to the current paper,to [TV17, Ch. 1, 2] for a more in-depth treatment, and to [BE17] for a detailed discussion of signsin the super setting.3.
General linear Lie supersalgebras over Frobenius superalgebras
Throughout this document A will denote a symmetric Frobenius superalgebra with parity-preserving trace map tr : A → k . Thus(3.1) tr( ab ) = ( − ¯ a ¯ b tr( ba ) = ( − ¯ a tr( ba ) = ( − ¯ b tr( ba ) , a, b ∈ A, where the second and third equalities follow from the fact that tr( ab ) = 0 unless ¯ a = ¯ b . Thedefinition of a Frobenius superalgebra gives that A possesses a homogeneous basis B A and a leftdual basis { b ∨ : b ∈ B A } such that(3.2) tr( b ∨ c ) = δ b,c , b, c ∈ B A . It follows that, for all a ∈ A , we have(3.3) a = X b ∈ B A tr( b ∨ a ) b = X b ∈ B A tr( ab ) b ∨ . Note that ¯ b = b ∨ , and that the left dual basis to { b ∨ : b ∈ B A } is given by(3.4) ( b ∨ ) ∨ = ( − ¯ b b. Remark 3.1.
More generally, a (not necessarily symmetric) Frobenius superalgebra is an associa-tive superalgebra with a trace map admitting dual bases, but where we do not require the tracemap to satisfy (3.1). One can show that there is an automorphism ψ of A , called the Nakayamaautomorphism , such that tr( ab ) = ( − ¯ a ¯ b tr( bψ ( a )) . More generally, one can also allow the tracemap to be parity reversing. Lemma 3.2.
For all homogeneous a, c ∈ A , we have (3.5) X b ∈ B A ( − ¯ b ¯ c abc ⊗ b ∨ = ( − ¯ a ¯ c X B A ( − ¯ b ¯ c b ⊗ cb ∨ a. Proof.
We have X b ∈ B A ( − ¯ b ¯ c abc ⊗ b ∨ (3.3) = X b,e ∈ B A ( − ¯ b ¯ c tr( e ∨ abc ) e ⊗ b ∨ = X b,e ∈ B A ( − ¯ c ¯ e +¯ c ¯ a e ⊗ tr( ce ∨ ab ) b ∨ (3.3) = ( − ¯ a ¯ c X e ∈ B A ( − ¯ c ¯ e e ⊗ ce ∨ a, FFINE ORIENTED FROBENIUS BRAUER CATEGORIES 5 where, in the second equality, we used the fact that tr( e ∨ abc ) = 0 unless ¯ e + ¯ a + ¯ b + ¯ c = 0 to simplifythe exponent of − . (cid:3) Fix m, n ∈ N . For ≤ i ≤ m + n , define p ( i ) ∈ Z / Z by(3.6) p ( i ) = ( ¯0 if ≤ i ≤ m, ¯1 if m + 1 ≤ i ≤ m + n. Let
Mat m | n ( A ) be the associative superalgebra consisting of ( m + n ) × ( m + n ) matrices with entriesin A , where multiplication is given by matrix multiplication and the Z / Z -grading is defined asfollows. For a ∈ A and ≤ i, j ≤ m + n , let a ( i,j ) ∈ Mat m | n ( A ) denote the matrix with a in the ( i, j ) position and in all other positions. Then, for homogeneous a ∈ A , we define a ( i,j ) = ¯ a + p ( i ) + p ( j ) . It is straightfoward to verify that
Mat m | n ( A ) is a symmetric Frobenius superalgebra, with trace tr m | n := tr ◦ str : Mat m | n ( A ) → k , where str is the usual supertrace. Thus, for ≤ i, j ≤ m + n and a ∈ A , we have tr m | n ( a ( i,j ) ) = δ i,j ( − p ( i ) tr( a ) . Let g = g m | n ( A ) denote the Lie superlagebra associated to Mat m | n ( A ) . Precisely, g is equal to Mat m | n ( A ) as a k -supermodule and the Lie superbracket is defined by [ M, N ] =
M N − ( − ¯ M ¯ N N M. for homogeneous
M, N ∈ g and extended by linearity. We have that(3.7) B g := { b ( i,j ) : b ∈ B A , ≤ i, j ≤ m + n } is a basis for g with left dual basis (with respect to tr m | n ) given by(3.8) ( b ( i,j ) ) ∨ = ( − p ( j ) b ∨ ( j,i ) . Note that, here and in what follows, we adopt the convention that we apply the symbol ∨ beforeconsidering subscripts. Thus, for example, b ∨ ( i,j ) = ( b ∨ ) ( i,j ) . Example 3.3.
When A = k , then g m | n ( k ) is the usual general linear superalgebra over k . Example 3.4. If A = k [ t ] / ( t l ) , l ≥ , then g m | n ( A ) is a truncated current superalgebra . When n = 0 , this is also known as a Takiff algebra . Example 3.5.
Let Cl denote the two-dimensional Clifford superalgebra generated by an odd ele-ment c satisfying c = 1 . Recall that the queer Lie superalgebra q ( n ) has even and odd parts q ( n ) ¯0 = (cid:26)(cid:18) M M (cid:19) : M ∈ Mat n ( k ) (cid:27) , q ( n ) ¯1 = (cid:26)(cid:18) MM (cid:19) : M ∈ Mat n ( k ) (cid:27) , where Mat n ( k ) denotes the set of n × n matrices with entries in k . Then it is straightforward toverify that we have an isomorphism of Lie superalgebras q ( m + n ) ∼ = −→ g m | n (Cl) given by (cid:18) ( i,j )
00 1 ( i,j ) (cid:19) c p ( i )+ p ( j ) ( i,j ) , (cid:18) ( i,j ) ( i,j ) (cid:19) c p ( i )+ p ( j )+1 ( i,j ) , ≤ i, j ≤ m + n. Note that Cl is not actually a symmetric Frobenius superalgebra in the sense discussed above(where we required the trace map to be parity preserving). However it is a Frobenius superalgebra;see Remark 3.1. Up to a scalar multiple, there are two choices of homogeneous trace map, oneparity preserving and one parity reversing. Under the parity-reversing trace map, Cl is symmetric, ALEXANDRA MCSWEEN AND ALISTAIR SAVAGE while under the parity-preserving trace map it has nontrivial Nakayama automorphism given by ψ ( c ) = − c .Let A m | n denote the k -supermodule equal to A m + n as a k -module, with Z / Z -grading determinedby ae i = ¯ a + p ( i ) , a ∈ A, ≤ i ≤ m + n, where e i denotes the element of A m | n with a in the i -th entry and all other entries equal to . Wewill also consider A m | n as a left A -supermodule with action a ( a , . . . , a m + n ) = ( aa , . . . , aa m + n ) . Let smod- g denote the category of right g -supermodules. Let V + = A m | n , written as row matrices,and let V − = A m | n , written as column matrices. Then V + is naturally a right g -supermodule withaction given by right matrix multiplication, while V − is a right g -supermodule with action(3.9) v · M := − ( − ¯ v ¯ M M v, v ∈ V − , M ∈ g . We define the k -bilinear form(3.10) B : V − ⊗ V + → k , B ( v ⊗ w ) := ( − ¯ v ¯ w tr( wv ) . It is straightforward to verify that B is a homomorphism of right g -supermodules.For a ∈ A and ≤ i ≤ m + n , let a i, ± denote the element of V ± with a in the i -th position and in every other position. Then(3.11) B + := { b i, + : b ∈ B A , ≤ i ≤ m + n } and B − := { b ∨ i, − : b ∈ B A , ≤ i ≤ m + n } are dual bases of V + and V − with respect to the bilinear form B . For v = b i, + ∈ B + , let v ∨ = b ∨ i, − ∈ B − . Thus, we have B ( v ∨ , w ) = δ v,w for v, w ∈ B + .Define(3.12) Ω := X M ∈ B g M ⊗ M ∨ ∈ g ⊗ g , τ := X b ∈ B A b ⊗ b ∨ ∈ A ⊗ A. The following result will be crucial in our computations of the categorical action of the affine orientedFrobenius Heisenberg category.
Lemma 3.6.
For all u, v ∈ V + , we have (3.13) τ ( u ⊗ v ) = ( − ¯ u ¯ v ( v ⊗ u )Ω . Proof.
It suffices to prove the result for u = a k, + and v = c l, + , where a, c ∈ A and ≤ k, l ≤ m + n .We have ( a k, + ⊗ c l, + )Ω = m + n X i,j =1 X b ∈ B A ( − (¯ b + p ( i )+ p ( j ))(¯ c + p ( l ))+ p ( j ) a k, + b ( i,j ) ⊗ c l, + b ∨ ( j,i ) = X b ∈ B g ( − (¯ b + p ( k ))(¯ c + p ( l ))+¯ cp ( l ) ( ab ) l, + ⊗ ( cb ∨ ) k, + (3.3) = X b,e ∈ B A ( − (¯ b + p ( k ))(¯ c + p ( l ))+¯ cp ( l ) tr( e ∨ ab ) e l, + ⊗ ( cb ∨ ) k, + (3.3) = ( − (¯ a + p ( k ))(¯ c + p ( l )) X e ∈ B A ( − ¯ e (¯ c + p ( l ))+¯ cp ( l ) e l, + ⊗ ( ce ∨ a ) k, + (3.3) = ( − (¯ a + p ( k ))(¯ c + p ( l )) X b,e ∈ B A ( − ¯ e (¯ c + p ( l ))+¯ cp ( l ) e l, + ⊗ tr( ce ∨ b )( b ∨ a ) k, + FFINE ORIENTED FROBENIUS BRAUER CATEGORIES 7(3.1) = ( − (¯ a + p ( k ))(¯ c + p ( l )) X b,e ∈ B A ( − (¯ e +¯ c )(¯ c + p ( l )) tr( e ∨ bc ) e l, + ⊗ ( b ∨ a ) k, + (3.3) = ( − (¯ a + p ( k ))(¯ c + p ( l )) X b ∈ B A ( − ¯ b (¯ c + p ( l )) ( bc ) l, + ⊗ ( b ∨ a ) k, + = ( − (¯ a + p ( k ))(¯ c + p ( l )) τ ( c l, + ⊗ a k, + ) . (cid:3) Affine oriented Frobenius Brauer categories
We introduce here our main object of study, the affine oriented Frobenius Brauer category. Thisis the central charge k = 0 case of the Frobenius Heisenberg category Heis k ( A ) introduced in [Sav19]and further studied in [BSW20a]. Definition 4.1.
The oriented Frobenius Brauer category OB ( A ) associated to the symmetric Frobe-nius superalgebra A is the strict monoidal supercategory generated by objects ↑ and ↓ and morphisms : ↑ ⊗ ↑ → ↑ ⊗ ↑ , a : ↑ → ↑ , a ∈ A, : → ↓ ⊗ ↑ , : ↑ ⊗ ↓ → , : → ↑ ⊗ ↓ , : ↓ ⊗ ↑ → , subject to the relations = , λ a + µ b = λa + µb , ba = ab , (4.1) = , = , a = a , (4.2) = , = , = = , (4.3) = , = , (4.4)for all a, b ∈ A and λ, µ ∈ k . In the above, the left and right crossings are defined by(4.5) := , := . The parity of a is ¯ a , and all the other generating morphisms are even. Remark 4.2.
One can define OB ( A ) equivalently as the strict monoidal supercategory generatedby the morphisms , , , , a , a ∈ A, subject to the relations (4.1), (4.2), and (4.4), and the first two equalities in (4.3). In this presen-tation, one defines the left cup and cap by = and = . We refer to the morphisms a as tokens . It follows from (4.1) and (4.2) that the map A → End
AOB ( A ) ( ↑ ) , a a , is a superalgebra homomorphism and, also using (2.4), that a b = a b = ( − ¯ a ¯ b a b , a = a . Define the teleporter (4.6) = = := X b ∈ B A b b ∨ (3.4) = X b ∈ B A b ∨ b . ALEXANDRA MCSWEEN AND ALISTAIR SAVAGE
We do not insist that the tokens in a teleporter (4.6) are drawn at the same horizontal level. Theconvention when this is not the case is that b is on the higher of the tokens and b ∨ is on the lowerone. We will also draw teleporters in larger diagrams. When doing so, we add a sign of ( − y ¯ b infront of the b summand in (4.6), where y is the sum of the parities of all tokens in the diagramvertically between the tokens labeled b and b ∨ . For example, a c = X b ∈ B A ( − (¯ a +¯ c )¯ b a cb b ∨ . This convention ensures that one can slide the endpoints of teleporters along strands: a c = a c = a c = a c . It follows from (3.5) that tokens can “teleport” across teleporters in the sense that, for a ∈ A , wehave(4.7) a = a , a = a , a = a , a = a . where the strings can occur anywhere in a diagram (i.e. they do not need to be adjacent). Theendpoints of teleporters slide through crossings and they can teleport too. For example we have(4.8) = , = = . Definition 4.3.
The affine oriented Frobenius Brauer category
AOB ( A ) associated to the Frobeniussuperalgebra A is the strict monoidal supercategory obtained from OB ( A ) by adjoining an evengenerator : ↑→↑ , which we call a dot , subject to the relations(4.9) − = , a = a , a ∈ A. It follows from (4.2) and (4.9) that we also have(4.10) − = . In addition, endpoints of teleporters slide through dots:(4.11) = . Remark 4.4 ( Z -grading) . If the symmetric Frobenius superalgebra A is Z -graded with the tracemap tr of degree − d A , then the categories OB ( A ) and AOB ( A ) are also naturally Z -graded. The Z -degrees of the generating morphisms are as follows: deg (cid:0) a (cid:1) = deg( a ) , deg (cid:0) (cid:1) = d A , deg (cid:0) (cid:1) = deg ( ) = deg ( ) = deg ( ) = ( ) = 0 . All of the results of the current paper can be carried out in the graded setting.
Example 4.5.
As noted in the introduction, when A = k , the categories OB ( k ) and AOB ( k ) are the oriented Brauer and affine oriented Brauer categories , respectively; see [BCNR17]. Theendomorphism algebras of OB ( k ) are oriented Brauer algebras , which are isomorphic to walledBrauer algebras . Remark 4.6.
The definitions of OB ( A ) and AOB ( A ) can be generalized to allow for A to be a (notnecessarily symmetric) Frobenius superalgebra with trace map of arbitrary parity. The only change FFINE ORIENTED FROBENIUS BRAUER CATEGORIES 9 to the relations is that the parity of the dot is equal to the parity tr of the trace map (i.e. the dotis odd if the trace map is parity reversing) and the second relation in (4.9) becomes a = ( − ¯ a tr ψ ( a ) , a ∈ A, where ψ is the Nakayama automorphism. (This level of generality was considered in [Sav19].)With this modification, we can take A to be the two-dimensional Clifford superalgebra Cl ; seeExample 3.5. Then OB (Cl) and AOB (Cl) are the oriented Brauer–Clifford and degenerate affineoriented Brauer–Clifford supercategories, respectively, introduced in [BCK19].We recall some other relations, which are proved in [Sav19, Th. 1.3], that follow from the definingrelations. In what follows, the relations not involving dots hold in both OB ( A ) and AOB ( A ) . Therelations (4.4) means that ↓ is right dual to ↑ . In fact, we also have(4.12) = , = , and so ↓ is also left dual to ↑ . Thus OB ( A ) and AOB ( A ) are rigid . Furthermore, we have that(4.13) := = , := = , a := a = a , a ∈ A. These relations mean that dots, tokens, and crossings slide over all cups and caps: a = a , a = a , = , = , (4.14) = , = , = , = . (4.15)More precisely, the cups and caps equip OB ( A ) and AOB ( A ) with the structure of strict pivotal supercategories; see [BSW20a, (5.16)]. It follows from the definition of the tokens on downwardstrands that ba = ( − ¯ a ¯ b ba . For r ≥ , we define r = (cid:0) (cid:1) ◦ r . We adopt the following conventions for bubbles with a negative number of dots:(4.16) ar = − δ r, − tr( a ) , a r = δ r, − tr( a ) if r < . For any homogeneous a ∈ A , we define(4.17) a † := X b ∈ B A ( − ¯ a ¯ b bab ∨ , which is well-defined independent of the choice of the basis B A .Then we have the infinite Grassmannian relation (4.18) X r + s = n a br − s − = − δ n, tr( ab )1 , and the bubble slide relations (4.19) a r = a r − X s,t ≥ r − s − t − s + ta † , ar = ar − X s,t ≥ r − s − t − s + ta † . The braid relation(4.20) = also holds for all orientations of the strands. We next recall the basis theorem for
AOB ( A ) . Recall that the cocenter C ( A ) of A is the quotientof A by the subspace spanned by ab − ( − ¯ a ¯ b ba for all homogeneous a, b ∈ A . For a ∈ A , we let ˙ a denote its canonical image in C ( A ) .We define Sym( A ) to be the symmetric superalgebra generated by the vector superspace C ( A )[ x ] ,where x here is an even indeterminate. For n ∈ Z and a ∈ A , let e n ( a ) ∈ Sym( A ) denote(4.21) e n ( a ) := if n < , tr( a ) if n = 0 , ˙ ax n − if n > . This defines a parity-preserving linear map e n : A → Sym( A ) . By [BSW20a, Lem. 7.1], for each n ∈ Z there is a unique parity-preserving linear map h n : A → Sym( A ) such that(4.22) X r + s = n X c ∈ B A ( − r e r ( ac ) h s ( c ∨ b ) = δ n, tr( ab ) , for all a, b ∈ A. In the special case that A = k , Sym( A ) may be identified with the algebra of symmetric functionsso that e n (1) corresponds to the n -th elementary symmetric function and h n (1) corresponds to the n -th complete symmetric function.It follows from (4.18) and (4.22) that we have an homomorphism of superalgebras(4.23) β : Sym( A ) → End
AOB ( A ) ( ) , e n ( a ) ( − n − a n − , h n ( a ) an − , n ≥ . Let X = X n ⊗ · · · ⊗ X and Y = Y m ⊗ · · · ⊗ Y be objects of AOB ( A ) for X i , Y j ∈ {↑ , ↓} . An ( X, Y ) -matching is a bijection between the sets { i : X i = ↑} ⊔ { j : Y j = ↓} and { i : X i = ↓} ⊔ { j : Y j = ↑} . A reduced lift of an ( X, Y ) -matching is a string diagram representing a morphism X → Y such that • the endpoints of each string are points which correspond under the given matching; • there are no floating bubbles and no dots or tokens on any string; • there are no self-intersections of strings and no two strings cross each other more than once.For each ( X, Y ) matching, fix a set D ( X, Y ) consisting of a choice of reduced lift for each ( X, Y ) -matching. Then let D ◦ ( X, Y ) denote the set of all morphisms that can be obtained from theelements of D ( X, Y ) by adding a nonnegative number of dots and one element of B A near to theterminus of each string (i.e. such that there are no crossings between the terminus and the dots andelements of B A ).Using the homomorphism β from (4.23), we have that, for X, Y ∈ AOB ( A ) , Hom
AOB ( A ) ( X, Y ) isa right Sym( A ) -supermodule under the action φθ := φ ⊗ β ( θ ) , φ ∈ Hom
AOB ( A ) ( X, Y ) , θ ∈ Sym( A ) . Theorem 4.7 ([BSW20a, Th. 7.2]) . For
X, Y ∈ AOB ( A ) , the morphism space Hom
AOB ( A ) ( X, Y ) is a free right Sym( A ) -supermodule with basis D ◦ ( X, Y ) . It follows from Theorem 4.7 that the map (4.23) is an isomorphism of superalgebras. By the n = 1 case of (4.18), we see that(4.24) a = a , a ∈ A. Furthermore, it follows from (4.19) that these bubbles are strictly central:(4.25) a = a , a ∈ A. FFINE ORIENTED FROBENIUS BRAUER CATEGORIES 11
For any linear map θ : C ( A ) → k , we can define the specialized oriented Frobenius Brauer category OB ( A, θ ) by imposing on OB ( A ) the additional relation(4.26) a = θ ( a ) , a ∈ A. Similarly, we can define the specialized affine oriented Frobenius Brauer category
AOB ( A, θ ) byimposing on AOB ( A ) the relation (4.26). We will see in (5.1) that, under the categorical actionto be defined in Section 5, the bubbles (4.24) act by multiplication by the supertrace of the map V + → V + , v av . Hence these actions factor through the corresponding specialized categories.5. Categorical action
We are now ready to define the action of the oriented Frobenius Brauer category and the affineoriented Frobenius Brauer category on the category of supermodules for g = g m | n ( A ) . Theorem 5.1.
We have a monoidal superfunctor ψ : OB ( A ) → smod- g given on objects by ↑ 7→ V + , ↓ 7→ V − and on morphisms by ψ ( ) : V + ⊗ V + → V + ⊗ V + , v ⊗ w ( − ¯ v ¯ w w ⊗ v,ψ ( a ) : V + V + , v av,ψ ( ) : k V − ⊗ V + , X v ∈ B + ( − ¯ v v ∨ ⊗ v,ψ ( ) : k V + ⊗ V − , X v ∈ B + v ⊗ v ∨ ,ψ ( ) : V + ⊗ V − → k , v ⊗ w ( − ¯ v ¯ w B ( w ⊗ v ) ,ψ ( ) : V − ⊗ V + → k , v ⊗ w B ( v ⊗ w ) . Proof.
We must show that ψ respects the relations (4.1) to (4.4). We first note that, using thedefinitions (4.5) and (4.13), the maps ψ ( ) : V + ⊗ V − → V − ⊗ V + , ψ ( ) : V − ⊗ V + → V + ⊗ V − , ψ ( ) : V − ⊗ V − → V − ⊗ V − are all given by v ⊗ w ( − ¯ v ¯ w w ⊗ v . Thus we have ψ (cid:18) (cid:19) : v X w ∈ B + ( − ¯ w w ∨ ⊗ w ⊗ v X w ∈ B − ( − ¯ w +¯ v ¯ w w ∨ ⊗ v ⊗ w v, verifying the third equality in (4.3). The remaining relations are similarly verified by direct compu-tation. (cid:3) Note that, for a ∈ A ,(5.1) ψ (cid:16) a (cid:17) = ψ (cid:16) a (cid:17) = X b ∈ B + ( − ¯ v B ( v ∨ , av )1 is multiplication by the supertrace of the map V + → V + , v av . In particular,(5.2) ψ (cid:16) (cid:17) = ψ (cid:16) (cid:17) = sdim( A )1 , where sdim( A ) = m dim( A ¯0 ) + n dim( A ¯1 ) − m dim( A ¯1 ) − n dim( A ¯0 ) is the super dimension of V + .For a supercategory C , let End k ( C ) denote the strict monoidal supercategory of superfunctorsand supernatural transformations. An action of a monoidal supercategory D on a supercategory C is a monoidal superfunctor D →
End k ( C ) . It follows immediately from Theorem 5.1 that OB ( A ) acts on smod- g by X ψ ( X ) ⊗ − , f ψ ( f ) ⊗ − for objects X in OB ( A ) and morphisms f in OB ( A ) . The following result extends this action to AOB ( A ) . Theorem 5.2.
We have a monoidal superfunctor
Ψ :
AOB ( A ) → End k ( smod- g ) given on objectsby ↑ 7→ V + ⊗ − , ↓ 7→ V − ⊗ − and on morphisms by Ψ( f ) = ψ ( f ) ⊗ − , f ∈ { , a , , , , : a ∈ A } , and Ψ( ) : V + ⊗ − → V + ⊗ − is the functor with components Ψ( ) W : V + ⊗ W → V + ⊗ W, v ⊗ w ( v ⊗ w )Ω , for W ∈ smod- gl d ( A ) , where Ω is the element defined in (3.12) .Proof. In light of Theorem 5.1, it suffices to check that
Ψ( ) is a supernatural transformation, whichis straightforward, and that Ψ respects the relations (4.9). To verify the first relation in (4.9), wecompute that Ψ (cid:0) (cid:1) W : V + ⊗ V + ⊗ W → V + ⊗ V + ⊗ W is the map given by u ⊗ v ⊗ w ( − ¯ u ¯ v v ⊗ u ⊗ w X M ∈ B g ( − ¯ u ¯ v (cid:16) ( − ¯ u ¯ M vM ⊗ uM ∨ ⊗ w + ( − (¯ u + ¯ w ) ¯ M vM ⊗ u ⊗ wM ∨ (cid:17) . Similarly, Ψ (cid:0) (cid:1) W : V + ⊗ V + ⊗ W → V + ⊗ V + ⊗ W is the map given by u ⊗ v ⊗ w X M ∈ B g ( − ¯ w ¯ M u ⊗ vM ⊗ wM ∨ X M ∈ B g ( − ¯ u (¯ v + ¯ M )+ ¯ w ¯ M vM ⊗ u ⊗ wM ∨ . Thus, Ψ (cid:0) − (cid:1) W ( u ⊗ v ⊗ w ) = ( − ¯ u ¯ v ( v ⊗ u )Ω ⊗ w (3.13) = Ψ (cid:16) (cid:17) ( u ⊗ v ⊗ w ) . To verify the second relation in (4.9) we compute that, for a ∈ A , we have Ψ (cid:16) a (cid:17) W ( v ⊗ w ) = X M ∈ B g ( − ¯ w ¯ M avM ⊗ wM ∨ = Ψ (cid:16) a (cid:17) W ( v ⊗ w ) . (cid:3) As explained in the introduction, when A = k , Theorems 5.1 and 5.2 recover known results.Furthermore, as noted in Remark 4.6, the definitions of OB ( A ) and AOB ( A ) can be generalizedto allow A to be the two-dimensional Clifford superalgebra. In this case, the actions described inTheorems 5.1 and 5.2 correspond to those described in [BCK19, §4.2 and Th. 4.4] on supermodulesfor the queer Lie superalgebra (see Example 3.5).The center Z ( End ( smod- g )) := End End ( smod- g ) ( ) of the category End ( smod- g ) can be naturallyidentified with Z ( U ( g )) via the map(5.3) ρ : Z ( U ( g )) ∼ = −→ Z ( End ( smod- g )) , u ρ u , where ρ u is the natural transformation whose W -component for W ∈ smod- g is ( ρ u ) W : W → W, w ( − ¯ u ¯ w wu. Then it follows from Theorem 5.2 and (4.23) that we have a homomorphism of superalgebras ρ − ◦ Ψ ◦ β : Sym( A ) → Z ( U ( g )) . The following proposition describes this map explicitly.
FFINE ORIENTED FROBENIUS BRAUER CATEGORIES 13
Proposition 5.3.
The element ρ − ◦ Ψ (cid:16) a r (cid:17) = ( − r ρ − ◦ Ψ ◦ β ( e r +1 ( a )) ∈ Z ( U ( g )) is given by X ≤ i ,...,i r ≤ db ,...,b r ∈ B A ( − a ¯ b r + P rk =1 ¯ b k ¯ b k +1 ( b b ) i ,i ( b b ∨ ) i ,i · · · ( b r b ∨ r − ) i r ,i r − ( b ∨ r +1 ab ∨ r ) i r +1 ,i r , where we adopt the convention that i r +1 = i and b r +1 = b .Proof. For W ∈ smod- gl d ( A ) , we compute that Ψ (cid:16) a r (cid:17) W is the map w X v ∈ B + ( − ¯ v v ∨ ⊗ v ⊗ w X v ∈ B + ( − ¯ v +¯ a ¯ v v ∨ ⊗ ( av ⊗ w )Ω r = X c ∈ B A ≤ k ≤ d X ≤ i ,...,i r ≤ d ≤ j ,...,j r ≤ db ,...,b r ∈ B A ( − ¯ a ¯ c + ¯ w P p ¯ b p + P p Canad. J. Math. , 71(5):1061–1101, 2019. arXiv:1706.09999 , doi:10.4153/cjm-2018-030-8 .[BCNR17] J. Brundan, J. Comes, D. Nash, and A. Reynolds. A basis theorem for the affine oriented Brauer categoryand its cyclotomic quotients. Quantum Topol. , 8(1):75–112, 2017. arXiv:1404.6574 , doi:10.4171/QT/87 .[BE17] J. Brundan and A. P. Ellis. Monoidal supercategories. Comm. Math. 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