aa r X i v : . [ m a t h . R T ] M a y SUPER KAC-MOODY 2-CATEGORIES
JONATHAN BRUNDAN AND ALEXANDER P. ELLIS
Abstract.
We introduce generalizations of Kac-Moody 2-categories in whichthe quiver Hecke algebras of Khovanov, Lauda and Rouquier are replaced bythe quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka. Introduction
Overview.
Kac-Moody 2-categories were introduced by Khovanov and Lauda[KL3] and Rouquier [R]. They have rapidly become accepted as fundamental ob-jects in representation theory, with intimate connections especially to quantumgroups, canonical bases and knot invariants. Rouquier gave a seemingly differentdefinition to Khovanov and Lauda: • Rouquier’s presentation starts from generators and relations for certainunderlying quiver Hecke algebras, adjoins right duals of all the generating1-morphisms, then imposes one more “inversion relation” at the level of2-morphisms. • The Khovanov-Lauda presentation incorporates various additional generat-ing 2-morphisms, and extra relations including biadjointness and cyclicity.These additional generators and relations are useful for various applica-tions, e.g. they are needed in order to extract a candidate for a basis ineach space of 2-morphisms.In [B], the first author has shown that the two versions are actually equivalent. Themain purpose of this article is to extend the computations made in [B] to include super Kac-Moody 2-categories . We will define these shortly following Rouquier’sapproach, starting from certain underlying quiver Hecke superalgebras which wereintroduced already by Kang, Kashiwara and Tsuchioka [KKT]. For the quiver withone odd vertex, the quiver Hecke superalgebra is the odd nilHecke algebra definedindependently in [EKL]; see also [Wa, § following [BE], since it leads to some conceptualsimplifications compared to the approach of [EL]. We proceed to the definitions. Fix once and for all a super-commutative ground ring k = k ¯0 ⊕ k . We are mainly interested in the situationthat k is a field concentrated in even parity. Definition 1.1. A superspace is a Z / k , k )-bimodule in which the left andright actions are related by cv = ( − | c || v | vc ; here and subsequently, | x | denotesthe parity of a homogeneous vector in a superspace. An even linear map betweensuperspaces is a parity-preserving k -module homomorphism. Mathematics Subject Classification : 17B10, 18D10.Research of J.B. supported in part by NSF grant DMS-1161094.
Let SV ec be the Abelian category of all (small) superspaces and even linearmaps. It is a symmetric monoidal category with tensor functor − ⊗ − : SV ec × SV ec → SV ec being the usual tensor product over k , and symmetric braiding defined on objectsby u ⊗ v ( − | u || v | v ⊗ u . (Our notation here follows [BE]: SV ec is the underlyingcategory to the monoidal super category SV ec whose morphisms are not necessarilyhomogeneous linear maps.) Definition 1.2. A supercategory means a SV ec -enriched category, i.e. each mor-phism space is a superspace and composition induces an even linear map. A superfunctor between supercategories is a SV ec -enriched functor, i.e. a functor F : A → B such that the function Hom A ( λ, µ ) → Hom B ( F λ, F µ ) , f F f is aneven linear map for all λ, µ ∈ ob A .Let SC at be the category of all (small) supercategories, with morphisms beingsuperfunctors. Given two supercategories A and B , we define A ⊠ B to be the super-category whose objects are ordered pairs ( λ, µ ) of objects of A and B , respectively,and Hom A ⊠ B (( λ, µ ) , ( σ, τ )) := Hom A ( λ, σ ) ⊗ Hom B ( µ, τ ) . Composition in A ⊠ B is defined using the symmetric braiding in SV ec , so that( f ⊗ g ) ◦ ( h ⊗ k ) = ( − | g || h | ( f ◦ h ) ⊗ ( g ◦ k ). Given superfunctors F : A → A ′ and G : B → B ′ , there is a superfunctor F ⊠ G : A ⊠ B → A ′ ⊠ B ′ sending( λ, µ ) ( F λ, Gµ ) and f ⊗ g F f ⊗ Gg . We have now defined a functor − ⊠ − : SC at × SC at → SC at which makes SC at into a monoidal category. Definition 1.3. A is a category enriched in SC at . See also [BE,Definition 2.2] for the definition of a between 2-supercategories. Remark 1.4.
In [BE, Definition 2.1], the 2-supercategories of Definition 1.3 arecalled strict λ, µ in a 2-supercategory A , there is givena supercategory H om A ( λ, µ ) of morphisms from λ to µ . Elements of Hom A ( λ, µ ) :=ob H om A ( λ, µ ) are in A . For 1-morphisms F, G ∈ Hom A ( λ, µ ), wealso use the shorthand Hom A ( F, G ) for the superspace Hom H om A ( λ,µ ) ( F, G ). Itselements are . We often represent x ∈ Hom A ( F, G ) by the picture xF λ.µ G (1.1)The composition y ◦ x of x with another 2-morphism y ∈ Hom A ( G, H ) is obtainedby vertically stacking pictures: xyF λ.µ HG
UPER KAC-MOODY 2-CATEGORIES 3
The composition law in A gives a coherent family of superfunctors T ν,µ,λ : H om A ( µ, ν ) ⊠ H om A ( λ, µ ) → H om A ( λ, ν )for λ, µ, ν ∈ ob A . Given 2-morphisms x : F → H, y : G → K between 1-morphisms F, H : λ → µ, G, K : µ → ν , we denote T ν,µ,λ ( y ⊗ x ) : T ν,µ,λ ( G, F ) → T ν,µ,λ ( K, H )simply by yx : GF → KH , and represent it by horizontally stacking pictures: xF λ.µ HyGν K When confusion seems unlikely, we will use the same notation for a 1-morphism F as for its identity 2-morphism. With this convention, we have that yH ◦ Gx = yx =( − | x || y | Kx ◦ yF , or in pictures: F λ.µ HGν K xy = F λ.µ HGν K xy = ( − | x || y | F λ.µ HGν K xy . This identity is the super interchange law . The presence of the sign here means thata 2-supercategory is not a 2-category in the usual sense. In particular, diagramsfor 2-morphisms in 2-supercategories are only invariant under rectilinear isotopymodulo signs. Consequently, care is needed with horizontal levels when workingwith odd 2-morphisms diagrammatically: a more complicated diagram such as ν KGyv µ HFxu λ should be interpreted by first composing horizontally then composing vertically.The example just given represents ( vu ) ◦ ( yx ) not ( v ◦ y )( u ◦ x ). Super Kac-Moody 2-categories.
With these foundational definitions behind us,we are ready to introduce the main object of study. We need to fix some additionaldata: • Let I be a (possibly infinite) index set equipped with a parity function I → Z / , i
7→ | i | ; we will say that i ∈ I is even or odd according to whether | i | = ¯0 or ¯1, respectively. If I has odd elements, we make the additionalassumption that 2 is invertible in the ground ring k . • Let ( − d ij ) i,j ∈ I be a generalized Cartan matrix, so d ii = − d ij ≥ i = j , and d ij = 0 ⇔ d ji = 0. We make the additional assumption that | i | = ¯1 ⇒ d ij is even. (1.2) • Pick a complex vector space h and linearly independent subsets { α i | i ∈ I } and { h i | i ∈ I } of h ∗ and h , respectively, such that h h i , α j i = − d ij for all i, j ∈ I . Let P := { λ ∈ h ∗ | h h i , λ i ∈ Z for all i ∈ I } be the weight lattice and Q := L i ∈ I Z α i be the root lattice . • Let g be the Kac-Moody algebra associated to this data with Chevalleygenerators { e i , f i , h i | i ∈ I } and Cartan subalgebra h . J. BRUNDAN AND A. ELLIS • Finally fix units t ij ∈ k × ¯0 such that t ii = 1 , d ij = 0 ⇒ t ij = t ji , (1.3)and scalars s pqij ∈ k ¯0 for 0 < p < d ij , 0 < q < d ji such that s pqij = s qpji , p | i | = ¯1 ⇒ s pqij = 0. (1.4)In case all elements of I are even, the following is the same as the Rouquier’s defi-nition of Kac-Moody 2-category from [R] (viewing the latter as a 2-supercategoryby declaring that all of its 2-morphisms are even). Definition 1.5.
The
Kac-Moody 2-supercategory is the 2-supercategory U ( g ) withobjects P , generating 1-morphisms E i λ : λ → λ + α i and F i λ : λ → λ − α i for each i ∈ I and λ ∈ P , and generating 2-morphisms x : E i λ → E i λ ofparity | i | , τ : E i E j λ → E j E i λ of parity | i || j | , η : 1 λ → F i E i λ of parity ¯0and ε : E i F i λ → λ of parity ¯0, subject to certain relations. To record therelations among these generators, we switch to diagrams, representing the identity2-morphisms of E i λ and F i λ by λ + α i ↑ i λ and λ − α i ↓ i λ , respectively, and the othergenerators by x = • i λ , τ = i jλ , η = i λ , ε = i λ . (1.5)(parity | i | ) (parity | i || j | ) (parity ¯0) (parity ¯0)We denote the n th power of x (under vertical composition) by x ◦ n = • in λ . (1.6)(parity | i | n )First, we have the quiver Hecke superalgebra relations from [KKT]: i jλ = i = j , t ij i j λ if d ij = 0, t ij i j λ • d ij + t ji i j λ • d ji + X
In (1.7), we have drawn multiple dots on the same horizontal level, which is poten-tially ambiguous: our convention for this is that it means the horizontal compositionof x ◦ p and x ◦ q , so that i j λ • • p q := i j λ • • p q . Note further by the assumption (1.4) that s pqij i j λ • • p q = s pqij i j λ • • p q . Similar remarks apply to (1.9) and all other such situations below.Next we have the right adjunction relations : i λ = i λ , i λ = i λ . (1.10)These imply that F i λ + α i is a right dual of E i λ .Finally there are some inversion relations . To formulate these, we first introducea new 2-morphism ji λ := i j λ : E j F i λ → F i E j λ . (1.11)(parity | i || j | )Then we require that the following (not necessarily homogeneous) 2-morphisms areisomorphisms: ji λ : E j F i λ ∼ → F i E j λ if i = j , (1.12) ii λ ⊕ h h i ,λ i− M n =0 i λn • : E i F i λ ∼ → F i E i λ ⊕ ⊕h h i ,λ i λ if h h i , λ i ≥ , (1.13) ii λ ⊕ −h h i ,λ i− M n =0 i λ n • : E i F i λ ⊕ ⊕−h h i ,λ i λ ∼ → F i E i λ if h h i , λ i ≤ . (1.14)Note that (1.13)–(1.14) are 2-morphisms in the additive envelope of U ( g ). Never-theless this defines some genuine relations for U ( g ) itself (rather than its additiveenvelope): we mean that there are some as yet unnamed generating 2-morphismsin U ( g ) which are the matrix entries of two-sided inverses to (1.13)–(1.14). Second adjunction.
Let | i, λ | := | i | ( h h i , λ i + 1) . (1.15)Since | i, λ | = | i, λ ± α j | for any j ∈ I , this only depends on the coset of λ modulo Q . In section 2, we will define some additional 2-morphisms η ′ : 1 λ → E i F i λ and J. BRUNDAN AND A. ELLIS ε ′ : F i E i λ → λ represented diagrammatically by leftward cups and caps: η ′ = iλ , ε ′ = iλ . (1.16)(parity | i, λ | ) (parity | i, λ | )Following the idea of [BHLW], we will normalize these in a different way to [CL, B],in order to salvage some cyclicity. Consequently, our definitions of ε ′ and η ′ dependon the additional choice of units c λ ; i ∈ k × ¯0 for each i ∈ I and λ ∈ P such that c λ + α j ; i = t ij c λ ; i . (1.17)In section 6, we will show that η ′ and ε ′ satisfy the following left adjunction relations : i λ = ( − | i,λ | i λ , i λ = i λ . (1.18)Consequently, Π | i,λ | F i λ + α i is a left dual of E i λ , working now in the Π-envelope U π ( g ) of U ( g ) from [BE, Definition 4.4]; cf. Definition 1.6 below. Further relations.
In sections 3–7, we also derive various other relations from thedefining relations, enough to see in particular that the inverses of the 2-morphisms(1.12)–(1.14) can be written as certain horizontal and vertical compositions of x, τ, ε, η, ε ′ and η ′ , i.e. the 2-morphisms named so far are enough to generate allother 2-morphisms in U ( g ). Some of our extra relations are as follows. • The super analog of Lauda’s infinite Grassmannian relation : Let Sym bethe algebra of symmetric functions over k . Recall Sym is generated bothby the elementary symmetric functions e r ( r ≥
0) and by the completesymmetric functions h s ( s ≥ ′ )], elementary and completesymmetric functions are related by the equationse = h = 1 , X r + s = n ( − s e r h s = 0 for all n > . Take i ∈ I , λ ∈ P and set h := h h i , λ i . If i is even, Lauda [L] observedalready that there exists a unique homomorphism β λ ; i : Sym → End U ( g ) (1 λ ) (1.19)such thate n c − λ ; i i λ • n + h − if n > − h , h n ( − n c λ ; i iλ • n − h − if n > h ,bearing in mind the new normalization of bubbles. The analog of this when i is odd is as follows. Let Sym[ d ] be the supercommutative superalgebraobtained from Sym by adjoining an odd generator d with d = 0. Thenthere exists a unique homomorphism β λ ; i : Sym[ d ] → End U ( g ) (1 λ ) (1.20)such thate n c − λ ; i i λ • n + h − if n > − h n ( − n c λ ; i iλ • n − h − if n > h d e n c − λ ; i i λ • n + h if n ≥ − h d h n ( − n c λ ; i iλ • n − h if n ≥ h . UPER KAC-MOODY 2-CATEGORIES 7
Furthermore, lettingSYM := O i even Sym ⊗ O i odd Sym[ d ] (1.21)where the tensor products are taken in some fixed order, there is surjective homomorphism β λ : SYM ։ End U ( g ) (1 λ ) , (1.22)defined by taking the product of the maps β λ ; i applied to the i th tensorfactor of SYM for all i ∈ I . • Centrality of odd bubbles : Assuming i ∈ I is odd, we introduce the odd2-morphism iλ := β λ ; i ( d ) . (1.23)We call this the odd bubble of color i . By the super interchange law itsquares to zero: (cid:18) i λ (cid:19) = 0 . (1.24)We show moreover that odd bubbles are central in U ( g ) in the sense that i j λ = ij λ , i j λ = ij λ (1.25)for all j ∈ I . (This means that it would be reasonable to set odd bubbles tozero, imposing additional relations iλ = 0 for all odd i ∈ I and λ ∈ P .) • Cyclicity properties : If i is even then • i λ = i λ • , (1.26)i.e. even dots are cyclic. However if i is odd we have that • i λ = 2 i i λ − i λ • . (1.27)In all cases, crossings satisfy ij λ = j i λ . (1.28) Nondegeneracy Conjecture.
Let
F, G : λ → µ be some 1-morphisms in U ( g ). Insection 8, we construct an explicit set (cid:8) f ( σ ) (cid:12)(cid:12) σ ∈ c M ( F, G ) (cid:9) of 2-morphisms whichgenerates Hom U ( g ) ( F, G ) as a right SYM-module; here the action of p ∈ SYM isby horizontally composing on the right with β λ ( p ). This puts us in position toformulate the following conjecture, which is the appropriate generalization of thenondegeneracy condition formulated by Khovanov and Lauda in [KL3, § F = G = 1 λ , it implies that the homomorphism β λ from (1.22) isan isomorphism. J. BRUNDAN AND A. ELLIS
Conjecture: Hom U ( g ) ( F, G ) is a free SYM -module with basis (cid:8) f ( σ ) (cid:12)(cid:12) σ ∈ c M ( F, G ) (cid:9) . We cannot prove this at present. We will discuss its signficance and some possibleapproaches to its proof later on in the introduction.
Gradings.
By a graded superspace , we mean a superspace equipped with an addi-tional Z -grading V = L n ∈ Z V n = L n ∈ Z V n, ¯0 ⊕ V n, ¯1 . Let GSV ec be the symmetricmonoidal category of graded superspaces and degree-preserving even linear maps.Mimicking Definition 1.2, a graded supercategory means a GSV ec -enriched category.Let GSC at be the monoidal category of all (small) graded supercategories. Finally,mimicking Definition 1.3, a graded 2-supercategory means a category enriched in GSC at . Thus, it is a 2-supercategory whose 2-morphism spaces are graded super-spaces, and horizontal and vertical composition respect these gradings. We willsoon need the following universal construction from [BE, Definition 6.10]: Definition 1.6.
Suppose that A is a graded 2-supercategory. Its ( Q, Π) -envelope A q,π is the graded 2-supercategory with the same objects as A , 1-morphisms definedfromHom A q,π ( λ, µ ) := (cid:8) Q m Π a F (cid:12)(cid:12) for all F ∈ Hom A ( λ, µ ), m ∈ Z and a ∈ Z / (cid:9) with the horizontal composition law ( Q n Π b G )( Q m Π a F ) := Q m + n Π a + b ( GF ), and2-morphisms defined fromHom A q,π ( Q m Π a F, Q n Π b G ) := (cid:8) x n,bm,a (cid:12)(cid:12) for all x ∈ Hom A ( F, G ) (cid:9) viewed as a superspace with operations x n,bm,a + y n,bm,a := ( x + y ) n,bm,a , c ( x n,bm,a ) :=( cx ) n,bm,a for c ∈ k , and grading deg( x n,bm,a ) := deg( x ) + n − m , (cid:12)(cid:12) x n,bm,a (cid:12)(cid:12) := | x | + a + b .Representing x n,bm,a by the picture x λµ GF banm for x as in (1.1), the vertical and horizontal composition laws for 2-morphisms in A q,π are defined in terms of the ones in A as follows: y cbnm ◦ x baml := xy canl , (1.29) y dcnm x balk := ( − c | x | + b | y | + ac + bc xy b + da + cl + nk + m . (1.30)For each object λ , there are distinguished 1-morphisms q λ := Q Π ¯0 λ , q − λ := Q − Π ¯0 λ and π λ := Q Π ¯1 λ in End A q,π ( λ ). Moreover, there are 2-isomorphisms σ λ : q λ ∼ → λ , ¯ σ λ : q − λ ∼ → λ and ζ λ : π λ ∼ → λ , all induced by the identity 2-morphism 1 λ . These give the required structure maps to make A q,π into a graded ( Q, Π) -2-supercategory in the sense of [BE, Definition 6.5].Assume for the remainder of the introduction that the Cartan matrix A is sym-metrizable, so that there exist positive integers ( d i ) i ∈ I such that d i d ij = d j d ji for UPER KAC-MOODY 2-CATEGORIES 9 all i, j ∈ I . Assume moreover that k is a field, and that the parameters chosenabove satisfy the following homogeneity condition : s pqij = 0 ⇒ pd ji + qd ij = d ij d ji . (1.31)Then we can put an additional Z -grading on U ( g ) making it into a graded 2-supercategory, by declaring that the generators from (1.5) and (1.16) are of thedegrees listed in the following table: x τ η ε η ′ ε ′ d i d i d ij d i (1 + h h i , λ i ) d i (1 − h h i , λ i ) d i (1 − h h i , λ i ) d i (1 + h h i , λ i )Let U q,π ( g ) denote the ( Q, Π)-envelope of U ( g ) in the sense of Definition 1.6. The un-derlying 2-category U q,π ( g ) consists of the same objects and 1-morphisms as U q,π ( g )but only its even 2-morphisms of degree zero. Also let ˙ U q,π ( g ) be the idempotentcompletion of the additive envelope of U q,π ( g ). Both of U q,π ( g ) and ˙ U q,π ( g ) are( Q, Π) -2-categories in the sense of [BE, Definition 6.14]. In particular, they areequipped with distinguished objects q = ( q λ ) and π = ( π λ ) in their Drinfeld cen-ters. Relation to the Ellis-Lauda 2-category.
Suppose that g is odd sl , i.e. I isan odd singleton. Then the 2-category ˙ U q,π ( g ) is 2-equivalent to the 2-categoryintroduced [EL]. We do not think that this is an important result going forward,so we will only give a rough sketch of its proof in the next paragraph. Our newapproach to the definition seems to be both conceptually more satisfactory and lessprone to errors when working with the relations. So our point of view really is that,henceforth, one should simply replace the object in [EL] with the one here.Briefly, the idea is simply to construct quasi-inverse 2-functors between the Ellis-Lauda 2-category U EL and our ˙ U q,π ( g ) by verifying relations. Let us write simply E, F and h for E i , F i and h i for the unique i ∈ I . Also we take d i := 1 andidentify P ↔ Z so λ ↔ h h, λ i . Then, the appropriate 2-functor in the direction U EL → ˙ U q,π ( g ) is the identity on the object set P . It sends the generating 1-morphisms E λ , F λ and Π λ from [EL, § E λ , Π ¯ λ +¯1 F λ and π λ , respectively. On the generating 2-morphisms from [EL, § • λ
7→ • λ ¯0¯102 λ λ ¯0¯10 − , λ
7→ − λ ¯0¯000 , • λ ( − λ +1 • λ ¯ λ ¯ λ +¯102 , λ
7→ − λ ¯0¯10 − , λ λ ¯1¯100 , λ λ ¯ λ ¯ λ , λ λ ¯ λ ¯ λ , λ λ ¯1¯100 , λ λ ¯ λ +¯1¯00 λ +1 , λ λ ¯ λ +¯1¯001 − λ , λ λ ¯0¯000 , λ λ − λ ¯ λ +¯1¯ λ +¯101 − λ , λ +1 λ λ ¯0¯00 λ +1 , λ λ ¯0¯000 . We leave it to the reader to compare the relations in [EL] with our relations, andto construct a quasi-inverse 2-functor in the other direction. In fact, when doingthis carefully, one uncovers some inconsistencies in the relations of [EL]; e.g. therelation [EL, (3.1)] is wrong in the case λ = 0 (due to an error in the last sentenceof the proof of [EL, Lemma 5.1] related to the nilpotency of the odd bubble). Decategorification Conjecture.
Recall finally that the
Grothendieck ring of anadditive 2-category A is K ( A ) := M λ,µ ∈ ob A K ( H om A ( λ, µ )) (1.32)where the K on the right hand side is the usual split Grothendieck group ofthe additive category H om A ( λ, µ ). It is a locally unital ring with distinguishedidempotents { λ | λ ∈ ob A } . If A is a ( Q, Π)-2-category, then K ( A ) is also linearover L := Z [ q, q − , π ] / ( π − q and π acting by multiplication by the classesof the distinguished objects q and π of the Drinfeld center.This discussion applies in particular to the ( Q, Π)-2-category ˙ U q,π ( g ), so that K ( ˙ U q,π ( g )) is a locally unital L -algebra with idempotents { λ | λ ∈ P } . Alsolet ˙ U q,π ( g ) L be the L -form of the idempotented version of the covering quantizedenveloping algebra associated to g introduced by Clark, Hill and Wang in [CHW1];see section 9. By similar arguments to those of [KL3], using also some results from[HW], we will show in section 11 that there is a surjective homomorphism of locallyunital L -algebras γ : ˙ U q,π ( g ) L ։ K ( ˙ U q,π ( g )) (1.33)sending e i λ and f i λ to [ E i λ ] and [ F i λ ], respectively. Moreover, also just like in[KL3], we will show in section 12 that the Nondegeneracy Conjecture formulatedabove, together with an additional assumption of bar-consistency on the Cartandatum, implies the truth of the following:Conjecture: γ is an isomorphism. Discussion.
In the purely even case, i.e. when all i ∈ I are even, the Nonde-generacy Conjecture (hence, the Decategorification Conjecture) was established byKhovanov and Lauda in [KL3, § g = sl n . In [W], Webster has proposeda proof of the Nondegeneracy Conjecture for all purely even types. There is also acompletely different proof of the Decategorification Conjecture based on results of[KK], which is valid in all finite types; see e.g. [BD, Corollary 4.21].Turning to the odd case, the Decategorification Conjecture for odd sl is provedin [EL, Theorem 8.4]. The only additional finite type possibilities come from odd b n , i.e. type b n with the element of I corresponding to the short simple rootchosen to be odd. For these, the Decategorification Conjecture may be deducedfrom [KKO1, KKO2]. We hope that Webster’s methods from [W] can be extendedto the super case to prove the Nondegeneracy Conjecture in general, but there is agreat deal of work still to be done in order to see this through. As a first step, wewould like to see the proof of the Nondegeneracy Conjecture from [KL3] extendedin order to include all odd b n , and hope to address this in subsequent work.Assuming the Decategorification Conjecture, one gets an interesting basis forthe covering quantum group ˙ U q,π ( g ) L coming from the isomorphism classes of theindecomposable objects of ˙ U q,π ( g ). In symmetric types, this should coincide (upto parity shift) with the canonical basis from [C, Theorem 4.14]. For odd b , thisassertion follows already from the results of [EL]. UPER KAC-MOODY 2-CATEGORIES 11
In a different direction, it should now be possible to develop super analogsof many of the foundational structural results proved by Chuang-Rouquier andRouquier in [CR, R]. Various applications, e.g. to spin representations of symmet-ric groups and to representations of the Lie superalgebra q ( n ), are expected.2. More generators
In sections 2–8, we assume that the ground ring k is as in Definition 1.1, and let U ( g ) be the Kac-Moody 2-supercategory from Definition 1.5. We begin by definingvarious additional 2-morphisms in U ( g ). Definition 2.1.
We have the downward dots and crossings , which are the rightmates of the upward dots and crossings: • i λ := • i λ , j iλ := j i λ , (2.1)(parity | i | ) (parity | i || j | ) • in λ := • i λ ◦ n = ( − | i |⌊ n ⌋ • i λn . (2.2)(parity | i | n )The sign in (2.2) is easily checked using the diagrammatics; see also [KKO2, Propo-sition 7.14]. Using (1.10) and (1.11), we deduce: i λ • n = ( − | i |⌊ n ⌋ • ni λ , i λ • n = ( − | i |⌊ n ⌋ • ni λ , (2.3) i j λ = i j λ , ij λ = i jλ , (2.4) i j λ = i j λ , ij λ = i jλ . (2.5) Definition 2.2.
We define the leftward crossing and various leftward cups and caps .First define jiλ : F i E j λ → E j F i λ , iλn ♦ : 1 λ → E i F i λ , iλn ♦ : F i E i λ → λ , (2.6)(parity | i || j | ) (parity | i | n ) (parity | i | n )by declaring that jiλ := (cid:18) ji λ (cid:19) − if i = j , (2.7) − iiλ ⊕ h h i ,λ i− M n =0 iλn ♦ := (cid:18) ii λ ⊕ h h i ,λ i− M n =0 i λn • (cid:19) − if h h i , λ i ≥ − iiλ ⊕ −h h i ,λ i− M n =0 iλn ♦ := (cid:18) ii λ ⊕ −h h i ,λ i− M n =0 i λn • (cid:19) − if h h i , λ i ≤ U ( g ). Then, remembering the scalars c λ ; i chosenfor (1.17), we set η ′ = iλ := c λ ; i iλ h h i ,λ i− ♦ if h h i , λ i > − | i,λ | c λ ; i i λ −h h i ,λ i • if h h i , λ i ≤
0, (2.10) ε ′ = i λ := c − λ ; i i λ −h h i ,λ i− ♦ if h h i , λ i < − ( − | i |h h i ,λ i c − λ ; i iλ h h i ,λ i • if h h i , λ i ≥
0, (2.11)both of which are of parity | i, λ | . The following are immediate from these definitions. i iλ = h h i ,λ i− X n =0 in ♦ i • n λ − i i λ , ii λ = −h h i ,λ i− X n =0 i n ♦ i • nλ − i i λ , (2.12) i λ = 0 , iλn • = 0 , i λ • n = δ n, h h i ,λ i− c λ ; i λ all for 0 ≤ n < h h i , λ i , (2.13) i λ = 0 , n • iλ = 0 , iλ • n = δ n, −h h i ,λ i− c − λ ; i λ all for 0 ≤ n < −h h i , λ i . (2.14) Definition 2.3.
We give meaning to negatively dotted bubbles by making the fol-lowing definitions for n < i λ • n := − ( − | i | ( n + h h i ,λ i +1) c λ ; i iλ • ♦ −h h i ,λ i− n − if n > h h i , λ i − c λ ; i λ if n = h h i , λ i − , n < h h i , λ i − , (2.15) UPER KAC-MOODY 2-CATEGORIES 13 iλ • n := − ( − | i | ( n + h h i ,λ i +1) c − λ ; i i λ • ♦h h i ,λ i− n − if n > −h h i , λ i − c − λ ; i λ if n = −h h i , λ i − , n < −h h i , λ i − . (2.16)Sometimes we will use the following convenient shorthand for dotted bubbles forany n ∈ Z : i λ • n + ∗ := i λ • n + h h i ,λ i− , iλ • n + ∗ := iλ • n −h h i ,λ i− , (2.17)both of which are of parity | i | n . Also, assuming that i ∈ I is odd, we introduce the odd bubble i λ := c − λ ; i i • h h i ,λ i λ if h h i , λ i ≥ c λ ; i λ • −h h i ,λ i i if h h i , λ i ≤
0. (2.18)There is no ambiguity in this definition in the case h h i , λ i = 0 thanks to the followingcalculation: c λ ; i iλ ( . ) = − c λ ; i i λ ( . ) = i λ ( . ) = − c − λ ; i i λ ( . ) = c − λ ; i iλ . The Chevalley involution
The next task is to construct an important symmetry of U ( g ). For this, we needsome preliminary lemmas. Lemma 3.1.
The following relations hold for all n ≥ : i jλ • n − ( − | i || j | n i jλ • n = δ i,j X r,s ≥ r + s = n − ( − | i | s i j λ •• sr , (3.1) i jλ • n − ( − | i || j | n i jλ • n = δ i,j X r,s ≥ r + s = n − ( − | i | s i j λ •• sr , (3.2) ij λ • n − ( − | i || j | n ij λ • n = δ i,j X r,s ≥ r + s = n − ( − | i | r ji λ • r • s , (3.3)( − | i || j | n ij λ • n − ij λ • n = δ i,j X r,s ≥ r + s = n − ( − | i | r ji λ • s • r , (3.4)( − | i || j | n i jλ • n − i jλ • n = δ i,j X r,s ≥ r + s = n − ( − | i | s i j λ •• sr , (3.5) ( − | i || j | n i jλ • n − i jλ • n = δ i,j X r,s ≥ r + s = n − ( − | i | s i j λ •• sr . (3.6) Proof.
The first two relations follow inductively from (1.8). The rest then follow byrotating clockwise, i.e. attach rightward caps to the top right strands and rightwardcups to the bottom left strands then use (2.3)–(2.5). (cid:3)
Lemma 3.2.
The following relations hold: ( − | i || j | i jλ = if i = j , t ij i j λ if d ij = 0 , ( − | i |⌊ dij ⌋ t ij i j λ • d ij + ( − | j |⌊ dji ⌋ t ji i j λ • d ji + X
Rotate (1.7) clockwise as explained in the proof of Lemma 3.1. (cid:3)
Lemma 3.3.
The following relations hold: j ki λ − j ki λ = X r,s ≥ r + s = d ij − ( − | i | s t ij kij λ •• r s + X
Rotate (1.9) clockwise. (cid:3)
Definition 3.4.
For a supercategory A , we write A sop for the supercategory withthe same objects, morphismsHom A sop ( λ, µ ) := Hom A ( µ, λ ) , UPER KAC-MOODY 2-CATEGORIES 15 and new composition law defined from f sop ◦ g sop := ( − | f || g | ( g ◦ f ) sop , wherewe denote a morphism f : λ → µ in A viewed as a morphism in A sop by f sop : µ → λ . For a 2-supercategory A , we write A sop for the 2-supercategory withthe same objects as A , and morphism categories defined from H om A sop ( λ, µ ) := H om A ( λ, µ ) sop . Horizontal composition in A sop is the same as in A . Here is thecheck of the super interchange law in A sop :( x sop y sop ) ◦ ( u sop v sop ) = ( xy ) sop ◦ ( uv ) sop = ( − ( | x | + | y | )( | u | + | v | ) (( uv ) ◦ ( xy )) sop = ( − | x || u | + | y || u | + | y || v | (( u ◦ x )( v ◦ y )) sop = ( − | x || u | + | y || u | + | y || v | ( u ◦ x ) sop ( v ◦ y ) sop = ( − | y || u | ( x sop ◦ u sop )( y sop ◦ v sop ) . We will often appeal to the following proposition to establish mirror images ofrelations in a horizontal axis. (This formulation is more convenient than the versionin [B, Theorem 2.3], since ω really is an involution of U ( g ) rather than a map toanother Kac-Moody 2-category.) Proposition 3.5.
There is a -supercategory isomorphism ω : U ( g ) ∼ → U ( g ) sop defined by the strict 2-superfunctor ω given on objects by ω ( λ ) := − λ , on generating1-morphisms by ω ( E i λ ) := F i − λ and ω ( F i λ ) := E i − λ , and on generating 2-morphisms by • i λ
7→ • i − λ sop , i jλ
7→ − ( − | i || j | i j − λ sop , i λ i − λ sop , i λ i − λ sop . Moreover we have that ω = id , as follows from the following describing the effectof ω on the other named -morphisms in U ( g ) : • i λ
7→ • i − λ sop , i jλ
7→ − ( − | i || j | i j − λ sop , ij λ
7→ − ( − | i || j | ji − λ sop , ijλ
7→ − ji − λ sop , iλn ♦ ( − | i | n i − λ n ♦ sop , iλn ♦ ( − | i | n i − λn ♦ sop , iλ ( − | i,λ | c λ ; i c − λ ; i i − λ sop , iλ ( − | i,λ | c − λ ; i c − − λ ; i i − λ sop , i λ • n ( − ( n +1) | i,λ | c λ ; i c − λ ; i i − λ • n sop , i λ ( − | i,λ | i sop − λ , iλ • n ( − ( n +1) | i,λ | c − λ ; i c − − λ ; i i − λ • n sop . Proof.
This is very similar to the proof of [B, Theorem 2.3] but the signs areconsiderably more subtle, so we include a few remarks. Note to start with that ω should send x ◦ n (vertical composition computed in U ( g )) to ω ( x ) ◦ n (vertical composition computed in U ( g ) sop ), so that ω • in λ = ( − | i |⌊ n ⌋ • in − λ sop . (3.10)It is important that the sign here matches the signs in (2.3). The proof of theexistence of ω amounts to checking relations. For example, to verify (1.9) in thecase i = k = j , one needs to show in view of (3.10) that( − | i || j | + | i | i j iλ − i j iλ = X r,s ≥ r + s = d ij − ( − | i | ( | j | + s + ⌊ r ⌋ + ⌊ s ⌋ ) t ij i j iλ • • r s + X
We proceed to prove analogs of the relations (2.3) and (3.3)–(3.4) for leftwardcups, caps and crossings.
Proposition 4.1.
The following relations hold for all n ≥ : jiλ • n − ( − | i || j | n jiλ • n = δ i,j X r,s ≥ r + s = n − ( − | i | s ij λ • r • s , (4.1)( − | i || j | n jiλ • n − jiλ • n = δ i,j X r,s ≥ r + s = n − ( − | i | ( h h i ,λ i + s ) ij λ • r • s , (4.2)( − | i |⌊ n ⌋ • niλ = • n iλ if | i | n = ¯0 , ( − h h i ,λ i • n iλ + 2 ii λ • n − if | i | n = ¯1 , (4.3) UPER KAC-MOODY 2-CATEGORIES 17 ( − | i |⌊ n ⌋ • n iλ = • niλ if | i | n = ¯0 , ( − h h i ,λ i • niλ + 2 ii λ • n − if | i | n = ¯1 . (4.4) Proof.
Let h := h h i , λ i . The relations (4.1)–(4.2) follow easily by induction startingfrom the case n = 1, which asserts: jiλ • − ( − | i || j | jiλ • = ( − | i || j | ( h +1) jiλ • − ( − | i || j | h jiλ • = δ i,j ij λ . It suffices to prove this under the assumption that h ≥
0; the case h < n = 1 case of (3.3)–(3.4) on top and bottom with aleftward crossing, then simplifies using (2.7) in case i = j or (2.3) and (2.10)–(2.14)in case i = j .For (4.3)–(4.4), we just need to prove the former, since the latter then followson applying ω . When i is even, (4.3) was already established in [B, Theorem 5.6],so let us assume for brevity that i is odd (though the argument here can easily beadapted to even i too). When n = 1 we must prove: • iλ = ( − h • iλ + 2 ii λ . If h < E i F i λ ⊕ ⊕− hλ ∼ → F i E i λ from (1.14) to reduce to checking i λ • = ( − h i λ • + 2 ii λ , iλ •• m = ( − h iλ • • m + 2 i • mi λ for all 0 ≤ m < − h . The first identity here is easily deduced from (3.3) and (2.14),while the second follows using (2.14) and the definition (2.18). Now assume that h ≥
0. Then we have: • iλ ( . ) = ( − h +1 c − λ ; i iλh • • ( . ) = ( − h c − λ ; i iλh +1 • + c − λ ; i ii λ • h ( . ) = c − λ ; i iλh • • + ii λ ( . ) = − c − λ ; i iλh •• + 2 ii λ ( . ) = ( − h • iλ + 2 ii λ . Thus we have proved the desired relation when n = 1. Applying it twice and using(1.24), we deduce that •• iλ = − •• iλ , which is the desired relation for n = 2.The general case follows easily from the two special cases established so far. (cid:3) Infinite Grassmannian relations
Recall the shorthand for dotted bubbles from (2.17), and that the odd bubble i λ squares to zero. Our next proposition implies that the homomorphisms β λ ; i from (1.19)–(1.20) in the introduction are well defined. In terms of these maps, itshows moreover that i λ • n + ∗ = c λ ; i β λ ; i (e n ) if | i | = ¯0 ,c λ ; i β λ ; i (cid:16) e ⌊ n ⌋ (cid:17) if | i | = ¯1 and n is even ,c λ ; i β λ ; i (cid:16) d e ⌊ n ⌋ (cid:17) if | i | = ¯1 and n is odd, (5.1) iλ • n + ∗ = c − λ ; i β λ ; i (( − n h n ) if | i | = ¯0 ,c − λ ; i β λ ; i (cid:16) ( − ⌊ n ⌋ h ⌊ n ⌋ (cid:17) if | i | = ¯1 and n is even ,c − λ ; i β λ ; i (cid:16) ( − ⌊ n ⌋ d h ⌊ n ⌋ (cid:17) if | i | = ¯1 and n is odd, (5.2)for all n ≥
0. This extends the infinite Grassmannian relation first introduced in[L]; see also [EL, Proposition 3.5] for a related result in the odd case.
Proposition 5.1.
The following relations hold: i λ • n + ∗ = 0 if n < , i λ • ∗ = c λ ; i λ , (5.3) iλ • n + ∗ = 0 if n < , iλ • ∗ = c − λ ; i λ . (5.4) Also the following hold for all t > : X r,s ≥ r + s = t i • r + ∗ • s + ∗ i λ = 0 if i is even , (5.5) X r,s ≥ r + s = t i • r + ∗ • s + ∗ i λ = 0 if i is odd. (5.6) Finally if i is odd, the following hold for all n ∈ Z : • n +1+ ∗ iλ = • n + ∗ i λi , • n +1+ ∗ iλ = • n + ∗ i λi . (5.7) Proof.
Let h := h h i , λ i . The equations (5.3)–(5.4) are implied by (2.13)–(2.16). Forthe rest, we first assume that h ≥ X r,s ∈ Z r + s = t − ( − | i | s • si i • rλ ( . ) = ( . ) h X n =0 ( − | i | ( n +1) • − n − i i • n + t − λ + X r ≥− ,s ≥ r + s = t − ( − | i | s • si i • rλ ( . ) = ( − | i | ( h +1) c − λ ; i i • h + t − − h − X n =0 ( − | i | ( h +1) c − λ ; i i • ♦ h ni • n + t − λ + X r ≥− ,s ≥ r + s = t − ( − | i | s • si i • rλ UPER KAC-MOODY 2-CATEGORIES 19( . ) = − ( − | i | ( h +1) c − λ ; ih • i λt − • + X r,s ≥ r + s = t − ( − | i | s • si i • rλ + ( − | i | ( t +1) • t − i i • − λ ( . ) = ( . ) ( − | i | i λt − • + X r,s ≥ r + s = t − ( − | i | s • si i • rλ + ( − | i | ( t +1) δ h, c λ ; i • t − iλ ( . ) = ( − | i | t i λt − • + ( − | i | ( t +1) δ h, c λ ; i • t − iλ ( . ) = ( . ) ( − | i | ( t +1) δ h, c λ ; i i λt − • + • t − iλ ( . ) = 0 . This establishes the first of the following identities, and the second follows fromthat on supercommuting the bubbles then applying the Chevalley involution fromProposition 3.5: for all t > X r,s ∈ Z r + s = t − ( − | i | s • si i • rλ = 0 if h ≥ X r,s ∈ Z r + s = t − ( − | i | r i • r • si λ = 0 if h ≤ i is even, (5.8) implies (5.5), and there is nothing more to be done.For the remainder of the proof we assume that i is odd. Take n > n + h + 1 is odd. We have that • ni λ ( . ) = ( − h •• n − i λ + 2 i λ • n − i ( . ) = − • ni λ + 2 i λ • n − i . This shows that • ni λ = i λ • n − i (5.9)assuming n > n + h + 1 is odd. A similar argument for clockwise bubblesshows that • n i λ = • n − i λi (5.10)assuming that n > n + h + 1 is odd. Now we proceed to show by ascendinginduction on n that (5.10) also holds when n ≤ n + h + 1 is odd. Thisstatement is vacuous if n < h , and it is also clear in case n = h thanks to thedefinition (2.18). So assume that h < n ≤ n + h + 1 is odd, and that (5.10) has been proved for all smaller n with n + h + 1 odd. From (5.8), we get that X r,s ∈ Z r + s = n − h − r + h +1 odd i • r • si λ − X r,s ∈ Z r + s = n − h − r + h +1 even i • r • si λ = 0 . The terms in the first summation here are zero unless s ≥ − h −
1, hence, r ≤ n . Inthe second summation we always have that s + h + 1 is odd, hence, terms here arezero unless s > − h − ≥
0. Applying (5.9) to each non-zero term in the secondsummation, we deduce that X r,s ∈ Z r + s = n − h − r + h +1 odd i • r • si λ − X r,s ∈ Z r + s = n − h − r + h +1 even i • ri • s − i λ = 0 . Now we reindex the second summation, replacing r by r − s by s + 1, todeduce that X r,s ∈ Z r + s = n − h − r + h +1 odd i • r • si λ − i • r − i • si λ = 0 . In view of the induction hypothesis, all of the terms here in which r < n vanish.This just leaves us with the term r = n , when s = − h − • si λ = c − λ ; i λ ,which we can cancel to establish the desired instance of (5.10). This completesthe induction. Hence, we have established the first equation from (5.7); the secondfollows from that using Proposition 3.5.It just remains to prove (5.6). We explain this assuming that h ≤
0; then onecan apply the Chevalley involution to get the other case. From (5.8) we get for any t > X r,s ≥ r + s =2 t ( − r i • r + h − • s − h − i λ = 0 . In all the terms of this summation we have that r ≡ s (mod 2). If both r and s are odd, we can apply (5.7) twice to pull out two odd bubbles, hence, these termsvanish thanks to (1.24). This leaves just the terms in which both r and s are even,which is exactly what is needed to establish the identity (5.6). (cid:3) Once we have proved the next two corollaries, we will not need to refer to thedecorated leftward cups and caps again.
Corollary 5.2.
The following relations hold: iλn ♦ = X r ≥ ( − | i | ( h h i ,λ i + n + r +1) iλi • − n − r − • r if ≤ n < h h i , λ i , (5.11) UPER KAC-MOODY 2-CATEGORIES 21 iλn ♦ = X r ≥ ( − | i | ( h h i ,λ i + n + r +1) i • ri λ • − n − r − if ≤ n < −h h i , λ i . (5.12) Proof.
We explain the proof of (5.11); the proof of (5.12) is entirely similar or itmay be deduced by applying ω using also (4.4). Let h := h h i , λ i >
0. Rememberingthe definition (2.8), it suffices to show that the vertical composition consisting of(1.13) on top of − iiλ ⊕ h − M n =0 X r ≥ ( − | i | ( h + n + r +1) iλi • − n − r − • r is equal to the identity. Using (2.12)–(2.13), this reduces to checking that X r ≥ ( − | i | ( h + n + r +1) i λ • r − n − r − • i = 0 if 0 ≤ n < h , (5.13) X r ≥ ( − | i | ( h + n + r +1) i • m + rλ • − n − r − i = δ m,n λ if 0 ≤ m, n < h . (5.14)For (5.13), each term in the summation is zero: if r ≥ h the counterclockwise dottedbubble is zero by (5.4); if 0 ≤ r < h one commutes the dots past the crossing using(3.3) then applies (2.13). To prove (5.14), note by (5.3)–(5.4) that in order for i • m + r λ to be non-zero we must have that r ≥ h − m −
1, while for iλ • − n − r − to be non-zero we must have r ≤ h − n −
1. Hence, we may assume that m ≥ n ,and are done for the same reasons in case m = n . If m > n the left hand side of(5.14) is equal to X r,s ≥ r + s = m − n ( − | i | ( m + n + r ) i • r + h − λ • s − h − i . Now one shows that this is zero using (5.5)–(5.7) and (1.24); when i is odd itis convenient when checking this to treat the cases m ≡ n (mod 2) and m n (mod 2) separately. (cid:3) Corollary 5.3.
The following relations hold: i iλ = h h i ,λ i− X n =0 X r ≥ ( − | i | ( h h i ,λ i + n + r +1) iλi • − n − r − • r i • n − i i λ , (5.15) ii λ = −h h i ,λ i− X n =0 X r ≥ ( − | i | ( h h i ,λ i + n + r +1) i • riλ • − n − r − • ni − i i λ . (5.16) Proof.
Substitute (5.11)–(5.12) into (2.12). (cid:3)
Corollary 5.4.
The following relations hold: i λ = h h i ,λ i X n =0 ( − | i | n i • ni λ • − n − , i λ = − −h h i ,λ i X n =0 ( − | i | ( n +1) i • − n − i λ • n . (5.17) Hence, for n ≥ we have: n • i λ = n + h h i ,λ i +2 X r =0 ( − | i |h h i ,λ i r i n − r − • i r • λ , (5.18) n • i λ = n + h h i ,λ i +2 X r =0 ( − | i | r i n − r − • i r • λ , (5.19) n • i λ = − n −h h i ,λ i X r =0 ( − | i | ( h h i ,λ i r +1) ir • in − r − • λ , (5.20) n • i λ = − n −h h i ,λ i X r =0 ( − | i | ( r +1) ir • in − r − • λ . (5.21) Proof.
We first prove (5.17). By our usual argument with the Chevalley involution,it suffices to prove the left hand relation. We are done already by (2.14) if h := h h i , λ i <
0. If h ≥ i λ ( . ) = − ( − | i | h c − λ ; i i λ • h ( . ) = ( − | i | h c − λ ; i i λ • h − X ≤ n
UPER KAC-MOODY 2-CATEGORIES 23 Left adjunction
The leftward cups and caps form the unit and counit of another adjunction.
Lemma 6.1.
The following relations hold: ii λ = ii λ if h h i , λ i ≤ − , (6.1) ii λ = iiλ if h h i , λ i ≥ − . (6.2) Proof.
Let h := h h i , λ i for short. First we prove (6.1), so h ≤ −
1. We claim that − i i λ = ii λ − δ h, − c − λ ; i i i λ . (6.3)To establish the claim, we vertically compose on the bottom with the isomorphism ii i λ ⊕ − h − M n =0 ii λ n • arising from (1.14) to reduce to showing equivalently that − ii λ = ii λ − δ h, − c − λ ; i i i λ , (6.4) − ii λn • = ii λn • − δ h, − c − λ ; i i λ for 0 ≤ n ≤ − h −
1. (6.5)Here is the verification of (6.4): − ii λ ( . ) = − ii λ ( . ) = − h +2 X n =0 ( − | i | ni i • ni λ • − n − = − h +2 X n =0 i i • n i λ • − n − . ) = − h +2 X n =0 i i • n λi • − n − . ) = ( . ) − δ h, − c − λ ; i i i • λ ( . ) = ( . ) − δ h, − c − λ ; i i i λ ( . ) = ( . ) ii λ − δ h, − c − λ ; i i i λ . For (6.5), by (5.4) and (1.10), the right hand side is c − λ ; i ↑ i λ if n = − h − >
0, andit is zero otherwise. On the other hand, the left hand side equals − ( − | i | n ii λn • ( . ) = − ( − | i | n i iλn • ( . ) = ( . ) X r,s ≥ r + s = n − ( − | i | ( rs + s ) r • s • i λ . This is obviously zero if n = 0. Assuming n >
0, we apply (5.18) to see that it iszero unless n = − h −
1, when the term with r = − h − , s = 0 contributes c − λ ; i ↑ i λ . This completes the proof of the claim. Now we can establish (6.1): ii λ ( . ) = ii λ − − h − X n =0 X r ≥ ( − | i | ( h + n + r +1) i • ri • − n − r − n • i λ ( . ) = ( . ) − i i λ ( . ) = ii λ − δ h, − c − λ ; i i i λ ( . ) = ii λ . The proof of (6.2) follows by a very similar argument; one first checks that − i i λ = i i λ − δ h, − c λ ; i ii λ when h ≥ − (cid:3) Proposition 6.2.
The following relations hold: i λ = ( − | i,λ | i λ , i λ = i λ . (6.6) Proof.
It suffices to prove the first equality; the second one then follows usingProposition 3.5. Let h := h h i , λ i for short, and recall that | i, λ | = | i | ( h + 1). If h ≥ i λ ( . ) = − ( − | i | h c − λ ; i i λh • ( . ) = − ( − | i | h c − λ ; i i λh • ( . ) = ( . ) ( − | i | ( h +1) i λ . If h ≤ − i λ ( . ) = ( − | i | ( h +1) c λ ; i i λ − h − • ( . ) = ( − | i | ( h +1) c λ ; i i λ − h − • ( . ) = ( . ) ( − | i | ( h +1) i λ . UPER KAC-MOODY 2-CATEGORIES 25
Finally if h = − − | i | ( h +1) i λ = i λ ( . ) = c λ ; i i λi ( . ) = ( . ) − c λ ; i i λi + i λ ( . ) = ( . ) − c λ ; i iλi + i λ ( . ) = ( . ) i λ . This completes the proof. (cid:3) Final relations
There are just a few more important relations to be derived.
Lemma 7.1.
The following hold for all i = j : ij λ = ij λ if h h i , λ i ≤ max( d ij − , , (7.1) ij λ = ijλ if h h i , λ i ≥ d ij . (7.2) Proof.
Let h := h h i , λ i . First we prove (7.1) assuming that 0 < h ≤ d ij − ji i λ , we reduceto proving that j i λ = j i λ . (7.3)Then to check this, we apply (5.16) to transform the left hand side into − ij λ ( . ) = − ij λ − X r,s ≥ r + s = d ij − ( − | i | s t ij r • s • ij λi − X
0. By(1.12) and (1.14), the following 2-morphism is invertible: j ii λ ⊕ − h − M n =0 ij λn • . Vertically composing with this on the bottom, we deduce that the relation we aretrying to prove is equivalent to the following relations: ij λ = ij λ , ij λn • = n • ij λ for 0 ≤ n < − h . (7.4)To establish the first of these, we pull the j -string past the ii -crossing: ij λ ( . ) = ij λ + X r,s ≥ r + s = d ij − ( − | i | s t ij ri • s • ij λ + X
r = d ij − t ij c − λ + α j ; i ij λ . Finally if h = d ij = 0, we only have thefirst term on the right hand side, which contributes t ij c − λ + α j ; i ij λ again thanksto (5.17), (5.4), (2.4) and (1.7). This is what we want because: ij λ ( . ) = ij λ ( . ) = ( . ) δ h, c − λ ; i ij λ ( . ) = δ h, t ij c − λ + α j ; i ij λ . We are just left with the right hand relations from (7.4) involving bubbles: ij λn • ( . ) = i jλn • ( . ) = ( − | i || j | n i jλn • ( . ) = ( − | i || j | n t ij i j λn + d ij • + a lin. comb. of i j λp • q • with n ≤ p < n + d ij ( . ) = δ n, − h − ( − | i || j | n t ij c − λ + α j ; i j ( . ) = ( . ) ( − | i || j | n n • ij λ ( . ) = n • ij λ . The relation (7.2) follows by very similar arguments to the previous paragraph;the first step is to vertically compose on the top with the isomorphism i ji λ ⊕ h − d ij − M n =0 i j λn • . (cid:3) Proposition 7.2.
The following relations hold for all i, j : ij λ = ij λ ij λ = ijλ , (7.5) UPER KAC-MOODY 2-CATEGORIES 27 ij λ = ( − | i || j | ij λ , ij λ = ( − | i || j | ijλ . (7.6) Proof.
We get (7.5) in half of the cases from Lemmas 6.1 and 7.1. To deduce theother half of the cases, attach leftward cups (resp. caps) to the two strands at thebottom (resp. the top) of the relations established in these two lemma, then simplifyusing (6.6). Finally (7.6) follows from (7.5) using Proposition 3.5 as usual. (cid:3)
The final two propositions of the section extend [KL3, Propositions 3.3–3.5].
Proposition 7.3.
The following hold for all n ≥ and λ ∈ P . (i) If i is even then i λ i n + ∗ • = X r ≥ ( r + 1) i n − r + ∗ • r • i λ , (7.7) in + ∗ • i λ = X r ≥ ( r + 1) r • i λ in − r + ∗ • . (7.8)(ii) If i is odd then i λ i n + ∗ • = X r ≥ (2 r + 1) i n − r + ∗ • r • i λ , (7.9) in + ∗ • i λ = X r ≥ (2 r + 1) r • i λ in − r + ∗ • . (7.10)(iii) For i = j with d ij > we have that j λ i n + ∗ • = t ij i n + ∗ • j λ + t ji i n − d ij + ∗ • d ji • j λ + X
Let h := h h i , λ i throughout the proof.(i)–(ii) When i is even, this was already established in [L]. So we just need toprove (ii), assuming i is odd. We observe to start with that the identities (7.9)and (7.10) (for fixed λ and all n ≥
0) are equivalent. To see this, let us rephrasethem in terms of power series. We make End U ( g ) ( E i λ ) into a k [ x ]-module so that x acts as by vertically composing on top with a dot. Let t be an indeterminateand e( t ) := P n ≥ e n t n , h( t ) := P n ≥ h n t n , which are power series in Sym[[ t ]]. Recalling (5.1)–(5.2), the identities (7.9) and (7.10) for all n ≥ i λ β λ ; i ((1 − d t )h( − t )) = X r ≥ (2 r + 1) x r t r β λ + α i ; i ((1 − d t )h( − t )) i λ ,β λ + α i ; i ((1 + d t )e( t )) i λ = X r ≥ (2 r + 1) x r t r i λ β λ ; i ((1 + d t )e( t )) , respectively, as follows by equating coefficients of t . Since e( t )h( − t ) = 1 in Symand d = 0, we deduce that (1 + d t )e( t ) and (1 − d t )h( − t ) are two-sided inverses.Using this, it is easy to see that the two generating function identities are indeedequivalent, e.g. multiplying the first on the right by β λ ; i ((1 + d t )e( t )) and on theleft by β λ + α i ; i ((1 − d t )e( t )) transforms it into the second.To complete the proof of (ii), we need to show that one of (7.9) or (7.10) holdsfor each fixed h . We proceed to verify (7.9) in case h ≤ −
1; a similar argumentestablishes (7.10) in case h ≥ −
1. So assume that h ≤ −
1. The identity to be provedis trivial in case n = 0 so suppose moreover that n >
0, so that n − h − ≥
1. Thenwe have that i λ i n + ∗ • ( . ) = ( . ) − ( − n − h − iλi n − h − • + δ h, − X r ≥ ( − r i − r − • n • i λr • ( . ) = ( . ) − ( − n − h − iλi n − h − • + δ h, − i ∗ • n • i λ ( . ) = ( . ) X r,s ≥ r + s = n − h − ( − rs + hs r • s • i λ + δ h, − i ∗ • n • i λ ( . ) = X r,s ≥ r + s = n − h − r + h +2 X t =0 ( − rs + hs + t i r − t − • i t • s • λ + δ h, − i ∗ • n • i λ = X r,s ≥ r + s = n − h − r + h +2 X t =0 ( − ( s +1) t i r − t − • i s + t • λ + δ h, − i ∗ • n • i λ = X r,s ≥ r + s = n − h − n X t = s ( − ( s +1) t i n − t + ∗ • i t • λ + δ h, − i ∗ • n • i λ = n X t =0 min( t,n − h − X s =0 ( − ( s +1) t i n − t + ∗ • i t • λ + δ h, − i ∗ • n • i λ UPER KAC-MOODY 2-CATEGORIES 29 = n X t =0 t X s =0 ( − ( s +1) t i n − t + ∗ • i t • λ = X t ≥ t even ( t + 1) i n − t + ∗ • i t • λ . This is what we wanted.(iii)–(iv) By an argument with generating functions similar to the one explainedin the proof of (ii) above, the identities (7.11) and (7.12) are equivalent, as are(7.13) and (7.14). Therefore it suffices just to prove one of them for each fixed h and all n ≥
0. For any n ≥
0, we have that j λi n • ( . ) = ( − | i || j | n j λi n • ( . ) = ( . ) ( − | i || j | n jλi n • ( . ) = jλi n • ( . ) = t ij j λi n + d ij • + t ji j λi n • d ji • + X
0. A similar argument establishes(7.12) and (7.14) for n ≥ d ij − h + 1, hence, we are completely done if h ≥ d ij .We are left with proving (7.11)–(7.12) when 1 ≤ h ≤ d ij −
1. We claim that (7.11)holds for all n ≤ d ij − h . The claim implies that (7.12) holds for all n ≤ d ij − h too,and we have already established (7.12) for n ≥ d ij − h + 1, so the claim is enoughto finish the proof. For the claim, we proceed by induction on n = 0 , , . . . , d ij − h .The base case n = 0 is trivial. For the induction step, take 1 ≤ n ≤ d ij − h . By(3.8), we have that ji λ • n − − ji λ • n − = X r,s ≥ r + s = d ij − ( − | i | s t ij iij λ •• r s • n − + X
1, we have proved that n X s =0 ( − | i | s t ij i n − s + ∗ • j is + ∗ • λ + X
1, keepingthe s = 0 terms on the left hand side, to obtain t ij i n + ∗ • j λ + t ji i n − d ij + ∗ • d ji • j λ + X
Corollary 7.4.
For i, j ∈ I with i odd, we have that i j λ = ij λ , i j λ = ij λ . (7.15) Proof.
Remembering the definition (2.18), the first relation follows from the n = 1cases of (7.9), (7.11) and (7.13); to see that the lower terms in (7.11) vanish, recallthat d ij is even. Hence, it satisfies d ij ≥
2, and s pqij = 0 if p = d ij −
1. The secondrelation follows from the first by applying ω . (cid:3) Remark 7.5.
One can invert the formulae in Proposition 7.3 to obtain also variousbubble slides in the other direction. For example, inverting (7.7)–(7.10) producesthe following, for i even, i even, i odd and i odd, respectively: i n + ∗ • λ i = i λ i n + ∗ • − • i λ i n − ∗ • + • i λ i n − ∗ • , (7.16) i λ in + ∗ • = in + ∗ • i λ − in − ∗ • • i λ + in − ∗ • • i λ , (7.17) i n + ∗ • λ i = i λ i n + ∗ • − • i λ i n − ∗ • + 4 X r ≥ ( − r r • i λ i n − r + ∗ • , (7.18) i λ in + ∗ • = in + ∗ • i λ − in − ∗ • • i λ + 4 X r ≥ ( − r in − r + ∗ • r • i λ . (7.19) UPER KAC-MOODY 2-CATEGORIES 31
Proposition 7.6.
The following relation holds: j i kλ − j i kλ = X r,s,t ≥ ( − | i | ( h h i ,λ i + r + s +1) i i λi • − r − s − t − • ri • s • t if i = j = k , + X r,s,t ≥ ( − | i | ( h h i ,λ i + r + s + t ) ii λi • − r − s − t − • si • r • t otherwise. (7.20) Proof.
Assuming either i = j = k or i = k , we attach crossings to the top left andbottom right pairs of strands of (3.8) to deduce that kij λ = kij λ . (7.21)When i = k , the lemma follows easily from this on simplifying using (2.7). A similarargument treats the case i = j , attaching crossings to the top right and bottom leftpairs of strands in the relation ij kλ = ij kλ which may be deduced by attaching a leftward cap to the top left and a leftwardcup to the bottom right of (1.9) and using (6.6) and (7.5). We are just left withthe case that i = j = k . For this we use (7.21) again to reduce to checking: iii λ = − i i iλ − X r,s,t ≥ ( − | i | ( h h i ,λ i + r + s +1) i i λi • − r − s − t − • ri • s • t , iii λ = − iii λ + X r,s,t ≥ ( − | i | ( h h i ,λ i + r + s + t ) ii λi • − r − s − t − • si • r • t . These two identities are proved in similar ways. One first uses (5.15)–(5.16) toreduce the double crossings, then (3.1)–(3.2) to pull the dots to the boundary,remembering also (2.13)–(2.14), (2.4) and (1.7). By now we can safely leave thedetails to the reader! (cid:3) The nondegeneracy conjecture
The main result of this section is a generalization of [KL3, Proposition 3.11]. Weneed some further notation which is adapted from [KL3]. Let Seq be the set of allwords in the alphabet {↑ i , ↓ i | i ∈ I } ; our words correspond to the signed sequences of [KL3]. For a = a m · · · a ∈ Seq, letwt( a ) := X i ∈ I (cid:16) { n = 1 , . . . , m | a n = ↑ i }− { n = 1 , . . . , m | a n = ↓ i } (cid:17) α i ∈ Q. (8.1)To λ ∈ P and a = a m · · · a ∈ Seq, we associate the 1-morphism E a λ := E a m · · · E a λ : λ → λ + wt( a ) (8.2)in U ( g ), with the convention that E ↑ i = E i and E ↓ i = F i . As λ and a vary, thesegive all of the 1-morphisms in U ( g ).Suppose that we are given a = a m · · · a and b = b n · · · b ∈ Seq. An ab -matching is a planar diagram with • m distinct vertices on a horizontal axis at the bottom labeled from right toleft by the letters a , . . . , a m ; • n distinct vertices on a horizontal axis at the top labeled from right to leftby the letters b , . . . , b n ; • ( m + n ) / I -colored strands drawn betweenthe horizontal axes whose endpoints are the given ( m + n ) vertices.We require moreover that: • the strands have only finitely many intersections and critical points (=points of slope zero); • there are no intersections at critical points, no triple intersections, and notangencies; • the colors and directions of the strands are consistent with the letters attheir endpoints.Note at least one ab -matching exists if and only if wt( a ) = wt( b ). Here is anexample with a = ↑ j ↓ j ↑ i ↓ k and b = ↑ i ↓ k ↑ i ↓ i : i k ij UPER KAC-MOODY 2-CATEGORIES 33
A matching is reduced if each strand has at most one critical point which shouldeither be a minimum or a maximum, there are no self-intersections of strands, anddistinct strands intersect at most once.Any ab -matching defines a pairing between the letters of the words a and b ,two letters being paired if they are endpoints of the same strand. We say thattwo matchings are equivalent if they define the same pairing. Every matchingis equivalent to at least one reduced matching. For example, here is a reducedmatching equivalent to the matching displayed above: i k ij A decorated ab -matching is an ab -matching whose strands have been decorated byfinitely many dots located away from intersections and critical points, each of whichis labeled by a non-negative integer. Given any decorated ab -matching σ and λ ∈ P ,there is a unique way to label the regions of σ by elements of P so that it becomesthe diagrammatic representation of a 2-morphism f ( σ, λ ) ∈ Hom U ( g ) ( E a λ , E b λ )as above.For each a , b ∈ Seq, we choose a set M ( a , b ) of representatives for the equiva-lence classes of reduced ab -matchings. For each element of M ( a , b ), we also choosea distinguished point on each of its strands located away from intersections andcritical points. Then let c M ( a , b ) be the set of decorated ab -matchings obtainedby taking each of the matchings in M ( a , b ) and putting a dot labeled with a non-negative integer at each of its distinguished points. Finally recall the homorphism β λ : SYM → End U ( g ) (1 λ ) from (1.22). Theorem 8.1.
Take a , b ∈ Seq with wt( a ) = wt( b ) and any λ ∈ P . Viewing Hom U ( g ) ( E a λ , E b λ ) as a right SYM -module so that p ∈ SYM acts by horizontallycomposing on the right with β λ ( p ) , the 2-morphisms (cid:8) f ( σ, λ ) (cid:12)(cid:12) σ ∈ c M ( a , b ) (cid:9) generate Hom U ( g ) ( E a λ , E b λ ) as a right SYM -module.Proof.
By the definitions, any 2-morphism in Hom U ( g ) ( E a λ , E b λ ) is a linear com-bination of diagrams obtained by horizontally and vertically composing the gener-ators x, τ, η, ε, η ′ and ε ′ . Now the point is that we have derived enough relationsabove to be able to algorithmically rewrite any 2-morphism represented by such adiagram as a linear combination of the 2-morphisms f ( σ, λ ) β λ ( p ) for σ ∈ c M ( a , b )and p ∈ SYM. This proceeds by induction on the total number of crossings in thediagram. We omit the details since it is essentially the same argument as used toprove [KL3, Proposition 3.11]. (cid:3)
Now we can properly state the Nondegeneracy Conjecture from the introduction:
Nondegeneracy Conjecture.
For all a , b ∈ Seq with wt( a ) = wt( b ) and any λ ∈ P , the superspace Hom U ( g ) ( E a λ , E b λ ) is a free right SYM -module with basis (cid:8) f ( σ, λ ) (cid:12)(cid:12) σ ∈ c M ( a , b ) (cid:9) . The covering quantum group
Henceforth, we assume that the Cartan matrix is symmetrized by positive inte-gers ( d i ) i ∈ I , and that the parameters are chosen to satisfy the homogeneity condi-tion (1.31). Let ( − , − ) be the symmetric bilinear form on the root lattice Q defined from ( α i , α j ) := − d i d ij . In this section, we recall the definition of the coveringquantum group ˙ U q,π ( g ) of Clark, Hill and Wang [CHW1, CHW2]. Our expositionis based mostly on [CFLW] and [C]. Note that our q is the parameter denoted q − in [CHW1, CHW2, CFLW], which is v − in [C]. We write e i λ and f i λ inplace of E i λ and F i λ ; we would also write k i for the generator K − i althoughwe won’t actually need this here. In [CHW2, CFLW, C], an additional assumptionof “bar-consistency” is made on the super Cartan datum; we do not insist on thisuntil later.Let L be the ring Q ( q )[ π ] / ( π − L := Z [ q, q − , π ] / ( π −
1) as in theintroduction. For n ∈ Z , we let[ n ] q,π := q n − ( πq ) − n q − ( πq ) − = (cid:26) q n − + πq n − + · · · + π n − q − n if n ≥ − π n ( q − n − + πq − n − + · · · + π − n − q n ) if n ≤ n ] ! q,π := [ n ] q,π [ n − q,π · · · [1] q,π , (cid:20) nr (cid:21) q,π := [ n ] ! q,π [ r ] ! q,π [ n − r ] ! q,π . We let − be the involution of L (or L ) with q = q − and π = π . Note this isdifferent from the bar involution used in [CFLW, C]; in particular, our quantumintegers are not bar invariant, but satisfy[ n ] q,π = π n − [ n ] q,π = − π [ − n ] q,π . (9.1)We have that (cid:2) nr (cid:3) q,π = π r ( n − r ) (cid:2) nr (cid:3) q,π , so that the quantum binomial coefficient isbar invariant if n is odd. For i ∈ I , we set q i := q d i , π i := π | i | . Let ˙ U q,π ( g ) be the locally unital L -algebra with mutually orthogonal idempotents { λ | λ ∈ P } , and generators e i λ = 1 λ + α i e i and f i λ = 1 λ − α i f i for all i ∈ I and λ ∈ P , subject to the following relations:( e i f j − π | i || j | f j e i )1 λ = δ i,j [ h h i , λ i ] q i ,π i λ , (9.2) d ij +1 X r =0 ( − r π r | j | + r ( r − / i (cid:20) d ij + 1 r (cid:21) q i ,π i e d ij +1 − ri e j e ri λ = 0 ( i = j ) , (9.3) d ij +1 X r =0 ( − r π r | j | + r ( r − / i (cid:20) d ij + 1 r (cid:21) q i ,π i f d ij +1 − ri f j f ri λ = 0 ( i = j ) . (9.4)Also let ˙ U q,π ( g ) L be the L -subalgebra of ˙ U q,π ( g ) generated by the divided powers e ( n ) i λ := e ni λ / [ n ] ! q i ,π i , f ( n ) i λ := f ni λ / [ n ] ! q i ,π i (9.5)for all i ∈ I, λ ∈ P and n ≥
1; see also [C, Lemma 3.5].We also need the antilinear (with respect to the bar involution of the groundring) algebra automorphisms ψ, ω : ˙ U q,π ( g ) → ˙ U q,π ( g ) and the linear algebra anti-automorphism ρ : ˙ U q,π ( g ) → ˙ U q,π ( g ), which are defined on generators by ω (1 λ ) = 1 − λ , ω ( e i λ ) = f i − λ , ω ( f i λ ) = e i − λ , (9.6) ψ (1 λ ) = 1 λ , ψ ( e i λ ) = e i λ , ψ ( f i λ ) = π | i,λ | f i λ , (9.7) UPER KAC-MOODY 2-CATEGORIES 35 ρ (1 λ ) = 1 λ , ρ ( e i λ ) = q −h h i ,λ i− i λ f i , ρ ( f i λ ) = q h h i ,λ i− i λ e i . (9.8)Note all of these are involutions. Let ∗ := ρ ◦ ψ and ! := ψ ◦ ρ . These are mutuallyinverse antilinear antiautomorphisms with1 ∗ λ = 1 λ , ( e i λ ) ∗ = q −h h i ,λ i− i λ f i , ( f i λ ) ∗ = q h h i ,λ i− i π | i,λ | λ e i , (9.9)1 ! λ = 1 λ , ( e i λ ) ! = π | i,λ | q h h i ,λ i i λ f i , ( f i λ ) ! = q −h h i ,λ i i λ e i . (9.10)The notation here varies somewhat across the literature, e.g. the counterparts ofour ω, ψ and ρ in the purely even setting are denoted by ω ◦ ψ, ψ and ¯ ρ in [KL3].In the remainder of the section, we are going to explain how to lift ω, ψ and ρ tothe Kac-Moody 2-supercategory.First, we must explain how to deal with antilinearity at the level of 2-categories.Let A be a graded supercategory. The supercategory A sop from Definition 3.4 is ac-tually a graded supercategory with the same grading as A , i.e. deg( f sop ) = deg( f ).Similarly, if A is a graded 2-supercategory then A sop is a graded 2-supercategory.If A is a graded ( Q, Π)-2-supercategory in the sense of [BE, Definition 6.5], withstructure maps σ λ : q λ ∼ → λ , ¯ σ λ : q − λ ∼ → λ and ζ λ : π λ ∼ → λ , we can regard A sop as a graded ( Q, Π)-2-supercategory by declaring that its structure maps are(¯ σ − λ ) sop : q − λ ∼ → λ , ( σ − λ ) sop : q λ ∼ → λ and ( ζ − λ ) sop : π λ ∼ → λ . The key pointhere is that we have interchanged the roles of q and q − . Lemma 9.1.
Suppose that A and B are graded 2-supercategories, and recall thedefinition of their ( Q, Π) -envelopes A q,π and B q,π from Definition 1.6. Given agraded 2-superfunctor φ : A → ( B q,π ) sop , there is a canonical induced graded 2-superfunctor ˜ φ : A q,π → ( B q,π ) sop .Proof. View ( B q,π ) sop as a graded ( Q, Π)-2-supercategory as explained above. Thenapply the universal property of ( Q, Π)-envelopes from [BE, Lemma 6.11(i)]. (cid:3)
Remark 9.2.
In the setup of Lemma 9.1, the construction from the proof of [BE,Lemma 6.11(i)] implies the following explicit description for ˜ φ . It is equal to φ onobjects. On a 1-morphism F in A with φ ( F ) = Q m ′ Π a ′ F ′ for a 1-morphism F ′ in B , we have that ˜ φ ( Q m Π a F ) = Q m ′ − m Π a + a ′ F ′ . Given another 1-morphism G in A with φ ( G ) = Q n ′ Π b ′ G ′ and x ∈ Hom A ( F, G ) with φ ( x ) = (cid:16) ( x ′ ) m ′ ,a ′ n ′ ,b ′ (cid:17) sop for x ′ ∈ Hom B ( G ′ , F ′ ), we have that˜ φ (cid:0) x n,bm,a (cid:1) = ( − a | x | + b | x | + ab + b (cid:16) ( x ′ ) m ′ − m,a + a ′ n ′ − n,b + b ′ (cid:17) sop . Note also that ˜ φ is not strict (even if φ itself is strict). Its coherence map˜ c Q n Π b G,Q m Π a F : ˜ φ ( Q n Π b G ) ˜ φ ( Q m Π a F ) ∼ → ˜ φ ( Q m + n Π a + b GF )is ( − ab (cid:16) f m ′ + n ′ − m − n,a + b + a ′ + b ′ k ′ − m − n,a + b + c ′ (cid:17) sop , where (cid:16) f m ′ + n ′ ,a ′ + b ′ k ′ ,c ′ (cid:17) sop denotes the coher-ence map c G,F : φ ( G ) φ ( F ) ∼ → φ ( GF ) for φ , for H ′ defined so that φ ( GF ) = Q k ′ Π c ′ H ′ and f ∈ Hom B ( H ′ , G ′ F ′ ).Since we are assuming now that the parameters satisfy (1.31), the Kac-Moody2-supercategory U ( g ) is a graded 2-supercategory with Z -grading defined as in theintroduction. Let U q,π ( g ) be its ( Q, Π)-envelope from Definition 1.6. We now pro-ceed to define the categorical counterparts of the antilinear automorphisms (9.6)–(9.7). Actually, the first was already defined in Proposition 3.5, but we need toextend this to the envelope.
Proposition 9.3.
There is an isomorphism of graded 2-supercategories ˜ ω : U q,π ( g ) ∼ → U q,π ( g ) sop defined on objects by λ
7→ − λ and 1-morphisms by Q m Π a E i λ Q − m Π a F i − λ , Q m Π a F i λ Q − m Π a E i − λ .Proof. If we compose the strict 2-superfunctor from Proposition 3.5 with the canon-ical inclusion U ( g ) sop → U q,π ( g ) sop , we obtain a strict graded 2-superfunctor ω : U ( g ) → U q,π ( g ) sop . This is defined on objects by λ λ , on 1-morphisms by E i λ Q Π ¯0 F i − λ , F i λ Q Π ¯0 E i − λ , and on 2-morphisms by the following: • i λ
7→ • − λ ¯0¯000 i sop , i jλ
7→ − ( − | i || j | − λ ¯0¯000 i j sop , i λ − λ ¯0¯000 i sop , i λ − λ ¯0¯000 i sop . It remains to apply Lemma 9.1 to get the desired graded 2-superfunctor ˜ ω (whichis no longer strict). (cid:3) Proposition 9.4.
Assume that there is a given element √− ∈ k ¯0 which squaresto − . Then there is an isomorphism of graded 2-supercategories ˜ ψ : U q,π ( g ) ∼ → U q,π ( g ) sop defined on objects by λ λ and 1-morphisms by Q m Π a E i λ Q − m Π a E i λ , Q m Π a F i λ Q − m Π a + | i,λ | F i λ .Proof. We claim that there is a strict graded 2-superfunctor ψ : U ( g ) → U q,π ( g ) sop which is defined on objects by λ λ , 1-morphisms by E i λ Q Π ¯0 E i λ , F i λ Q Π | i,λ | F i λ , and 2-morphisms by the following: • i λ • λ ¯0¯000 i sop if | i | = ¯0, √− • λ ¯0¯000 i sop if | i | = ¯1 , i jλ λ ¯0¯000 j i sop if | i || j | = ¯0 , √− λ ¯0¯000 j i sop if | i || j | = ¯1 , i λ λ | i,λ | ¯000 i sop , i λ ( − | i,λ | λ | i,λ | ¯000 i sop . UPER KAC-MOODY 2-CATEGORIES 37
To prove the claim, one needs to verify the relations. Note to start with that • in λ • n λ ¯0¯000 i sop if n | i | = ¯0, √− • n λ ¯0¯000 i sop if n | i | = ¯1 . Using this, the quiver Hecke superalgebra relations (1.7)–(1.9) are straightforward.The inversion relations (1.12)–(1.14) are also fine. The adjunction relations (1.10)need a little more care since the signs coming from (1.30) play a role. Then applyLemma 9.1 to get the desired graded 2-superfunctor ˜ ψ (which is no longer strict). (cid:3) Definition 9.5.
Let A be a graded 2-supercategory. Define A srev to be the graded2-supercategory with the same objects as A , and morphism categoriesHom A srev ( µ, λ ) := Hom A ( λ, µ ) . We write F srev : µ → λ (resp. x srev : F srev → G srev ) for the 1-morphism (resp.2-morphism) in A srev defined by the 1-morphism F : λ → µ (resp. x : F → G ) in A . Then, horizontal composition in A srev is defined on 1-morphisms by( F srev )( G srev ) := ( GF ) srev and on homogeneous 2-morphisms by ( x srev )( y srev ) :=( − | x || y | ( yx ) srev . Vertical composition of 2-morphisms in A srev is the same as in A . Here is the check of the super interchange law in A srev :( x srev y srev ) ◦ ( u srev v srev ) = ( − | x || y | + | u || v | ( yx ) srev ◦ ( vu ) srev = ( − | x || y | + | u || v | (( yx ) ◦ ( vu )) srev = ( − | x || y | + | u || v | + | x || v | (( y ◦ v )( x ◦ u )) srev = ( − | y || u | ( x ◦ u ) srev ( y ◦ v ) srev = ( − | y || u | ( x srev ◦ u srev )( y srev ◦ v srev ) . If A is a graded ( Q, Π)-2-supercategory with structure maps σ λ : q λ ∼ → λ , ¯ σ λ : q − λ ∼ → λ and ζ λ : π λ ∼ → λ , we can regard A srev as a graded ( Q, Π)-2-supercategoryby declaring that its structure maps are ( σ λ ) srev : ( q λ ) srev ∼ → (1 λ ) srev , (¯ σ λ ) srev :( q − λ ) srev ∼ → (1 λ ) srev and ( ζ λ ) srev : ( π λ ) srev ∼ → (1 λ ) srev . Lemma 9.6.
Suppose that A and B are graded 2-supercategories. Given a graded 2-superfunctor φ : A → ( B q,π ) srev , there is a canonical induced graded 2-superfunctor ˜ φ : A q,π → ( B q,π ) srev .Proof. View ( B q,π ) srev as a graded ( Q, Π)-2-supercategory as explained above.Then apply [BE, Lemma 6.11(i)]. (cid:3)
Remark 9.7.
In the setup of Lemma 9.6, the construction from the proof of[BE, Lemma 6.11(i)] implies the following explicit description for ˜ φ . It is equalto φ on objects. On a 1-morphism F in A with φ ( F ) = ( Q m ′ Π a ′ F ′ ) srev for a 1-morphism F ′ in B , we have that ˜ φ ( Q m Π a F ) = ( Q m + m ′ Π a + a ′ F ′ ) srev . Given another1-morphism G in A with φ ( G ) = ( Q n ′ Π b ′ G ′ ) srev and x ∈ Hom A ( F, G ) with φ ( x ) = (cid:16) ( x ′ ) n ′ ,b ′ m ′ ,a ′ (cid:17) srev for x ′ ∈ Hom B ( F ′ , G ′ ), we have that˜ φ (cid:0) x n,bm,a (cid:1) = ( − aa ′ + bb ′ (cid:16) ( x ′ ) n + n ′ ,b + b ′ m + m ′ ,a + a ′ (cid:17) srev . The coherence map˜ c Q n Π b G,Q m Π a F : ˜ φ ( Q n Π b G ) ˜ φ ( Q m Π a F ) ∼ → ˜ φ ( Q m + n Π a + b GF )is ( − a ( b + b ′ )+( a + b )( a ′ + b ′ + c ′ ) (cid:16) f m + n + k ′ ,a + b + c ′ m + n + m ′ + n ′ ,a + b + a ′ + b ′ (cid:17) srev , where (cid:16) f k ′ ,c ′ m ′ + n ′ ,a ′ + b ′ (cid:17) srev denotes the coherence map c G,F : φ ( G ) φ ( F ) ∼ → φ ( GF ) for φ , for H ′ defined so that φ ( GF ) = Q k ′ Π c ′ H ′ and f ∈ Hom B ( F ′ G ′ , H ′ ). Proposition 9.8.
Assume that there is a given element √− ∈ k ¯0 which squaresto − . Then there is an isomorphism of graded 2-supercategories ˜ ρ : U q,π ( g ) ∼ → U q,π ( g ) srev such that λ λ and Q m Π a E i λ ( Q m − d i (1+ h h i ,λ i ) Π a λ F i ) srev , Q m Π a F i λ ( Q m − d i (1 −h h i ,λ i ) Π a λ E i ) srev . Proof.
We claim that there is a strict graded 2-superfunctor ρ : U ( g ) → U q,π ( g ) srev defined on objects by λ λ , 1-morphisms by E i λ ( Q − d i (1+ h h i ,λ i ) Π ¯0 λ F i ) srev , F i λ ( Q − d i (1 −h h i ,λ i ) Π | i,λ | λ E i ) srev , and 2-morphisms by the following: • i λ • λ ¯0¯0 − d i (1+ h h i ,λ i ) − d i (1+ h h i ,λ i ) i srev if | i | = ¯0, √− • λ ¯0¯0 − d i (1+ h h i ,λ i ) − d i (1+ h h i ,λ i ) i srev if | i | = ¯1 , i jλ λ ¯0¯0 − d i (1+ h h i ,λ i ) − d j (1+ h h j ,λ i ) − ( α i ,α j ) − d i (1+ h h i ,λ i ) − d j (1+ h h j ,λ i ) − ( α i ,α j ) i j srev if | i || j | = ¯0 , −√− λ ¯0¯0 − d i (1+ h h i ,λ i ) − d j (1+ h h j ,λ i ) − ( α i ,α j ) − d i (1+ h h i ,λ i ) − d j (1+ h h j ,λ i ) − ( α i ,α j ) i j srev if | i || j | = ¯1 , i λ λ ¯0¯000 i srev , i λ λ ¯0¯000 i srev . To prove the claim, one needs to verify the relations. The quiver Hecke relations arethe most complicated; for these, use (3.5)–(3.6), (3.7) and (3.9). (Note the degreeshifts actually play no role in this argument; they are included to match (9.8).)Finally, apply Lemma 9.6 to get ˜ ρ . (cid:3) Suppose in this paragraph that k = k ¯0 is a field. Then the underlying 2-category U q,π ( g ) is a ( Q, Π)-2-category in the sense of [BE, Definition 6.14], as is its additiveKaroubi envelope ˙ U q,π ( g ). The Grothendieck ring K ( ˙ U q,π ( g )) is a locally unital L -algebra with distinguished idempotents { λ | λ ∈ P } . The analogous Grothendieckring arising from U q,π ( g ) sop may be identified with K ( ˙ U q,π ( g )) as a ring, but now q acts as q − . This means that the isomorphisms ˜ ω and ˜ ψ from Propositions 9.3–9.4induce antilinear locally unital algebra automorphisms[˜ ω ] , [ ˜ ψ ] : K ( ˙ U q,π ( g )) → K ( ˙ U q,π ( g )) . Also, the Grothendieck ring arising from U q,π ( g ) srev may be identified with theopposite K ( ˙ U q,π ( g )) op , so that the isomorphism ˜ ρ from Proposition 9.8 induces a UPER KAC-MOODY 2-CATEGORIES 39 linear algebra antiautomorphism[˜ ρ ] : K ( ˙ U q,π ( g )) → K ( ˙ U q,π ( g )) . The epimorphism γ : ˙ U q,π ( g ) L ։ K ( ˙ U q,π ( g )) to be constructed in Theorem 11.7below intertwines the maps ω, ψ and ρ from (9.6)–(9.8) with [˜ ω ] , [ ˜ ψ ] and [˜ ρ ]. Remark 9.9.
One can also consider the compositions ˜ ρ ◦ ˜ ψ and ˜ ψ ◦ ˜ ρ . Both ofthese maps can be defined directly on generators, revealing that they actually do notrequire the existence of √− ∈ k , unlike ˜ ρ and ˜ ψ themselves. Just as discussed in[KL3, (3.46)–(3.47], these maps may also be interpreted as taking right duals/matesand left duals/mates, respectively. They decategorify to the maps ∗ and ! from(9.9)–(9.10). 10. The sesquilinear form
Continue with the assumptions from section 9. Let f be the L -superalgebra ongenerators { θ i | i ∈ I } with | θ i | := | i | , subject to relations d ij +1 X r =0 ( − r π r | j | + r ( r − / i (cid:20) d ij + 1 r (cid:21) q i ,π i θ d ij +1 − ri θ j θ ri = 0 (10.1)for all i = j . There is a Q -grading f = L α ∈ Q f α on f compatible with the Z / θ i is of degree α i . Viewing f ⊗ f as analgebra with the twisted multiplication ( x ⊗ y )( x ′ ⊗ y ′ ) := π | y || x ′ | q − ( β,α ′ ) xx ′ ⊗ yy ′ for homogeneous x ∈ f α , y ∈ f β , x ′ ∈ f α ′ , y ′ ∈ f β ′ , we let r : f → f ⊗ f be thesuperalgebra homomorphism defined from r ( θ i ) = θ i ⊗ ⊗ θ i for each i ∈ I .By [CHW1, Proposition 3.4.1], there is a (non-degenerate) symmetric bilinear form( − , − ) on f defined by the following properties: • ( θ i , θ j ) = δ i,j / (1 − π i q i ); • ( xy, z ) = ( x ⊗ y, r ( z )); • ( x, yz ) = ( r ( x ) , y ⊗ z ).Here, the form on f ⊗ f is defined from ( x ⊗ y, x ′ ⊗ y ′ ) := ( x, x ′ )( y, y ′ ). Note that f α and f β are orthogonal for α = β . Theorem 10.1 (Lusztig, Clark) . There is a unique sesquilinear form ( = antilinearin the first argument, linear in the second) h− , −i : ˙ U q,π ( g ) × ˙ U q,π ( g ) → L such thatthe following hold:(1) h µ x λ , µ ′ x ′ λ ′ i = 0 if λ = λ ′ or µ = µ ′ ;(2) h xy, z i = h y, x ∗ z i ;(3) h e i d · · · e i λ , e j d · · · e j λ i = ( θ i · · · θ i d , θ j · · · θ j d ) .Moreover:(4) h x, y i = h ψ ( y ) , ψ ( x ) i ;(5) h x, yz i = h y ! x, z i .Assuming in addition that the bar-consistency assumption of [C, Definition 2.1(d)] holds, i.e. d i ≡ | i | (mod 2) for each i ∈ I, (10.2) the form h− , −i is non-degenerate.Proof. There is clearly at most one sesquilinear form on ˙ U q,π ( g ) satisfying properties(1)–(3). To see that there is indeed such a form, we appeal to [C, Proposition 5.8], which defines a bilinear form ( − , − ) ′ on ˙ U q,π ( g ) satisfying four properties. Ourform h− , −i is obtained from Clark’s form by setting h x, y i := ( σ ( ψ ( u )) , σ ( v )) ′ , (10.3)where ψ is the antilinear automorphism from (9.7) and σ is the linear antiautomor-phism defined by declaring that σ (1 λ ) = 1 λ , σ ( e i λ ) = 1 λ f i and σ ( f i λ ) = 1 λ e i .We leave it as an exercise to the reader to check that Clark’s four properties trans-late into our properties (1)–(4); actually, one needs the opposite formulation ofClark’s second property which may be derived from [C, Proposition 5.3], notingthat Clark’s τ is our σ ◦ ∗ ◦ ψ ◦ σ . The property (5) is immediate from (2) and (4)plus the definition of (9.10). Finally, assuming bar-consistency, the non-degeneracyfollows from [C, Theorem 5.12]. (cid:3) Remark 10.2.
One could also define a bilinear (rather than sesquilinear) form( − , − ) on ˙ U q,π ( g ) by setting ( x, y ) := h ψ ( x ) , y i . This is a generalization of Lusztig’sform from [Lu, Theorem 26.1.2] which is slightly different from the one introducedin [C]. Theorem 10.1 implies that ( − , − ) is symmetric and it satisfies ( xy, z ) =( y, ρ ( x ) z ).The next theorem gives a graphical description of the form h− , −i in the spiritof [KL3, Theorem 2.7]. Recall the notation Seq from section 8. For a , b ∈ Seq,let c M ( a , b ) be chosen as in Theorem 8.1. For σ ∈ c M ( a , b ) and λ ∈ P , define the degree deg( σ, λ ) and the parity | σ, λ | to be the degree and parity of the homogeneous2-morphism f ( σ, λ ), i.e. we sum the degrees and parities of all of the generatingdots, cups, caps and crossings in the diagram for f ( σ, λ ) as listed in the followingtable:Generator Degree Parity Generator Degree Parity • i λ d i | i | • i λ d i | i | i jλ − ( α i , α j ) | i || j | ij λ | i || j | i jλ − ( α i , α j ) | i || j | ijλ | i || j | i λ d i (1 + h h i , λ i ) ¯0 iλ d i (1 − h h i , λ i ) | i, λ | i λ d i (1 − h h i , λ i ) ¯0 iλ d i (1 + h h i , λ i ) | i, λ | Just as we did in (8.2), a word a = a m · · · a ∈ Seq defines a monomial e a λ := e a m · · · e a λ ∈ ˙ U q,π ( g ) , (10.4)where e ↑ i := e i and e ↓ i := f i . Clearly, these monomials taken over all a ∈ Seq andall λ ∈ P span ˙ U q,π ( g ). Theorem 10.3.
The sesquilinear form h− , −i from Theorem 10.1 satisfies h e a λ , e b µ i = δ λ,µ X σ ∈ c M ( a , b ) q deg( σ,λ ) π | σ,λ | (10.5) UPER KAC-MOODY 2-CATEGORIES 41 for each a , b ∈ Seq and λ, µ ∈ P .Proof. This argument parallels the proof of [KL3, Theorem 2.7] closely. We canclearly assume µ = λ . Let h a , b i λ denote the expression on the right hand side of(10.5). Note to start with that h a , b i λ does not depend on the particular choicemade for c M ( a , b ). This follows because one can pass between any two choices ofdecorated reduced matchings by a sequence of isotopies which do not change degreesor parities of diagrams. (This is similar to the proof of Theorem 8.1, which appliedmore complicated relations which are the same as these isotopies plus terms withfewer crossings.) To complete the proof of the theorem, we must show: h e a λ , e b λ i = h a , b i λ . (10.6)We proceed with a series of claims, which mimic [KL3, Lemmas 2.8–2.12]. Claim 1.
The identity (10.6) is true in case a and b are positive, i.e. they onlyinvolve upward arrows.To see this, if a = ↑ i c · · · ↑ i and b = ↑ j d · · · ↑ j , then M ( a , b ) is empty unless c = d ,in which case its elements are in bijection with permutations w ∈ S d such that i w ( r ) = j r for each r = 1 , . . . , d , and we have that h a , b i λ = δ c,d X w ∈ S d (cid:18) d Y r =1 δ i w ( r ) ,j r − π i r q i r (cid:19)(cid:18) Y ≤ rw ( s ) π | i r || i s | q − ( α ir ,α is ) (cid:19) . Using Theorem 10.1(iv), it remains to check that this equals ( θ i · · · θ i c , θ j , . . . , θ j d ).This follows by the explicit definition of the latter form on f . Claim 2. h e i e a λ , e b λ i = h↑ i a , b i λ ⇔ h e a λ , f i e b λ i = h a , ↓ i b i λ . Claim 3. h f i e a λ , e b λ i = h↓ i a , b i λ ⇔ h e a λ , e i e b λ i = h a , ↑ i b i λ .The proofs of these are the same as for [KL3, Lemma 2.9]. For example, for Claim 2,one considers the bijection between c M ( ↑ i a , b ) and c M ( a , ↓ i b ) obtained by attachinga cup on the bottom left. On the algebraic side, one uses Theorem 10.1(2) and(9.9). Claim 4 . h e a e i f j e b λ , λ i = h a ↑ i ↓ j b , ∅ i λ ⇔ h e a f j e i e b λ , λ i = h a ↓ j ↑ i b , ∅ i λ , as-suming i = j .Since h e a e i f j e b λ , λ i = π | i || j | h e a f j e i e b λ , λ i by (9.2), we must show that h a ↑ i ↓ j b , ∅ i λ = π | i || j | h a ↓ j ↑ i b , ∅ i λ . This follows by considering the bijection between c M ( a ↑ i ↓ j b , ∅ ) and c M ( a ↓ j ↑ i b , ∅ )obtained attaching a rightward crossing under the ↑ i ↓ j to convert it to ↓ j ↑ i ; see theproof of [KL3, Lemma 2.11] for further explanations. The only difference for us isthat the crossing is odd in case | i || j | = ¯1. Claim 5 . Assuming that h e a e b λ , λ i = h ab , ∅ i λ , we have that h e a e i f i e b λ , λ i = h a ↑ i ↓ i b , ∅ i λ ⇔ h e a f i e i e b λ , λ i = h a ↓ i ↑ i b , ∅ i λ .Define µ so that e b λ = 1 µ e b . In view of (9.2), we must show that h a ↑ i ↓ i b , ∅ i λ − π | i | h a ↓ i ↑ i b , ∅ i λ = [ h h i , µ i ] q i ,π i h ab , ∅ i λ . (10.7)To see this, we divide the decorated matchings in c M ( a ↑ i ↓ i b , ∅ ) and c M ( a ↓ i ↑ i b , ∅ )into three classes exactly as explained in the proof of [KL3, Lemma 2.12]. It isthen easy to see that the contributions to the left hand side of (10.7) from the firsttwo classes cancel. The third classes arise from decorated matchings in c M ( ab ) by inserting a cap (clockwise or counterclockwise in the two cases) between a and b .Hence, like in the proof of [KL3, Lemma 2.12] remembering also the sesquilinearityof h− , −i , we see that the left hand side of (10.7) expands to h q −h h i ,µ i i / (1 − π i q i ) − π i π | i,µ | q h h i ,µ i i / (1 − π i q i ) i h ab , ∅ i λ . This simplifies to the right hand side of (10.7).Now we can complete the proof of (10.6) in general. Using Claims 2 and 3, wereduce to checking (10.6) in the special case that b = ∅ . Under this assumption,we then proceed by induction on the length of a . Using Claims 4 and 5 plus theinduction hypothesis, we can rearrange a to assume that all ↓ ’s appear to the leftof all ↑ ’s. Then we use Claims 3 and 1 to finish the proof. (cid:3) Example 10.4 (cf. [C, Example 5.7]) . h e ( r ) i λ , e ( r ) i λ i = h f ( r ) i λ , f ( r ) i λ i = r Y s =1 − ( π i q i ) s , h e i f i λ , λ i = π | i,λ | h λ , e i f i λ i = q −h h i ,λ i i − π i q i , h e i f i λ , f i e i λ i = h f i e i λ , e i f i λ i = π i + q i (1 − π i q i ) . Surjectivity of γ In this section, we continue with the assumptions of §
9, and also assume that k = k ¯0 is a field. For a graded superalgebra A , we write A - GSM od for the Abeliancategory of graded left A -supermodules with morphisms that preserve degree andparity. Let Q and Π denote the grading and parity shift functors on A - GSM od , sothat ( QV ) n = V n − and (Π V ) a = V a +¯1 . Let A - GSP roj be the full subcategory of A - GSM od consisting of the finitely generated projective supermodules. Let K ( A )denote the split Grothendieck group of A - GSP roj . It is naturally an L -modulewith q and π acting by [ Q ] and [Π], respectively. For a detailed discussion of thefollowing basic facts, we refer the reader to [KL3, §§ • Assume the graded superalgebra A is Laurentian , i.e. its graded pieces arefinite-dimensional and are zero in sufficiently negative degree. Then, theKrull-Schmidt property holds in A - GSP roj . Moreover, K ( A ) is free as an L -module, with basis as a free Z -module given by the isomorphism classesof indecomposable projectives in A - GSP roj . • If α : A → B is a homomorphism of graded superalgebras, there is aninduced L -module homomorphism [ α ] : K ( A ) → K ( B ). If A and B arefinite-dimensional and α is surjective, then [ α ] is surjective. • Assume A is Laurentian, and let I be a two-sided homogeneous ideal thatis non-zero only in strictly positive degree. Then, the canonical quotientmap A ։ A/I induces an isomorphism K ( A ) ∼ → K ( A/I ). • If A and B are finite-dimensional graded superalgebras all of whose irre-ducible graded supermodules are absolutely irreducible of type M , then thereis an isomorphism K ( A ) ⊗ L K ( B ) ∼ → K ( A ⊗ B ) , [ P ] ⊗ [ Q ] [ P ⊗ Q ].For more background about K for supercategories, see [BE, § UPER KAC-MOODY 2-CATEGORIES 43
We also need to review some basic facts about quiver Hecke superalgebras es-tablished in [KL1, KL2] in the even case, and in [HW] in general. Note in [HW]that the additional assumption (10.2) of bar-consistency is made throughout, butit is not needed for the proofs of the particular results from [HW] cited below.The quiver Hecke supercategory H is the (strict) monoidal supercategory gener-ated by objects I and morphisms • i : i → i and i j : i ⊗ j → j ⊗ i of parities | i | and | i || j | , respectively, subject to the relations (1.7)–(1.9) (omitting the label λ fromthese diagrams). For objects i = i n ⊗ · · · ⊗ i ∈ I ⊗ n and j = j m ⊗ · · · ⊗ j ∈ I ⊗ m ,there are no non-zero morphisms i → j in H unless m = n . The graded endomor-phism superalgebra H n := M i , j ∈ I ⊗ n Hom H ( i , j ) (11.1)is the quiver Hecke superalgebra from [KKT]. Let H q,π be the ( Q, Π)-envelope ofthe monoidal supercategory H , which is defined like in Definition 1.6 rememberingthat monoidal supercategories are 2-supercategories with one object; see also [BE,Definition 1.16]. Let H q,π be the underlying monoidal category (same objects, evenmorphisms of degree zero). The idempotent completion of the additive envelope of H q,π is denoted ˙ H q,π as usual. It is equivalent to the category L n ≥ H n - GSP roj ,hence, we may identify K ( ˙ H q,π ) = M n ≥ K ( H n ) . (11.2)In particular, this means that the L -module on the right hand side of (11.2) isactually an L -algebra; its multiplication comes from the usual induction product − ◦ − on graded H n -supermodules.Fix i ∈ I and consider the idempotent 1 i n := 1 i ⊗ i ⊗···⊗ i ∈ H n . The gradedsubalgebra 1 i n H n i n is a copy of the nil-Hecke algebra in case | i | = ¯0, or the odd nil-Hecke algebra in case | i | = ¯1. In either case, we write simply X r for thedot on the r th strand and T r for the crossing of the r th and ( r + 1)th strands(numbering strands by 1 , . . . , n from right to left). The elements D r := − T r X r from [HW, (5.20)] are homogeneous idempotents which satisfy the braid relationsof the symmetric group S n . Hence, for each w ∈ S n there is an element D w definedas usual from a reduced expression for w . Letting w be the longest element of S n ,we define 1 i ( n ) := D w ∈ i n H n i n . (11.3)This is known to be a primitive homogeneous idempotent, hence, P ( i ( n ) ) := Q − d i n ( n − / H n i ( n ) (11.4)is an indecomposable projective graded H n -supermodule. Lemma 11.1.
There is a graded supermodule isomorphism H n i n ∼ = P ( i ( n ) ) ⊕ [ n ] ! qi,πi (meaning the obvious direct sum of copies of P ( i ( n ) ) with parity and degree shiftsmatching the expansion of [ n ] ! q i ,π i ).Proof. This is well known in the even case, and is noted after [HW, (5.28)] in theodd case. A different convention for ( q, π )-integers is adopted in [HW], which wehave taken into account by changing the parity shift in (11.4) compared to [HW,(5.28)]. (cid:3)
Next suppose that we are given two different elements i, j ∈ I . For r, s ≥
0, thetensor product in H gives a superalgebra embedding H r ⊗ H ⊗ H s ֒ → H r + s +1 . Let1 i ( r ) ji ( s ) denote the image of 1 i ( r ) ⊗ j ⊗ i ( s ) under this map, then set P ( i ( r ) ji ( s ) ) := Q − d i r ( r − / − d i s ( s − / H r + s +1 i ( r ) ji ( s ) . (11.5)In other words, P ( i ( r ) ji ( s ) ) = P ( i ( r ) ) ◦ P ( j ) ◦ P ( i ( s ) ). This is a graded projective H r + s +1 -supermodule. Proposition 11.2 (Khovanov-Lauda, Rouquier, Hill-Wang) . For i = j ∈ I , let n := d ij +1 . Then there exists a split exact sequence of graded H r + s +1 -supermodules −→ P ( i ( n ) j ) −→ · · · −→ Π r ( r − | i | + r | i || j | P ( i ( n − r ) ji ( r ) ) −→ · · ·−→ Π n ( n − | i | + n | i || j | P ( ji ( n ) ) −→ . In particular, there is an isomorphism ⌊ n +12 ⌋ M k =0 Π k | i | P ( i ( n − k ) ji (2 k ) ) ∼ = ⌊ n ⌋ M k =0 Π k | i | + | i || j | P ( i ( n − k − ji (2 k +1) ) . Proof.
See [HW, Theorem 5.9]. (cid:3)
Recall the L -algebra f defined at the beginning of section 10. Let f L be the L -subalgebra generated by the divided powers θ ( n ) i := θ ni / [ n ] ! q i ,π i for all i ∈ I and n ≥
1. Using Lemma 11.1 and Proposition 11.2, it follows that there is a unique L -algebra homomorphism¯ γ : f L → M n ≥ K ( H n ) , θ ( n ) i [ P ( i ( n ) )] . (11.6) Theorem 11.3 (Khovanov-Lauda, Hill-Wang) . The homomorphism ¯ γ from (11.6)is an isomorphism.Proof. See [HW, Theorem 6.14]. (cid:3)
Corollary 11.4.
Every irreducible graded H n -supermodule is absolutely irreducibleof type M .Proof. The absolute irreducibility follows from Theorem 11.3; see the proof of [KL1,Corollary 3.19]. They are all of type M by [HW, Proposition 6.15]. (cid:3) Now we are going upgrade some of these results to U ( g ). For each λ ∈ P , thereis a graded superalgebra homomorphism α n,λ : H n → M i , j ∈ I ⊗ n Hom U ( g ) ( E i λ , E j λ ) , (11.7)where for i = i n ⊗ · · · ⊗ i we write E i λ for E i n · · · E i λ . In diagrammaticterms, α n,λ takes the string diagram for an element of H n to the 2-morphismwhose diagram is obtained by adding the label λ on the right hand edge. Applyingthis to 1 i ( n ) , we obtain the homogeneous idempotent α n,λ (1 i ( n ) ) ∈ End U ( g ) ( E ni λ ).Then define the divided power E ( n ) i λ to be the 1-morphism in the idempotentcompletion ˙ U q,π ( g ) associated to the idempotent (cid:0) α n,λ (1 i ( n ) ) (cid:1) , ¯00 , ¯0 in the ( Q, Π)-envelope. Composing with the isomorphism ω from Proposition 3.5, we get also a UPER KAC-MOODY 2-CATEGORIES 45 graded superalgebra homomorphism α ′ n,λ := ω ◦ α n,λ : H sop n → M i , j ∈ I ⊗ n Hom U ( g ) ( F i λ , F j λ ) , (11.8)where F i λ := F i n · · · F i λ . Let F ( n ) i λ be the 1-morphism in ˙ U q,π ( g ) associatedto the idempotent (cid:0) α ′ n,λ (1 i ( n ) ) (cid:1) , ¯00 , ¯0 . Lemma 11.5. In K ( ˙ U q,π ( g )) , we have that [ Q Π ¯0 E ni λ ] = [ n ] ! q i ,π i [ E ( n ) i λ ] and [ Q Π ¯0 F ni λ ] = [ n ] ! q i ,π i [ F ( n ) i λ ] .Proof. This follows from the definitions and Lemma 11.1. To give some moredetail, Lemma 11.1 means that the idempotent 1 i n ∈ H n splits as a sum of n !idempotents, each of which is conjugate via some unit in H n to 1 ( i n ) . These unitsare homogeneous of various degrees and parities encoded in the ( q, π )-factorial[ n ] ! q i ,π i . When we apply the homomorphism α n,λ to this decomposition, we deducethat the 2-morphism 1 E ni λ splits as a sum of n ! idempotents, each of which isconjugate by some homogeneous unit in End U ( g ) ( E ni λ ) to α n,λ (1 i ( n ) ). Passing to˙ U q,π ( g ), we get from this an isomorphism Q Π ¯0 E ni λ ∼ → E ( n ) i ⊕ [ n ] qi,πi λ by taking thedirect sum of these units appropriately shifted so that they become even of degreezero. (cid:3) Lemma 11.6. In K ( ˙ U q,π ( g )) , we have that [ Q Π ¯0 E i F j λ ] − [ Q Π | i || j | F j E i λ ] = δ i,j [ h h i , λ i ] q i ,π i [1 λ ] , d ij +1 X r =0 ( − r π r | j | + r ( r − / i [ E ( d ij +1 − r ) i E (1) j E ( r ) i λ ] = 0 ( i = j ) , d ij +1 X r =0 ( − r π r | j | + r ( r − / i [ F ( d ij +1 − r ) i F (1) j F ( r ) i λ ] = 0 ( i = j ) . Proof.
The first identity follows from the inversion relations (1.12)–(1.14). Forexample, to prove it in the case i = j and h h i , λ i ≤
0, we use (1.14) to see thatthere is an isomorphism in ˙ U q,π ( g ) Q Π | i | E i F i λ ⊕ −h h i ,λ i− M n =0 Q d i ( −h h i ,λ i− − n ) Π n | i | λ ∼ → Q Π ¯0 F i E i λ . Since [ h h i , λ i ] q i ,π i = − π i P −h h i ,λ i− n =0 q −h h i ,λ i− − ni π ni , this gives what we need onpassing to the Grothendieck group.The second two identities are consequences of Proposition 11.2. One needs tointerpret the isomorphism there first in terms of idempotents, then apply the ho-momorphisms α n +1 ,λ and α ′ n +1 ,λ . (cid:3) Theorem 11.7.
There is a unique surjective L -algebra homomorphism γ : ˙ U q,π ( g ) L ։ K ( ˙ U q,π ( g )) sending λ , e ( n ) i λ and f ( n ) i λ to [1 λ ] , [ E ( n ) i λ ] and [ F ( n ) i λ ] , respectively.Proof. To establish the existence of the homomorphism γ , note to start with thatthere is an L -algebra homomorphism ˙ U q,π ( g ) → L ⊗ L K ( ˙ U q,π ( g )) sending 1 λ , e ( r ) i λ and f ( r ) i λ to [1 λ ], [ E ( r ) i λ ] and [ F ( r ) i λ ], respectively. To see this, we just have tocheck the defining relations of ˙ U q,π ( g ) from (9.2)–(9.4), which follow by Lemma 11.6.Then we restrict this homomorphism to ˙ U q,π ( g ) L , observing that the image of therestriction lies in K ( ˙ U q,π ( g )) thanks to Lemma 11.5.It remains to prove that γ is surjective. The proof of this is essentially the sameas the proof in the purely even case given in [KL3, § n, n ′ ≥ λ ∈ P , we let H n,n ′ ,λ := M i , j ∈ I ⊗ n i ′ , j ′ ∈ I ⊗ n ′ Hom U ( g ) ( E i F i ′ λ , E j F j ′ λ ) . Idempotents in this algebra are idempotent 2-morphisms in U ( g ), hence, there is acanonical homomorphism δ n,n ′ ,λ : K ( H n,n ′ ,λ ) → K ( ˙ U q,π ( g )) . Moreover, there is an L -algebra homomorphism α n,n ′ ,λ : H n ⊗ H sop n ′ ⊗ SYM → H n,n ′ ,λ sending a ⊗ a ′ ⊗ p to α n,µ ( a ) α ′ n ′ ,λ ( a ′ ) β λ ( p ), where µ is the weight labeling the lefthand edge of the diagram α ′ n ′ ,λ ( a ′ )1 λ . Let I n,n ′ ,λ be the two-sided ideal of H n,n ′ ,λ spanned by all string diagrams which involve a U-turn, i.e. they involve at leastone arc whose endpoints are both on the top edge; cf. [KL3, Proposition 3.17]. Let β n,n ′ ,λ : H n,n ′ ,λ ։ H n,n ′ ,λ /I n,n ′ ,λ be the canonical quotient map. The composition γ n,n ′ ,λ := β n,n ′ ,λ ◦ α n,n ′ ,λ issurjective. We get induced a commutative diagram at the level of Grothendieckgroups: K ( H n ⊗ H sop n ′ ⊗ SYM) K ( H n,n ′ ,λ /I n,n ′ ,λ ) K ( H n,n ′ ,λ ) [ γ n,n ′ ,λ ][ α n,n ′ ,λ ] [ β n,n ′ ,λ ] . Following the proof of [KL3, Proposition 3.36], using the facts summarized at thestart of this section plus the fact that H n is finite as a module over its center, oneshows that [ γ n,n ′ ,λ ] is onto, hence, so too is [ β n,n ′ ,λ ].Now let X be an indecomposable object in ˙ U q,π ( g ). Define its width to be thesmallest N ≥ X is isomorphic to a summand of Q m Π b E a λ for some a ∈ Seq of length N and some m ∈ Z , b ∈ Z / λ ∈ P . We are going to showby induction on width that each [ X ] is in the image of γ . For the base case, if X isof width zero, we claim that it is isomorphic to some Q m Π b λ . To see this, recallthat End U ( g ) (1 λ ) is a quotient of SYM, which is strictly postively graded with k indegree zero. Hence, 1 λ is either indecomposable or zero, which implies our claim.Since [1 λ ] is in the image of γ , the base of the induction is now established.For the induction step, take X of width N >
0. We can find some n, n ′ ≥ n + n ′ = N and i ∈ I ⊗ n , i ′ ∈ I ⊗ n ′ such that X is isomorphic to a summand of Q m Π b E i F i ′ λ . This is a consequence of the relations (1.12)–(1.14); cf. the proof of[KL3, Lemma 3.38]. It follows that [ X ] is in the image of δ n,n ′ ,λ , i.e. there is some Y ∈ H n,n ′ ,λ - GSP roj such that δ n,n ′ ,λ ([ Y ]) = [ X ]. The minimality in the definitionof width ensures that β n,n ′ ,λ ([ Y ]) = 0. Pick Z ∈ H n ⊗ H sop n ′ ⊗ SYM -
GSP roj such
UPER KAC-MOODY 2-CATEGORIES 47 that [ γ n,n ′ ,λ ]([ Z ]) = [ β n,n ′ ,λ ]([ Y ]). Then one argues explicitly with idempotents asin [KL3, § α n,n ′ ,λ ]([ Z ]) = [ Y ] + [ Y ′ ]for Y ′ ∈ H n,n ′ ,λ - GSP roj with [ β n,n ′ ,λ ]([ Y ′ ]) = 0. By induction, δ n,n ′ ,λ ([ Y ′ ]) isin the image of γ . Hence, to show that [ X ] = δ n,n ′ ,λ ([ Y ]) is so, we are reducedto showing that δ n,n ′ ,λ ([ α n,n ′ ,λ ]([ Z ])) is in the image of γ . This follows using thefollowing commutative diagram: f L ⊗ L f L ˙ U q,π ( g ) L K ( H n ) ⊗ L K ( H n ′ ) K ( ˙ U q,π ( g )) K ( H n ⊗ H sop n ′ ⊗ SYM) K ( H n,n ′ ,λ ) i λ γ ¯ γ − ⊗ ¯ γ − [ α n,n ′ ,λ ] j n,n ′ δ n,n ′ ,λ . Here, ¯ γ is the isomorphism from Theorem 11.3. the isomorphism j n,n ′ exists becauseof Corollary 11.4, and i λ sends θ i · · · θ i n ⊗ θ j · · · θ j m e i · · · e i n f j · · · f j m λ (cid:3) The decategorification conjecture
We continue to assume the homogeneity condition (1.31) holds and that k = k ¯0 is a field. Let us restate the Decategorification Conjecture from the introduction: Decategorification Conjecture.
The surjective homomorphism γ from Theo-rem 11.7 is an isomorphism. The proof of the following theorem mimics [KL3, § Theorem 12.1.
Assume that the Nondegeneracy Conjecture holds and moreoverthat the Cartan datum is bar-consistent, i.e. (10.2) holds. Then the Decategorifi-cation Conjecture holds as well.Proof.
For a graded superspace V , we let dim q,π V := P n ∈ Z P a ∈ Z / (dim V n,a ) q n π a .For example, viewing the algebra SYM from (1.21) as a graded superalgebra so thatthe isomorphism (1.22) preserves degrees and parities, we have that S := dim q,π SYM = Y i ∈ I Y r ≥ − ( π i q i ) r ∈ Z [[ q ]][ π ] / ( π − . The Nondegeneracy Conjecture implies (indeed, is equivalent to) the assertion that h e a λ , e b λ i = S − dim q,π Hom U ( g ) ( E a λ , E b λ ) (12.1)for a , b ∈ Seq with wt( a ) = wt( b ) and λ ∈ P .Now consider the sesquilinear form on K ( ˙ U q,π ( g )) defined by letting h [ X ] , [ Y ] i bezero if X, Y are 1-morphisms in ˙ U q,π ( g ) whose domains or codomains are different,and setting h [ X ] , [ Y ] i := S − X n ∈ Z X a ∈ Z / dim Hom ˙ U q,π ( g ) ( Q n Π a X, Y ) q n π a if X and Y have the same domain and codomain. Equivalently, for 1-morphisms X, Y : λ → µ in U q,π ( g ), we have that h [ X ] , [ Y ] i = S − dim q,π Hom U q,π ( g ) ( X, Y ) . Comparing with (12.1), using also Theorem 10.3, we deduce that the forms h− , −i on ˙ U q,π ( g ) L and K ( ˙ U q,π ( g )) are intertwined by the homomorphism γ in the sensethat h x, y i = h γ ( x ) , γ ( y ) i .Finally, suppose that x ∈ ˙ U q,π ( g ) L is in the kernel of γ . By the previous para-graph, we have that h x, y i = 0 for all y ∈ ˙ U q,π ( g ) L . In view of the non-degeneracyof the form h− , −i from Theorem 10.1, this implies that x = 0. (cid:3) Remark 12.2.
The assumption of bar-consistency made in both of Theorems 10.1and 12.1 is probably unnecessary. We have included it because we have appealedto [C, Theorem 5.12], where it is assumed from the outset. Providing one allowsthat the canonical basis should be bar-invariant only up to multiplication by π , weexpect that the arguments of [C] should still be valid without bar-consistency, butwe have not checked this assertion in detail. Example 12.3.
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