aa r X i v : . [ m a t h . QA ] J a n CATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS
LAURENT VERAA
BSTRACT . Let U be a quantized enveloping algebra. We consider the adjoint action of an sl -subalgebra of U on a subalgebra of U + that is maximal integrable for this action. We categorify this representation in thecontext of quiver Hecke algebras. We obtain an action of the 2-category associated with sl on a category ofmodules over certain quotients of quiver Hecke algebras. Our approach is similar to that of Kang-Kashiwara[KK12] for categorifications of highest weight modules via cyclotomic quiver Hecke algebras. One of themain new features is a compatibility of the categorical action with the monoidal structure, categorifying thenotion of derivation on an algebra. As an application of some of our results, we categorify the higher orderquantum Serre relations, extending results of Stoˇsi´c [Sto15] to the non simply-laced case. C ONTENTS
1. Introduction 12. Quantum groups 43. KLR algebras 74. Categorification of the adjoint action of E i sl action 316. Projective resolutions 38References 431. I NTRODUCTION
The study of categorified quantum groups began, in its current form, with the work of Chuang andRouquier [CR08]. While categorifications of representations of Lie algebras and their quantized ver-sions had already appeared (for instance in [Ari96], [BFK99] or [HK01] among others), the key noveltyin [CR08] was the introduction of Hecke algebra actions at the level of natural transformations in theaxiomatic. The resulting notion is called an sl -categorification. This was later generalized to arbitrarysymmetrizable Kac-Moody types by Khovanov and Lauda in [KL11], [KL10], and Rouquier in [Rou08].More precisely, let C be a symmetrizable generalized Catan matrix and U the corresponding quantumgroup. In [KL11] and [Rou08], a family of graded algebra ( H β ) β ∈ Q + attached to C is introduced, where Q + denotes the cone of linear combinations of simple roots with coefficients in Z > . These algebrasare now known as the Khovanov-Lauda-Rouquier algebras (or simply KLR algebras, or quiver Heckealgebras). It is shown by Khovanov and Lauda in [KL11] that over a field, the direct sum over β ∈ Q + of the Grothendieck groups of the category of finitely generated graded projective H β -modules is iso-morphic to the integral positive part of U . The multiplication of U corresponds to an induction producton the KLR algebras side. Furthermore, Khovanov-Lauda and Rouquier introduce 2-categories whichcategorify Beilinson-Lusztig-MacPherson’s idempotent version of the quantum group ˙ U . While the def-initions of the 2-categories in [KL10] and [Rou08] differ on a few points, it was shown by Brundan in[Bru16] that they are actually isomorphic. Hence there is an essentially unique 2-quantum group U asso-ciated to C . Rouquier proved in [Rou08] that in the case C = ( ) , the 2-representations of the 2-category U recover the sl -categorifications of [CR08].In [KL09], Khovanov and Lauda also conjecture that the irreducible module of U of highest weight Λ is categorified by the cyclotomic KLR algebras ( H Λ β ) β ∈ Q + , which are certain quotients of the KLRalgebras. This was proved by Kang and Kashiwara in [KK12] (see also [Web17]). We recall briefly theresults and the strategy of proof of [KK12]. Kang and Kashiwara prove that the 2-category U acts on thedirect sum of the categories of H Λ β -modules, for β ∈ Q + . Given a simple root i , the Chevalley generators F i and E i of U act as some induction and restriction functors F Λ i and E Λ i between the algebras H Λ β . Their proof that these functors yield an action of U is based on the following key result. Given a module M over H Λ β , they show that there is an exact sequence0 → F i ( M ) → F i ( M ) → F Λ i ( M ) →
0. (1.1)where F i is a “left i -induction” functor and F i a “right i -induction” functor between KLR algebras. Fur-thermore this exact sequence is natural in M . From this they can recover many of the properties neededto construct a representation of U , such as the exactness of the functors F Λ i and E Λ i , and a categorificationof the Lie algebra relation [ e i , f i ] = h i . They also prove that at the Grothendieck group level, this actionof U gives the irreducible module of U of highest weight Λ .In this paper, we categorify part of the adjoint action of U using a similar approach. By adjoint action,we mean the left adjoint action of U on itself arising from the Hopf algebra structure. Explicitly, for asimple root i , the adjoint action of the Chevalley generators e i , f i , k i on a element y of U of weight β ∈ Q + takes the form ad e i ( y ) = e i y − q h i ∨ , β i i ye i , ad f i ( y ) = ( f i y − y f i ) k i , ad k i ( y ) = q h i ∨ , β i i y . (1.2)where i ∨ denotes the coroot associated to i , and h , i is the pairing between the dual weight and theweight lattices. Note that the adjoint action of U on itself is not integrable. Our approach does not yielda categorification of the complete adjoint action of U , but rather of the action of a given sl -subalgebra U i of U on a subalgebra U + [ i ] of U + which is integrable for the action of U i , and maximal for thisproperty. More explicitly, for a fixed simple root i , U i is the subalgebra of U generated by e i , f i and k i . The subalgebra U + [ i ] of U + is generated by the ad ( n ) e i ( e j ) for n > j a simple root not equalto i . Here ad ( n ) e i denotes the n th divided power of ad e i . The subalgebra U + [ i ] is studied by Lusztigin [Lus10, Chapter 38], as part of his study of the braid group action. We prove in Proposition 2.4that the adjoint action induces an integrable representation of U i on U + [ i ] , and that U + [ i ] is the largestsubspace of U + with this property. It is this representation that we categorify. To do so, we start bycategorifying the algebra U + [ i ] . For β ∈ Q + , we define an algebra H i β as the quotient of H β by the two-sided ideal generated by the idempotent 1 β − i , i . These algebras are the analogues of the cyclotomic KLRalgebras in our context. However, they behave quite differently: for instance, they are typically infinitedimensional (see Proposition 4.10) while the cyclotomic KLR algebras are always finite dimensional.Then, we define the category H [ i ] as the direct sum of the categories of finitely generated graded H i β -modules, for β ∈ Q + . Our category H [ i ] is a Serre and monoidal full subcategory of the category of allmodules over the KLR algebras, and serves as a categorical analogue of the subalgebra U + [ i ] . On H [ i ] we consider an endofunctor ad E i , which can be defined as an induction functor between the algebras H i β , in a way similar to the functors F Λ i of [KK12]. In Proposition 4.11, we endow the powers of ad E i with an action of the affine nil Hecke algebras. In particular, we obtain well-defined divided powersad ( n ) E i . This action is one of the axioms to establish a structure of 2-representation of sl on H [ i ] . Theother axioms are more delicate to check, and we prove them using a similar approach to [KK12]. Ourfirst main result is the following. Theorem 1.1 (Theorem 4.18) . For all M ∈ H [ i ] of weight β ∈ Q + , there is a short exact sequence → q h i ∨ , β i i ME i → E i M → ad E i ( M ) → Furthermore, this sequence is natural in M.
Here, the coefficient q h i ∨ , β i i denotes a grading shift. The short exact sequence of Theorem 1.1 canbe seen as a categorification of equation (1.2), and it shows in particular that the functor ad E i lifts theoperator ad e i to the categorical setting. It is an analogue of the short exact sequence (1.1) from [KK12]in our framework. From this, we can deduce as in [KK12] that the functor ad E i is exact (Corollary 4.20).At this stage, the main axiom left to check is the categorification of the relation [ e i , f i ] = h i . This is donein Theorem 5.6. The conclusion of this work is that the action of U i on U + [ i ] lifts to an sl categoricalaction on H [ i ] . This is our second main result. Theorem 1.2 (Theorem 5.3) . The endofunctor ad E i induces a 2-representation of sl on H [ i ] . A key new feature of our work is a compatibility of the 2-representation with the monoidal structure.In general, the adjoint action of a Hopf algebra on itself is compatible with the multiplication. In the case
ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 3 of the quantum group U , this takes the simple form ad e i ( yz ) = ad e i ( y ) z + q h i ∨ , β i i y ad e i ( z ) for y , z ∈ U and y of weight β ∈ Q + . We prove that this formula has a categorical analogue, in the form of a shortexact sequence. Theorem 1.3 (Corollary 4.21) . For M , N ∈ H [ i ] with M of weight β ∈ Q + , there is a short exact sequence → q h i ∨ , β i i M ad E i ( N ) → ad E i ( MN ) → ad E i ( M ) N → Furthermore, this sequence is natural in M , N. This feature does not appear for cyclotomic KLR algebras (there is no monoidal structure in that case)and, to the author’s knowledge, this is the first example of a 2-representation with such structure. Weuse this in a crucial way to simplify the computations in our proofs. Iterating the short exact sequenceof Theorem 1.3, we obtain an interesting filtration of ad nE i ( MN ) . We describe it in detail in Proposition4.23. As one of the consequences of Theorem 1.3, we can prove the following result, which categorifiesthe fact that U + [ i ] is generated by the ad ( n ) e i ( e j ) for n > j ∈ I \ { i } as a subalgebra of U + . Theorem 1.4 (Theorem 4.30, Corollary 4.31) . The category H [ i ] is generated by • the modules ad ( n ) E i ( E j . . . E j r ) for n > and j , . . . , j r ∈ I \ { i } as a Serre subcategory, • the modules ad ( n ) E i ( E j ) for n > and j ∈ I \ { i } , as a Serre and monoidal subcategory. Thanks to this theorem, we are able to reduce some explicit computations to modules of the formad ( n ) E i ( E j . . . E j r ) , which can be understood quite well.As we mentioned above, the action of U i on U + [ i ] is an integrable representation. As another con-sequence of Theorems 1.1 and 1.3, we prove in Corollary 4.25 that the functor ad E i is locally nilpotent.More precisely, we prove in Proposition 4.24 that the algebra H i β is zero precisely when s i ( β ) / ∈ Q + ,where s i is the reflection of the root lattice corresponding the the simple root i . In particular, the algebra H i β + ni is zero when n is large enough, which proves that ad E i is locally nilpotent.Finally, as an application of the above results, we construct projective resolutions of the modulesad ( n ) E i ( E j . . . E j r ) and prove a categorification of the higher order quantum Serre relations. This general-izes results of Stoˇsi´c [Sto15] to the non simply laced case. At the decategorified level, the higher orderquantum Serre relations state that ad ( n ) e i ( e mj ) = i = j and n > − m h i ∨ , j i . One canwrite this more explicitly as n ∑ k = ( − ) k q k ( n + m h i ∨ , j i− ) i e ( n − k ) i e mj e ( k ) i = i , n > M an H i β -module, we define acomplex of H β + ni -modules of the formAd ( n ) E i ( M ) = → q n ( n + h i ∨ , β i− ) i ME ( n ) i → . . . → q k ( n + h i ∨ , β i− ) i E ( n − k ) i ME ( k ) i → . . . → E ( n ) i M → E ( n ) i M is in cohomological degree 0. We prove in Theorem 6.6 that the cohomology ofAd ( n ) E i ( M ) is concentrated in degree 0, and equal to ad ( n ) E i ( M ) . By our vanishing criterion on the algebras H i β + ni , we also know that this cohomology is zero when n is large enough. When M = E mj , the bound forthe vanishing of the cohomology is simply n > − m h i ∨ , j i . In that case, we get a complex of projectivemodules with zero cohomology, hence a null-homotopic complex (Theorem 6.8). This categorifies thehigher order quantum Serre relation. Our method is completely different from that of [Sto15], wherethe result is proved by constructing explicit homotopies, using the thick diagrammatic calculus for KLRalgebras (see [KLMS12], [Sto19]). Our approach has the drawback of not providing explicit homotopies,but has the advantage of working for any type and minimizing the amount of computations done in thehomotopy category of KLR algebras. Similar projective resolutions also appear in [BKM14] and recentlyin [BKS19].We now describe the structure of the paper. In Section 2, we recall the main definitions regardingquantum groups following [Lus10], and we briefly discuss their adjoint representation. In Section 3, wedefine the KLR algebras, and recall a few basic results (including the PBW theorem and a description LAURENT VERA of the center). We also explain how they categorify the positive part of quantum groups. In Section 4,we define and study the category H [ i ] and the functor ad E i . After proving some elementary properties,we state our main theorem 4.18 and deduce from it various consequences, such as the exactness of ad E i (Corollary 4.20), the compatibility with the monoidal structure (Corollary 4.21) and the vanishing crite-rion for the algebras H i β (Proposition 4.24). The end of Section 4 is devoted to the proof of Theorem 4.18.In Section 5, we complete the proof that H [ i ] is endowed with a structure of sl -categorification. Thisentails checking the categorification of the Lie algebra relation [ e i , f i ] = h i , which amounts to provingsome Mackey-type decompositions for the algebras H i β . Finally, Section 6 is devoted to the constructionof projective resolutions and to the proof of the categorical higher quantum Serre relations. Section 6does not rely on Section 5. Acknowledgments.
I would like to thank my advisor, Rapha¨el Rouquier for his support and manyhelpful discussions. 2. Q
UANTUM GROUPS
Definitions.
Root datum.
We refer to [Lus10] for a detailed introduction to quantum groups. For the rest ofthis paper, we fix a
Cartan datum ( I , · ) . This means that I is a non empty set and · : Z I ⊗ Z Z I → Z is asymmetric bilinear form such that • for all i ∈ I , we have i · i ∈ Z > , • for all i = j ∈ I , we have i · ji · i ∈ Z .We also fix a root datum of type ( I , · ) . This is the data of finitely generated free abelian groups X (the weight latice ) and Y (the dual weight latice ), together with a perfect pairing h· , ·i : Y ⊗ Z X → Z andinjective set maps ( I ֒ → X , i i ) , ( I ֒ → Y , i i ∨ ) satisfying the condition (cid:10) i ∨ , j (cid:11) = i · ji · i for all i , j ∈ I . Let c i , j = h i ∨ , j i and d i = i · i . For all i ∈ I , there is an automorphism s i of the lattice X defined by s i ( λ ) = λ − h i ∨ , λ i i for λ ∈ X . The root lattice is Q = ⊕ i ∈ I Z i , and we let Q + = ⊕ i ∈ I Z > i .Given an element β = ∑ i ∈ I n i i of Q + , the integer n = ∑ i ∈ I n i is called the height of β , and denoted | β | .We also let I β = n ( j , . . . , j n ) ∈ I n , n ∑ k = j k = β o .2.1.2. Quantum groups.
Let us start by introducing some notation in Q ( q ) . For all i ∈ I , we let q i = q d i .For k ∈ Z , the quantum integer [ k ] i is given by [ k ] i = q ki − q − ki q i − q − i .If k >
0, the quantum factorial [ k ] i ! is given by [ k ] i ! = k ∏ l = [ l ] i . Definition 2.1.
The quantum group U associated to the above root datum is the unital Q ( q ) -algebra onthe generators e i , f i for i ∈ I , and k µ for µ ∈ Y subject to the relations(1) k = k µ k µ ′ = k µ + µ ′ for all µ , µ ′ ∈ Y ,(2) k µ e i = q h µ , i i e i k µ and k µ f i = q −h µ , i i f i k µ for all i ∈ I and µ ∈ Y , ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 5 (3) for all i , j ∈ I e i f j − f j e i = δ i , j k i − k − i q i − q − i where k i = k d i i ∨ ,(4) for all i = j ∈ I , the quantum Serre relations: − c i , j + ∑ ℓ = ( − ) ℓ e ( ℓ ) i e j e ( − c i , j + − ℓ ) i = − c i , j + ∑ ℓ = ( − ) ℓ f ( ℓ ) i f j f ( − c i , j + − ℓ ) i = e ( ℓ ) i and f ( ℓ ) i are called the divided powers and are defined by e ( ℓ ) i = [ ℓ ] i ! e ℓ i , f ( ℓ ) i = [ ℓ ] i ! f ℓ i .The algebra U is Q -graded with e i in degree i and f i in degree − i for all i ∈ I , and k µ in degree 0 forall µ ∈ Y . Furthermore, U has a structure of Q -graded Hopf algebra, with coproduct ∆ and antipode S defined by the following formulas: ∆ ( e i ) = e i ⊗ + k i ⊗ e i , ∆ ( f i ) = f i ⊗ k − i + ⊗ f i , ∆ ( k µ ) = k µ ⊗ k µ , S ( e i ) = − k − i e i , S ( f i ) = − f i k i , S ( k µ ) = k − µ ,for all i ∈ I and µ ∈ Y .For i ∈ I , we denote by U i the subalgebra of U generated by e i , f i and k i . We also denote by U + thesubalgebra of U generated by the e j for j ∈ I . There is a non-degenerate bilinear form y ⊗ z ( y , z ) on U + . To describe it, we start by endowing U + ⊗ Q ( q ) U + with an algebra structure by defining themultiplication ( y ⊗ z )( y ′ ⊗ z ′ ) = q β · γ ( yy ′ ) ⊗ ( zz ′ ) for y , y ′ , z , z ′ ∈ U + with z , y ′ homogeneous of respective degrees β , γ ∈ Q + . There is a morphism ofalgebras r : U + → U + ⊗ Q ( q ) U + defined by r ( e i ) = e i ⊗ + ⊗ e i for all i ∈ I . Then there is a uniquebilinear form y ⊗ z ( y , z ) on U + satisfying the following properties (see [Lus10, Proposition 1.2.3]):(1) (
1, 1 ) = ( e i , e j ) = δ i , j − q i for all i , j ∈ I ,(3) ( y , zz ′ ) = ( r ( y ) , z ⊗ z ′ ) for all y , z , z ′ ∈ U + ,(4) ( yy ′ , z ) = ( y ⊗ y ′ , r ( z )) for all y , y ′ , z ∈ U + .Finally, we define the integral form of U . Let A = Z (cid:2) q , q − (cid:3) . We let A U be the sub- A -algebra of U generated by the k µ for µ ∈ Y , and e ( n ) i , f ( n ) i for i ∈ I and n >
0. We have subalgebras of A U defined asabove: A U + and A U i for i ∈ I .2.1.3. Representations of quantum groups. A weight representation of U is a U -module V that decomposesas V = M λ ∈ X V λ where V λ = (cid:8) v ∈ V | ∀ µ ∈ Y , k µ v = q h λ , µ i v (cid:9) .An integrable representation of U is a weight representation V on which the action of e i and f i is locallynilpotent for all i ∈ I . More explictly, a weight representation V is an integrable representation if andonly if for all v ∈ V and i ∈ I , there exists an integer n > e ni v = f ni v = Adjoint representation. If A is a Hopf algebra with coproduct ∆ and antipode S , the (left) adjointrepresentation of A on itself is defined byad z ( y ) = ∑ z ( ) yS ( z ( ) ) for all y , z ∈ A . Here we have used Sweedler’s notation for the coproduct ∆ ( z ) = ∑ z ( ) ⊗ z ( ) . LAURENT VERA
The adjoint action is compatible with the product of A , in the following sense: for all y , y ′ , z ∈ A wehave ad z ( yy ′ ) = ∑ ad z ( ) ( y ) ad z ( ) ( y ′ ) .In the case of the quantum group U , there are simple formulas for the adjoint action of the algebragenerators of U . For y ∈ U homogeneous of degree β ∈ Q + , i ∈ I and µ ∈ Y we havead e i ( y ) = e i y − q h i ∨ , β i i ye i , ad f i ( y ) = ( f i y − y f i ) k i , ad k µ ( y ) = q h µ , β i y .The compatibility with the product takes the form of the following “ q -Leibniz formulas”ad e i ( yz ) = ad e i ( y ) z + q h i ∨ , β i i y ad e i ( z ) ,ad f i ( yz ) = q h i ∨ , γ i i ad f i ( y ) z + y ad e i ( z ) , (2.1)for all i ∈ I and y , z ∈ U homogeneous of respective degrees β , γ ∈ Q + . Hence ad e i and ad f i can bethought of as “ q -derivations” of the algebra U . We can define divided powers for ad e i and ad f i as above:ad ( n ) e i = [ n ] i ! ad ne i , ad ( n ) f i = [ n ] i ! ad nf i .The quantum Serre relations take a particularly simple form in terms of the adjoint representation.Namely, we have ad ( − c i , j + ) e i ( e j ) =
0, ad ( − c i , j + ) f i ( f j ) = i , j ∈ I . These can be generalized to higher order quantum Serre relations (see [Lus10, Proposi-tion 7.1.5]). We have ad ( n ) e i ( e mj ) =
0, ad ( n ) f i ( f mj ) = i = j ∈ I and n , m ∈ Z > such that n > − mc i , j .The adjoint representation is a weight representation of U . However, it is not a integrable represen-tation of U because the operators ad e i and ad f i are not locally nilpotent. For instance, for every integer m > me i ( e i ) = (cid:18) m ∏ k = ( − q mi ) (cid:19) e m + i =
0. (2.2)Nevertheless, an integrable representation can be obtained by looking at the action of a given sl -subalgebra on a certain subalgebra of U + . More precisely, we fix i ∈ I and consider the adjoint actionof the subalgebra U i on U . Following Lusztig [Lus10, Chapter 38], we let U + [ i ] be the subalgebra of U + generated by the elements ad ( n ) e i ( e j ) , for j ∈ I \ { i } and n >
0. The subalgebra U + [ i ] can also bedescribed in terms of the inner product of U + , as follows. Proposition 2.2 ([Lus10, Proposition 38.1.6]) . The subalgebra U + [ i ] consists of the elements y ∈ U + suchthat ( ze i , y ) = for all z ∈ U + . We also have the following useful result.
Proposition 2.3 ([Lus10, Lemmas 38.1.2 and 38.1.5]) . The multiplication map U + [ i ] ⊗ Q ( q ) Q ( q )[ e i ] → U + is an isomorphism. We now prove the main property of U + [ i ] regarding the adjoint action. Proposition 2.4.
The subalgebra U + [ i ] is stable under the adjoint action of U i . The adjoint action of U i on U + [ i ] is an integrable representation. Furthermore, U + [ i ] is maximal in the following sense: if V ⊆ U + is an integrableU i -submodule of U for the adjoint action, then V ⊆ U + [ i ] .Proof. We need to prove that U + [ i ] is stable under ad e i and ad f i . Since ad e i and ad f i satisfy formulas(2.1), it suffices to prove that ad e i ( y ) , ad f i ( y ) ∈ U + [ i ] for y an algebra generator of U + [ i ] .Let y = ad ne i ( e j ) for n > j ∈ I \ { i } . Then ad e i ( y ) = ad n + e i ( e j ) ∈ U + [ i ] . Hence U + [ i ] is stableunder ad e i . To prove that ad f i ( x ) ∈ U + [ i ] , we remark thatad f i ( e j ) = f i ( y ) = ad e i ( ad f i ( ad n − e i ( e j ))) − [ c i , j + ( n − )] i ad n − e i ( e j ) , ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 7 the second equation following from the relation e i f i − f i e i = k i − k − i q i − q − i . The result now follows by animmediate induction on n .Hence we have a representation of U i on U + [ i ] . Let us check that this representation is integrable.The set of elements y ∈ U + [ i ] on which ad e i and ad f i act nilpotently is a sub- U i -module, and also asubalgebra given relations (2.1). By the quantum Serre relations, ad e i and ad f i act nilpotently on every e j for j ∈ I \ { i } . Since these generate U + [ i ] as an algebra and U i -module, we conclude that the action of U i on U + [ i ] is integrable.Finally, we prove the maximality of U + [ i ] with respect to these properties. Let y ∈ U + be such thatad me i ( y ) = m >
0. By Proposition 2.3, we can write y in the form y = N ∑ ℓ = y ℓ e ℓ i for some elements y , . . . , y N of U + [ i ] with y N =
0. Assume N >
0. Using equations (2.1) and (2.2), wesee that for all ℓ > me i ( y ℓ e ℓ i ) ∈ a ℓ y ℓ e ℓ + mi + ∑ k <ℓ + m U + [ i ] e ki for some a ℓ ∈ Q ( q ) non-zero. Hence0 = ad me i ( y ) ∈ a N y N e N + mi + ∑ k < N + m U + [ i ] e ki .By the uniqueness part of Proposition 2.3, we have y N = N =
0, andthe result follows. (cid:3)
3. KLR
ALGEBRAS
In this section, we define the KLR algebras and recall how they categorify the positive part of thequantum group. We follow the approach and definitions of Rouquier [Rou08], [Rou12]. We also refer to[Bru13] for a survey of the subject. At the end of the section, we prove some elementary computationalresults about KLR algebras that will be useful in the proofs of our main results.For the rest of this paper, we fix a commutative ring K . We use the symbol ⊗ for ⊗ K . If M is a graded K -module with degree d component M d , we denote by qM the shifted module whose grading is definedby ( qM ) d = M d + . For a = ∑ ℓ ∈ Z a ℓ q ℓ ∈ N [ q , q − ] we let aM = M ℓ ∈ Z ( q ℓ M ) ⊕ a ℓ .3.1. Polynomial rings and nil Hecke algebras.
Symmetric groups.
The symmetric group on n letters is denoted by S n . For k = ℓ two integers in {
1, . . . , n } , we let s k , ℓ be the permutation ( k ℓ ) ∈ S n , and we put s k = s k , k + . Given a finite sequence k = ( k , . . . , k r ) of elements of {
1, . . . , n − } , we let s k = s k . . . s k r . If k ℓ , we denote by [ k ↑ ℓ ] the sequence ( k , k +
1, . . . , ℓ ) and by [ ℓ ↓ k ] the sequence ( ℓ , ℓ −
1, . . . k ) . If k > ℓ , [ k ↑ ℓ ] and [ ℓ ↓ k ] areunderstood to be the empty sequence ∅ , in which case s ∅ = ω ∈ S n , a reduced decomposition of ω is a finite sequence k = ( k , . . . , k r ) such that s k = ω and r is minimal (the integer r is the length of ω , denoted by l ( ω ) ). The longest element of S n is denoted by ω [ n ] . If k < ℓ ∈ {
1, . . . , n } , we denote by ω [ k , ℓ ] the longest element of the parabolic subgroup of S n generated by s k , . . . , s ℓ − . The elements ω [ k , ℓ ] satisfy the relations ω [ k , ℓ + ] = ω [ k , ℓ ] s [ ℓ ↓ k ] = s [ k ↑ ℓ ] ω [ k , ℓ ]= ω [ k + ℓ + ] s [ k ↑ ℓ ] = s [ ℓ ↓ k ] ω [ k + ℓ + ] . (3.1) LAURENT VERA
Demazure operators.
Let P n = K [ x , . . . x n ] . We consider P n graded with x k in degree 2 for all k ∈ {
1, . . . , n } . The symmetric group S n acts on P n by permuting x , . . . , x n . For k = ℓ two integers in {
1, . . . , n − } , we define the Demazure operator , or divided difference operator , ∂ k , ℓ on P n by ∂ k , ℓ ( f ) = f − s k , ℓ ( f ) x ℓ − x k .For k ∈ {
1, . . . , n − } , we put ∂ k = ∂ k , k + . The Demazure operators are P S n n -linear. They are also “skewderivations” of the algebra P n for the automorphism s k , ℓ , in the sense that the satisfy the followingskewed Leibniz formula: ∂ k , ℓ ( f g ) = ∂ k , ℓ ( f ) g + s k , ℓ ( f ) ∂ k , ℓ ( g ) (3.2)for all k = ℓ ∈ {
1, . . . , n − } and f , g ∈ P n . The Demazure operators are all conjugate under the actionof S n : for k = ℓ and ω ∈ S n we have ω∂ k , ℓ ω − = ∂ ω ( k ) , ω ( ℓ ) . We also have the following relations: ∂ k = k ∈ {
1, . . . n − } , ∂ k ∂ ℓ = ∂ ℓ ∂ k for all k , ℓ ∈ {
1, . . . n − } such that | k − ℓ | > ∂ k + ∂ k ∂ k + = ∂ k ∂ k + ∂ k for all k ∈ {
1, . . . n − } .3.1.3. Affine nil Hecke algebras.
We follow [Rou08, Subsection 3.1].
Definition 3.1.
The affine nil Hecke algebra of rank n is the K -algebra H n on the generators x , . . . , x n and τ , . . . , τ n − subject to the following relations(1) x k x ℓ = x ℓ x k for all k , ℓ ∈ {
1, . . . , n } ,(2) τ k = k ∈ {
1, . . . , n − } ,(3) τ k x ℓ − x s k ( ℓ ) τ k = ℓ = k + − ℓ = k k ∈ {
1, . . . , n − } and ℓ ∈ {
1, . . . , n } ,(4) τ k τ ℓ = τ ℓ τ k for all k , ℓ ∈ {
1, . . . , n − } such that | k − ℓ | > τ k + τ k τ k + − τ k τ k + τ k = k ∈ {
1, . . . , n − } .The algebra H n is graded, with x ℓ in degree 2 for all ℓ ∈ {
1, . . . , n } and τ k in degree − k ∈ {
1, . . . , n − } . There is an isomorphism of graded K -algebras ([Rou08, Proposition 3.4]) H n ∼ −→ End P S nn ( P n ) , x k x k , τ k ∂ k . (3.3)Given a finite sequence k = ( k , . . . , k r ) of elements of {
1, . . . , n − } , we put τ k = τ k . . . τ k r and ∂ k = ∂ k . . . ∂ k r . For all ω ∈ S n , we can define an element τ ω ∈ H n in the following way: take k a reducedexpression of ω , and let τ ω = τ k . By relations (4) and (5) in Definition 3.1, this does not depend onthe choice of the reduced expression. There is a similarly defined operator ∂ ω on P n . By relation (2) inDefinition 3.1, for all ω , ω ∈ S n we have τ ω τ ω = (cid:26) τ ω ω if l ( ω ω ) = l ( ω ) + l ( ω ) ,0 otherwise.Since P n is a free graded P S n n -module of graded rank q − n ( n − ) [ n ] !, the K -algebra H n is a matrix algebraover P S n n . Let e n = x x . . . x n − n τ ω [ n ] . Then e n is a primitive idempotent of H n , and there is anisomorphism of graded ( H n , P S n n ) -bimodules (cid:26) P n ∼ −→ q − n ( n − ) H n e n P P τ ω [ n ] (3.4)From this, we deduce that there is an isomorphism of graded ( H n , P S n n ) -bimodules H n ≃ q − n ( n − ) [ n ] ! ( H n e n ) . (3.5) ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 9
KLR algebras.
Definitions.
For all i , j ∈ I , we fix a polynomial Q i , j ( u , v ) ∈ K [ u , v ] . Assume that this data satisfiesthe following conditions: • for all i ∈ I , Q i , i ( u , v ) = • for all i , j ∈ I , Q i , j ( u , v ) = Q j , i ( v , u ) , • for all i = j ∈ I , there are some invertible elements t i , j and t j , i of K such that Q i , j ( u , v ) has theform Q i , j ( u , v ) ∈ t i , j u − c i , j + t j , i v − c j , i + ∑ s , t > d i s + d j t = − i · j Ku s v t .The KLR algebras are a family of K -algebras attached to the data of these polynomials. Definition 3.2.
Let β be an element of Q + of height n . The Khovanov-Lauda-Rouquier algebra H β is theunital K -algebra with generators 1 ν for ν ∈ I β , x , . . . , x n and τ , . . . , τ n − subject to the following rela-tions:(1) 1 ν ν ′ = δ ν , ν ′ ν for all ν , ν ′ ∈ I β , and ∑ ν ∈ I β ν = β , where 1 β is the unit of H n ,(2) x k ν = ν x k for all ν ∈ I β and k ∈ {
1, . . . , n } ,(3) τ k ν = s k ( ν ) τ k for all ν ∈ I β and k ∈ {
1, . . . , n − } ,(4) x k x ℓ = x ℓ x k for all k , ℓ ∈ {
1, . . . , n } ,(5) τ k ν = Q ν k , ν k + ( x k , x k + ) ν for all ν ∈ I β and k ∈ {
1, . . . , n − } ,(6) (cid:16) τ k x ℓ − x s k ( ℓ ) τ k (cid:17) ν = ν if ℓ = k + ν k = ν k + − ν if ℓ = k and ν k = ν k + k ∈ {
1, . . . , n − } , ℓ ∈ {
1, . . . n } and ν ∈ I β ,(7) τ k τ ℓ = τ ℓ τ k for all k , ℓ ∈ {
1, . . . , n − } such that | k − ℓ | > ( τ k + τ k τ k + − τ k τ k + τ k ) ν = δ ν k , ν k + ∂ k , k + (cid:0) Q ν k , ν k + ( x k + , x k + ) (cid:1) ν for all k ∈ {
1, . . . , n − } and ν ∈ I β . Convention.
We will number components of tuples ν ∈ I n from the right to the left, to match with thegraphical calculus interpretation of the KLR algebras. Namely, we will write ν = ( ν n , . . . , ν ) for ν ∈ I n and we let the symmetric group S n act on I n accordingly. For instance, if ν = ( c , b , a ) ∈ I , we have s ( ν ) = ( c , a , b ) and s ( ν ) = ( b , c , a ) .For a sequence k = ( k , . . . , k r ) of elements of {
1, . . . , n − } , we put τ k = τ k . . . τ k r . We also define a(possibly non-unital) K -algebra H n by H n = M β ∈ Q + | β | = n H β .The algebra H n is unital only when I is finite. There is a grading on H n defined in the following way: • ν is in degree 0, • x k ν is in degree ν k · ν k , • τ ℓ ν is in degree − ν l · ν l + ,for all ν ∈ I n , k ∈ {
1, . . . , n } and ℓ ∈ {
1, . . . , n − } . There is an anti-automorphism rev n of H n definedby rev n (cid:0) ( ν n ,..., ν ) (cid:1) = ( ν ,..., ν n ) , rev n ( x k ) = x n − k , rev n ( τ ℓ ) = τ n − − ℓ ,for all ( ν n , . . . , ν ) ∈ I n , k ∈ {
1, . . . , n } and ℓ ∈ {
1, . . . , n − } . Hence H n ≃ H op n . Let H = M β ∈ Q + H β − modbe the category of finitely generated graded modules over the KLR algebras. The morphisms in H aregiven by degree preserving module maps. The category H is K -linear and abelian. Furthermore it isgraded, in the sense that it is equipped with the grading shift functor M qM .3.2.2. PBW theorem.
The KLR algebras satisfy a PBW type theorem.
Theorem 3.3 ([Rou08, Theorem 3.7]) . For all β ∈ Q + , the K-algebra H β is free as a K-module. Let n = | β | and let T be a complete set of reduced decompositions of elements of S n . Then the following sets are bases over Kfor H β : n τ ω x a . . . x a n n ν | ν ∈ I β , a , . . . , a n > ω ∈ T o , n x a . . . x a n n τ ω ν | ν ∈ I β , a , . . . , a n > ω ∈ T o .Let P β be the subalgebra of H β generated by x , . . . , x n and 1 ν for ν ∈ I β . As a consequence ofTheorem 3.3, there is an isomorphism of algebras P β ≃ M ν ∈ I β P n ν .Furthermore, H β is free of rank n ! as a left (or right) module over P β , with basis { τ ω | ω ∈ T } , for T anyset of reduced expressions for the elements of S n .3.2.3. Center.
Let β ∈ Q + of height n . The symmetric group S n acts naturally on I β by permutation ofthe components. Thus it also acts on the algebra P β by permuting x , . . . , x n and the 1 ν , ν ∈ I β . Proposition 3.4 ([Rou08, Proposition 3.9]) . The center of H β is P S n β . Monoidal structure.
The category H has a monoidal structure, which we describe now.3.3.1. Inductions between KLR algebras.
Assume that I is finite. For n , m >
0, there is a morphism of K -algebras r n + mn : H n → H n + m (called right inclusion ) defined by r n + mn ( ν ) = ∑ µ ∈ I m µν , r n + mn ( x k ) = x k , r n + mn ( τ ℓ ) = τ ℓ ,for all ν ∈ I n , k ∈ {
1, . . . , n } and ℓ ∈ {
1, . . . , n − } . Here, µν denotes the concatenation of µ ∈ I m with ν ∈ I n . We also have a morphism of K -algebras l n + mn : H n → H n + m (called left inclusion ) defined by l n + mn ( ν ) = ∑ µ ∈ I m νµ , l n + mn ( x k ) = x m + k , l n + mn ( τ ℓ ) = τ m + ℓ ,for all ν ∈ I n , k ∈ {
1, . . . , n } and ℓ ∈ {
1, . . . , n − } . By Theorem 3.3, right and left inclusion are injective.Let H m , n = H m ⊗ H n . There is an injective morphism of K -algebras (cid:26) H m , n → H n + m , y ⊗ z y ⋄ z = l n + mm ( y ) r n + mn ( z ) .This defines an associative binary operation ⋄ on L k > H k with unit element 1 ∈ H = K . This mor-phism also endows H n + m with a structure of H m , n -bimodule. By Theorem 3.3, H n + m is free of rank ( n + m ) ! n ! m ! as a left (resp. right) H m , n -module, and there are decompositions H n + m = M ω ∈ T H m , n τ ω = M ω ∈ T ′ τ ω H m , n , (3.6)where T (resp. T ′ ) is a complete set of reduced decompositions for minimal length representatives ofright (resp. left) cosets of S m × S n in S n + m . Hence there is a biexact bifunctor defined by (cid:26) H m − mod × H n − mod → H n + m − mod, ( M , N ) MN = H n + m ⊗ H m , n ( M ⊗ N ) . ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 11
This defines the monoidal structure of H .It will be convenient to work with the algebras H β rather than with H n . Let us describe the monoidalstructure at their level. For β , . . . , β r ∈ Q + , we put 1 β ,..., β r = β ⋄ . . . ⋄ β r , an idempotent of H β + ... + β r .Let β , γ ∈ Q + of respective heights m , n , and let H β , γ = H β ⊗ H γ . There is a non-unital morphism of K -algebras (cid:26) H β , γ → H β + γ , y ⊗ z y ⋄ z .The image of the unit of H β , γ by this morphism is the idempotent 1 β , γ of H β + γ . This endows H β + γ β , γ (resp. 1 β , γ H β + γ ) with a structure of ( H β + γ , H β , γ ) -bimodule (resp. ( H β , γ , H β + γ ) -bimodule). It is free asa right (resp. left) H β , γ -module, with basis { τ ω , ω ∈ T } (resp. { τ ω , ω ∈ T ′ } ) with T (resp. T ′ ) as above.If M ∈ H β − mod and N ∈ H γ − mod, then MN is an H β + γ module given by MN = H β + γ β , γ ⊗ H β , γ ( M ⊗ N ) .This also defines the monoidal structure in the case of I infinite.Finally, we describe some useful induction and restriction functors. For i ∈ I , we denote by E i thefree H i -module of rank 1. The left i-induction functor is the functor defined by E i ( − ) : (cid:26) H → H M E i M It is exact. Since E i is free of rank 1, for M ∈ H β − mod we simply have E i M ≃ H β + i i , β ⊗ H β M .The left i-restriction functor F i is the right adjoint to the left i -induction functor. For M ∈ H β − mod, F i ( M ) = i , β − i M (viewed as an H β − i -module, via the right inclusion H β − i → i , β − i H β i , β − i ). When β − i / ∈ Q + , we understand 1 i , β − i =
0. The functor F i is also exact. There is a similarly defined pair ofadjoint functors called right i -induction and right i -restriction, but they will not be as useful to us.3.3.2. Divided powers.
Let n > i ∈ I . The algebra H ni is isomorphic to the affine nil Hecke algebra H n (up to a grading dilatation by d i ). The H ni -module E ni is free of rank 1. Its endomorphism ring (as an H ni -module without grading) is H opp ni , acting by right multiplication. Recall the primitive idempotent e n = x . . . x n − n τ ω [ n ] of H n . We define the n th divided power E ( n ) i by E ( n ) i = q − n ( n − ) i E ni e n .As a graded H ni -module, we have simply E ( n ) i ≃ q − n ( n − ) i P ni by the isomorphism (3.4). By the structuretheory of the affine nil Hecke algebra (3.5) there is an isomorphism of graded H ni -modules E ni ≃ [ n ] i ! E ( n ) i .3.4. Categorified quantum groups.
Consider the full subcategory H− proj of H consisting of projec-tive modules. This is an additive and graded category. The split Grothendieck group K ( H− proj ) is thequotient of the free abelian group on isomophism classes [ M ] of objects M ∈ H− proj by the relations [ M ] + [ N ] = [ M ⊕ N ] for all M , N ∈ H− proj. Since H− proj is graded, we can endow K ( H− proj ) witha structure of Z [ q , q − ] -module as follows: for p ∈ Z [ q , q − ] and M ∈ H− proj, we let p [ M ] = [ pM ] . Fur-thermore, there is a product on K ( H− proj ) induced from the moinoidal structure: if M , N ∈ H− proj,we define [ M ][ N ] = [ MN ] . Hence K ( H− proj ) has the structure of a Z [ q , q − ] -algebra. There is also anon-degenerate bilinear form on K ( H− proj ) defined as follows: for M , N ∈ H− proj let ([ M ] , [ N ]) = ∑ k ∈ Z rk ( Hom H ( M , q k N )) q k .Khovanov and Lauda proved that H− proj categorifies half of the integral quantum group A U + . Theorem 3.5 ([KL11, Theorem 8]) . Assume that K is a field. Then there is a unique isomorphism of Z [ q , q − ] -algebras A U + ∼ −→ K ( H− proj ) sending e ( n ) i to (cid:2) E ( n ) i (cid:3) for all i ∈ I and n > . Furthermore, this isomorphism is an isometry for the inner productson A U + and K ( H− proj ) . Some computational lemmas.
We now recall some elementary results in KLR algebras that willbe used in the proofs below. First, we recall some basic results about computations in affine nil Heckealgebras.
Lemma 3.6. (1) Let n > , let k ∈ {
1, . . . , n − } and let P ∈ P n . In H n we have τ k P − s k ( P ) τ k = ∂ k ( P ) . (2) Let n > , let k be a finite sequence of elements of {
1, . . . , n − } and let P ∈ P n . In H n we have τ k P τ ω [ n ] = ∂ k ( P ) τ ω [ n ] . Proof.
For statement (1), consider the two maps (cid:26) P n → H n P P τ k − s k ( P ) τ k , (cid:26) P n → H n P ∂ k ( P ) .They are both skew derivations, in the sense that they satisfy relation (3.2) with respect to the automor-phism s k . Furthermore, by relation (3) in Definition 3.1, they coincide on x , . . . , x n , which are algebragenerators of P n . Hence, they are equal and we have τ k P − s k ( P ) τ k = ∂ k ( P ) for all P ∈ P n .For statement (2), by induction on the length of the finite sequence k , it suffices to treat the case where k has length 1. If k ∈ {
1, . . . , n − } , then for all P ∈ P n we have τ k P τ ω [ n ] = s k ( P ) τ k τ ω [ n ] + ∂ k ( P ) τ ω [ n ] by statement (1) of the lemma. However, τ k τ ω [ n ] = l ( s k ω [ n ]) < l ( ω [ n ]) . Hence we have τ k P τ ω [ n ] = ∂ k ( P ) τ ω [ n ] , which completes the proof. (cid:3) Lemma 3.7.
Let m , n > , and let T be a complete set of reduced expressions of minimal length representatives ofleft cosets of S m × S n in S m + n . Let ω be the longest element in T. Then for all y ∈ H n and z ∈ H m we have ( y ⋄ z ) τ ω − τ ω ( z ⋄ y ) ∈ ∑ ω ∈ T \{ ω } τ ω H m , n . Proof.
There is a filtration of H n + m with 1 ν , x ℓ in degree 0 for all ν ∈ I n + m and ℓ ∈ {
1, . . . , n + m } , and τ k in degree 1 for all k ∈ {
1, . . . , n + m − } . In the associated graded gr ( H n + m ) , relations (6) and (8) ofDefinition 3.2 become τ k x ℓ − x s k ( ℓ ) τ k = τ k + τ k τ k + − τ k τ k + τ k = ( H n + m ) , we have ( y ⋄ z ) τ ω − τ ω ( z ⋄ y ) = y ∈ { x , . . . , x n , τ , . . . , τ n − , 1 ν , ν ∈ I n } and z ∈ { x , . . . , x m , τ , . . . , τ m − , 1 ν , ν ∈ I m } . However,the linear span of the elements y ⊗ z ∈ H n , m for which equation (3.7) holds in gr ( H n + m ) is a subalgebraof H n , m . So equation (3.7) holds in gr ( H n + m ) for all y ⊗ z ∈ H n , m . Hence for all y ⊗ z ∈ H n , m we have ( y ⋄ z ) τ ω − τ ω ( z ⋄ y ) ∈ ∑ l ( ω ) < l ( ω ) τ ω ( H n + m ) where ( H n + m ) is the piece of the filtration in degree 0 (that is, ( H n + m ) = ⊕ ν ∈ I n + m P n + m ν ). Usingdecomposition (3.6), we can write the right hand side as wanted. (cid:3) We now generalize relations (5) and (8) from Definition 3.2. For this, we need to introduce somenotation. For i ∈ I and ν = ( ν , . . . , ν n ) ∈ I n , we define Q i , ν ( u , v , . . . , v n ) = ∏ k n ν k = i Q i , ν k ( u , v k ) ∈ K [ u , v . . . , v n ] .These polynomials generalize the polynomials Q i , j ( u , v ) , and will appear many times in the followingsections. ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 13
Lemma 3.8.
Let n > , let i ∈ I and let ν ∈ I n be such that ν k = i for all k ∈ {
1, . . . , n } . In H n + the followingrelations hold:(1) τ [ n ↓ ] τ [ ↑ n ] i , ν = Q i , ν ( x n + , x , . . . , x n ) i , ν ,(2) τ [ ↑ n ] τ [ n ↓ ] ν , i = Q i , ν ( x , x , . . . , x n + ) ν , i .In H n + the following relation holds:(3) (cid:16) τ [ n + ↓ ] τ τ [ ↑ n + ] − τ [ ↑ n ] τ n + τ [ n ↓ ] (cid:17) i , ν , i = ∂ n + ( Q i , ν ( x n + , x , . . . , x n + )) i , ν , i .Proof. Relation (2) follows from relation (1) by applying the anti-automorphism rev n + , so it suffices toprove (1) and (3). The proof is by induction on n . The case n = ν and consider ν ′ = ( j , ν ) for some j = i . We have τ [ n + ↓ ] τ [ ↑ n + ] i , ν ′ = τ n + τ [ n ↓ ] τ [ ↑ n ] τ n + i , j , ν = τ n + Q i , ν ( x n + , x , . . . , x n ) τ n + i , j , ν where the second equality comes from the inductive assumption. Using relation (6) from Definition 3.2,we can commute the polynomial Q i , ν ( x n + , x , . . . , x n ) to the left of τ n + and we get τ [ n + ↓ ] τ [ ↑ n + ] i , ν ′ = Q i , ν ( x n + , x , . . . , x n ) τ n + i , j , ν = Q i , ν ( x n + , x , . . . , x n ) Q i , j ( x n + , x n + ) i , j , ν = Q i , ν ′ ( x n + , x , . . . , x n + ) i , ν ′ ,where the second equality is relation (5) of Definition 3.2. This proves relation (1), and hence also relation(2).Assume that relation (3) is proved for ν and consider ν ′ = ( j , ν ) for some j = i . We have τ [ n + ↓ ] τ τ [ ↑ n + ] i , ν ′ , i = τ n + τ [ n + ↓ ] τ τ [ ↑ n + ] τ n + i , ν ′ , i = τ n + (cid:16) τ [ ↑ n ] τ n + τ [ n ↓ ] + ∂ n + ( Q i , ν ( x n + , x , . . . , x n + )) (cid:17) τ n + i , ν ′ , i the second equality coming from the inductive assumption. Let us simplify each of the two terms. Forthe term τ n + τ [ ↑ n ] τ n + τ [ n ↓ ] τ n + i , ν ′ , i , both factors τ n + can be commuted across the factors τ [ ↑ n ] and τ [ n ↓ ] using relation (6) from Definition 3.2, which yields τ n + τ [ ↑ n ] τ n + τ [ n ↓ ] τ n + i , ν ′ , i = τ [ ↑ n ] τ n + τ n + τ n + τ [ n ↓ ] i , ν ′ , i = τ [ ↑ n ] (cid:0) τ n + τ n + τ n + + ∂ n + n + ( Q i , j ( x n + , x n + )) (cid:1) τ [ n ↓ ] i , ν ′ , i = τ [ ↑ n + ] τ n + τ [ n ↓ ] i , ν ′ , i + ∂ n + ( Q i , j ( x n + , x n + )) τ [ ↑ n ] τ [ n ↓ ] i , ν ′ , i = τ [ ↑ n + ] τ n + τ [ n ↓ ] i , ν ′ , i + ∂ n + ( Q i , j ( x n + , x n + )) Q i , ν ( x , x , . . . , x n + ) i , ν ′ , i The second equality comes from relation (8) in Definition 3.2. The third is deduced by commuting thepolynomial ∂ n + n + ( Q i , j ( x n + , x n + ) to the left of τ [ ↑ n ] using relation (6) in Definition 3.2. Finally forthe last equality we use relation (2) of the lemma which was proved above to compute τ [ ↑ n ] τ [ n ↓ ] .Now for the term τ n + ∂ n + ( Q i , ν ( x n + , x , . . . , x n + )) τ n + i , ν ′ , i , we can commute the polynomial tothe left of the factor τ n + by relation (6) in Definition 3.2 to obtain τ n + ∂ n + ( Q i , ν ( x n + , x , . . . , x n + )) τ n + i , ν ′ , i = s n + ( ∂ n + ( Q i , ν ( x n + , x , . . . , x n + ))) τ n + i , ν ′ , i = ∂ n + ( Q i , ν ( x n + , x , . . . , x n + )) Q i , j ( x n + , x n + ) i , ν ′ , i .For the last equality, we have used the conjugation of the Demazure operators to rewrite the polynomial,and relation (5) in Definition 3.2 to simplify τ n + . With the two terms written as such, we deduce (cid:16) τ [ n + ↓ ] τ τ [ ↑ n + ] − τ [ ↑ n + ] τ n + τ [ n ↓ ] (cid:17) i , ν ′ , i = ∂ n + ( Q i , j ( x n + , x n + )) Q i , ν ( x , x , . . . , x n + ) i , ν ′ , i + ∂ n + ( Q i , ν ( x n + , x , . . . , x n + )) Q i , j ( x n + , x n + ) i , ν ′ , i = ∂ n + (cid:0) Q i , j ( x n + , x n + ) Q i , ν ( x n + , x , . . . , x n + ) (cid:1) i , ν ′ , i = ∂ n + (cid:0) Q i , ν ′ ( x n + , x , . . . , x n + (cid:1) i , ν ′ , i where the second equality is the fact that ∂ n + is a skewed derivation (3.2). This concludes the proof of(3). (cid:3)
4. C
ATEGORIFICATION OF THE ADJOINT ACTION OF E i We now fix i ∈ I , and explain how to categorify the action of U i on U + [ i ] . We start by introducingthe category H [ i ] , our categorical analogue of U + [ i ] . Then, we define an endofunctor ad E i of H [ i ] ,which categorifies the action of ad e i on U + [ i ] . In the first two subsections, we prove some elementaryproperties of these objects. In particular, we prove that the affine nil Hecke algebra acts on powers ofad E i in Proposition 4.11. Another important point is the realization of ad E i as the cokernel of a naturaltransformation τ E i , ( − ) , which we construct in 4.2.3. This sets up the necessary preliminaries for Theorem4.18 which states that τ E i , ( − ) is injective, providing the key short exact sequence for the study of ad E i . InSubsection 4.3, we give various corollaries of this theorem, such as the exactness of the functor ad E i anda compatibility of ad E i with the monoidal structure. We give the proof of Theorem 4.18 in Subsection4.4.4.1. The category H [ i ] . Definitions.
Definition 4.1.
For β ∈ Q + , let H i β be the graded K -algebra defined by H i β = H β / ( β − i , i ) , where ( β − i , i ) is the two-sided ideal of H β generated by 1 β − i , i (when β − i / ∈ Q + , we put 1 β − i , i = n >
0, let H in be the graded K -algebra defined by H in = M β ∈ Q + | β | = n H i β .We define the category H [ i ] by H [ i ] = M β ∈ Q + H i β − mod.Equivalently, for n > H in as the quotient of H n by the two sided ideal generatedby the elements 1 ν i where ν ∈ I n − . If I is finite and n , m >
0, the right inclusion r n + mn : H n → H n + m satisfies r n + mn ( ν i ) = ∑ µ ∈ I m µν i ∈ (cid:0) ρ i (cid:1) ρ ∈ I n + m − .Hence we have an induced morphism r n + mn : H in → H in + m . The left inclusions however do not inducemaps at the level of the algebras H in .We view H [ i ] as a full subcategory of H . If M ∈ H β − mod, M is an object of H [ i ] if and only if1 β − i , i M = Example 4.2. (1) If j ∈ I \ { i } , we have H ii + j ≃ K [ x , x ] / Q i , j ( x , x ) . In particular, H ii + j is eitherzero (in the case where i · j =
0) or not finitely generated as a K -module. We will see below thatthis generalizes to every H i β for β ∈ Q + . This contrasts with cyclotomic KLR algebras, which arefinitely generated as K -modules.(2) If j ∈ I \ { i } , we have E j ∈ H [ i ] . Furthermore, if M ∈ H i β − mod, we have 1 j , β − j M ∈ H [ i ] since1 β − i , i j , β − j M = j , β − j β − i , i M = Serre subcategory if it is non-empty, fulland closed under subquotients, extensions and degree shifts.
Proposition 4.3.
The category H [ i ] is a Serre and monoidal subcategory of H .Proof. Let β ∈ Q + . The functor (cid:26) H β − mod β − i , i H β β − i , i − mod M β − i , i M is exact. Hence its kernel is a Serre subcategory of H β − mod. If follows that H [ i ] is a Serre subcategoryof H . ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 15
Let M ∈ H i β − mod and N ∈ H i γ − mod, where β , γ ∈ Q + have respective heights m , n . Then by (3.6) MN = M ω ∈ T τ ω β , γ ⊗ H β , γ ( M ⊗ N ) ,where T is a complete set of reduced decompositions for minimal length representatives of left cosets of S m × S n in S m + n . If ω ∈ T , then s − ω ( ) ∈ { n } . Hence we have1 β + γ − i , i τ ω β , γ = (cid:26) τ ω β − i , i , γ if s − ω ( ) = n , τ ω β , γ − i , i if s − ω ( ) = β − i , i , γ and 1 β , γ − i , i act by zero on M ⊗ N , we conclude that 1 β + γ − i , i MN =
0, and MN ∈ H [ i ] .Thus H [ i ] is a monoidal subcategory of H . (cid:3) The following proposition can be seen as the categorification of the description of U + [ i ] in terms ofthe bilinear form (see Proposition 2.2). It is a first justification of why H [ i ] is a categorical analogue of U + [ i ] . This will be justified further below when we give generators of H [ i ] as a Serre and monoidalcategory. Proposition 4.4.
Let M ∈ H . Then M ∈ H [ i ] if and only if for all N ∈ H , Hom H ( NE i , M ) = .Proof. Assume that M is an H β -module for β ∈ Q + . Then the condition 1 β − i , i M = H β − i ( N , 1 β − i , i M ) = N ∈ H β − i − mod. By adjunction between right i -induction and right i -restriction, this last condition is equivalent to Hom H β ( NE i , M ) = N ∈ H β − i − mod, and theproposition follows. (cid:3) The inclusion functor H [ i ] ֒ → H has a left adjoint π i : H → H [ i ] . Explicitly, for a module M ∈ H β − mod, we have π i ( M ) = M / ( β − i , i M ) where ( β − i , i M ) denotes the H β -submodule of M generated by 1 β − i , i M . Note that π i ( M ) = M if M ∈H [ i ] . We also have Hom H ( π i ( M ) , π i ( M )) ≃ Hom H ( M , π i ( M )) . So at the decategorified level, π i isan orthogonal projection. The following lemma gives compatibilities between the functor π i and the j -induction functors. It will be used repeatedly below. Lemma 4.5.
For all j ∈ I, there is a canonical isomorphism which is natural in M ∈ H [ i ] π i ( E j M ) ≃ (cid:26) E j π i ( M ) if j = i , π i ( E i π i ( M )) if j = i . Proof.
Assume that M ∈ H β − mod. There is a canonical quotient map π i ( E j M ) ։ π i ( E j π i ( M )) , whichwe prove is an isomorphism. Let N ∈ H [ i ] . We have isomorphismsHom H β + j ( π i ( E j M ) , N ) ≃ Hom H β + j ( E j M , N ) ≃ Hom H β ( M , 1 j , β N ) .the first one coming from the adjunction between π i and the inclusion of H [ i ] in H , and the secondone from the adjunction between left j -induction and left j -restriction. The same argument appliedto π i ( M ) yields a canonical isomorphism Hom H β + j ( π i ( E j π i ( M )) , N ) ≃ Hom H β ( π i ( M ) , 1 j , β N ) . Since1 j , β N ∈ H [ i ] , we have Hom H β ( π i ( M ) , 1 j , β N ) ≃ Hom H β ( M , 1 j , β N ) . Hence we have a diagramHom H β + j ( π i ( E j π i ( M )) , N ) ∼ / / can (cid:15) (cid:15) Hom H β ( M , 1 j , β N ) Hom H β + j ( π i ( E j M ) , N ) ∼ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ in which the isomorphisms are given by adjunctions as explained above. It is straightforward to checkthat this diagram commutes. Hence the vertical map in the diagram is an isomorphism. So the canonicalquotient map π i ( E j M ) ։ π i ( E j π i ( M )) is an isomorphism by Yoneda’s lemma. In the case where j = i ,we have E j ∈ H [ i ] , so E j π i ( M ) ∈ H [ i ] and π i ( E j π i ( M )) = E j π i ( M ) , completing the proof. (cid:3) Polynomial annihilators.
Let β ∈ Q + . Since H i β is a quotient of H β , it is naturally an H β -bimodule,and in particular a P β -bimodule. A key element to understanding the algebra H i β is to understand whichpolynomials of P β act by zero on H i β . The goal of this paragraph is to give explicitly such elements of P β .We start with some general results about affine nil Hecke algebras. For a K -algebra A , consider the K -algebra A ⊗ H n . It contains the K -algebra A [ x , . . . , x n ] = A ⊗ P n as a subalgebra. The action of H n on P n given by the isomorphism (3.3) induces actions of H n and A ⊗ H n on A [ x , . . . , x n ] . Lemma 4.6.
Let J be a two-sided ideal of A ⊗ H n . Then J ∩ A [ x , . . . , x n ] is an H n -submodule of A [ x , . . . , x n ] .Proof. The isomorphism of H n -modules (cid:26) A [ x , . . . , x n ] ∼ −→ (cid:0) A ⊗ H n (cid:1) e n , P Pe n , (4.1)induces an injection J ∩ A [ x , . . . , x n ] ֒ → Je n . Let us prove that this is also a surjection. Let h ∈ J . Wehave h = ∑ ω ∈ S n h ω τ ω for some h ω ∈ P n . Since J is an ideal, we have h ω τ ω [ n ] = h τ ω − ω [ n ] ∈ J .However, in H n we have 1 ∈ P n τ ω [ n ] P n , so h ω ∈ P n h ω τ ω [ n ] P n ⊆ J . In particular, he n = h e n is in theimage of J ∩ A [ x , . . . , x n ] . So the isomorphism of H n -modules (4.1) maps J ∩ A [ x , . . . , x n ] bijectivelyto Je n . Since Je n is an H n -submodule of H n e n , it follows that J ∩ A [ x , . . . , x n ] is an H n -submodule of A [ x , . . . , x n ] . (cid:3) Corollary 4.7. (1) Let p ∈ Z ( A )[ x , . . . , x n ] , and let J be the two-sided ideal of A ⊗ H n generated by p.Then as a left A [ x , . . . , x n ] -module we haveJ = M ω ∈ S n (( A ⊗ H n ) · p ) τ ω where ( A ⊗ H n ) · p denotes the (cid:0) A ⊗ H n (cid:1) -submodule of A [ x , . . . , x n ] generated by p.(2) Let M be a left (resp. right) (cid:0) A ⊗ H n (cid:1) -module, and let P ∈ A [ x , . . . , x n ] be such that PM = (resp.MP = ). Then ( H n · P ) M = (resp. M ( H n · P ) = ), where H n · P denotes the H n submodule ofA [ x , . . . , x n ] generated by P.Proof. For (1), we saw in the proof of Lemma 4.6 that J = M ω ∈ S n ( J ∩ A [ x , . . . , x n ]) τ ω .It also follows from Lemma 4.6 that J ∩ A [ x , . . . , x n ] contains ( A ⊗ H n ) · p . Conversely, given h , h inH n ,we see easily that the coefficient of h ph on A [ x , . . . , x n ] is in ( A ⊗ H n ) · p . Hence, J ∩ A [ x , . . . , x n ] =( A ⊗ H n ) · p , which completes the proof.For (2), let J be the annihilator of M in A ⊗ H n . Then J is a two-sided ideal of A ⊗ H n . Hence J ∩ A [ x , . . . , x n ] is an H n -submodule of A [ x , . . . , x n ] by Lemma 4.6. The result follows. (cid:3) Recall that for ν ∈ I n , we have defined a polynomial Q i , ν ( u , v , . . . , v n ) = ∏ k n ν k = i Q i , ν k ( u , v k ) . Proposition 4.8.
Let n > , let ν ∈ I n and let a ∈ {
1, . . . , n } be such that ν a = i. Then Q i , ν ( x a , x , . . . , x n ) ν acts by zero on H in .Proof. The proof is by induction on n . Assume that n =
1. If ν = i , 1 i is zero in H i , so the result holds.If ν = i , there is no a such that ν a = i , and the result holds too.Assume that the result is proved for n >
1. Let ν ∈ I n + . We write ν in the form ν = ( ν n + , ν ′ ) with ν ′ ∈ I n . We consider two cases depending on whether ν n + = i or ν n + = i .Case 1: if ν n + = i , then we have a < n + Q i , ν ( x a , x , . . . , x n + ) ν = Q i , ν n + ( x a , x n + ) Q i , ν ′ ( x a , x , . . . , x n ) ν = Q i , ν n + ( x a , x n + ) r n + n ( Q i , ν ′ ( x a , x , . . . , x n ) ν ′ ) ν .By induction, Q i , ν ′ ( x a , x , . . . , x n ) ν ′ is zero in H in . Hence Q i , ν ( x a , x , . . . , x n + ) ν is zero in H in + . ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 17
Case 2: If ν n + = i , then we consider three subcases: (i) a < n +
1, (ii) a = n + ν n = i , and (iii) a = n + ν n = i . • Sub-case (i): if a < n +
1, we have Q i , ν ( x a , x , . . . , x n + ) ν = Q i , ν ′ ( x a , x , . . . , x n ) ν = r n + n ( Q i , ν ′ ( x a , x , . . . , x n ) ν ′ ) ν .By induction, Q i , ν ′ ( x a , x , . . . , x n ) ν ′ is zero in H in so Q i , ν ( x a , x , . . . , x n + ) ν is zero in H in + . • Sub-case (ii): if a = n + ν n = i , we have Q i , ν ( x n + , x , . . . , x n + ) ν = Q i , ν ′ ( x n + , x , . . . , x n − , x n + ) ν = s n ( Q i , ν ′ ( x n , x , . . . , x n )) ν .By induction Q i , ν ′ ( x n , x , . . . , x n ) ν = r n + n ( Q i , ν ′ ( x n , x , . . . , x n ) ν ′ ) ν is zero in H in + . Since ν n + = ν n = i , there is a natural structure of left (cid:0) H ⊗ K [ x , . . . , x n − ] (cid:1) -module on 1 ν H in + , for whichthe polynomial Q i , ν ′ ( x n , x , . . . , x n ) acts by zero. By Corollary 4.7, s n ( Q i , ν ′ ( x n , x , . . . , x n )) alsoacts by zero, thus Q i , ν ( x n + , x , . . . , x n + ) ν is zero in H in + . • Sub-case (iii): if a = n + ν n = i , let ν ′′ = ( i , ν n − , . . . , ν ) . Then we have Q i , ν ( x n + , x , . . . , x n + ) ν = Q i , ν n ( x n + , x n ) Q i , ν ′′ ( x n + , x , . . . , x n − , x n + ) ν = τ n Q i , ν ′′ ( x n + , x , . . . , x n − , x n + ) ν = τ n Q i , ν ′′ ( x n , x , . . . , x n − , x n ) τ n ν = τ n r n + n ( Q i , ν ′′ ( x n , x , . . . , x n ) ν ′′ ) τ n ν By induction, Q i , ν ′′ ( x n , x , . . . , x n ) ν ′′ is zero in H in , hence Q i , ν ( x n + , x , . . . , x n + ) ν is zero in H in + . (cid:3) The polynomials Q i , ν ( u , v , . . . , v n ) have important symmetry properties. For β ∈ Q + of height n , let Q i , β ( u , x , . . . , x n ) = ∑ ν ∈ I β Q i , ν ( u , x , . . . , x n ) ν ∈ P β [ u ] . Lemma 4.9.
Let β ∈ Q + . As an element of P β [ u ] , the polynomial Q i , β has coefficients in the center of H β .Proof. The symmetric group S n acts on P β [ u ] by acting on the coefficients. By Proposition 3.4, Z ( H β ) = P S n β so it suffices to check that that Q i , β ( u , x , . . . , x n ) is invariant under the action of S n . Let ω ∈ S n .We have ω ( Q i , β ( u , x , . . . , x n )) = ∑ ν ∈ I β (cid:18) ∏ k n ν k = i ω ( Q i , ν k ( u , x k )) (cid:19) ω ( ν )= ∑ ν ∈ I β (cid:18) ∏ k n ν k = i Q i , ν k ( u , x ω ( k ) ) (cid:19) ω ( ν ) .Doing the re-indexation ν ′ = ω ( ν ) in this last sum yields ω ( Q i , β ( u , x , . . . , x n )) = ∑ ν ′ ∈ I β (cid:18) ∏ k n ν ′ ω ( k ) = i Q i , ν ′ ω ( k ) ( u , x ω ( k ) ) (cid:19) ν ′ = ∑ ν ′ ∈ I β (cid:18) ∏ ℓ n ν ′ ℓ = i Q i , ν ′ ℓ ( u , x ℓ ) (cid:19) ν ′ where the second equality follows from doing the re-indexation ℓ = ω ( k ) in the product. Hence Q i , β ( u , x , . . . , x n ) is invariant under the action of S n and the result is proved. (cid:3) As another application of Corollary 4.7, we now prove that the algebras H i β are not finitely generatedas K -modules in general. Proposition 4.10.
Let β ∈ Q + \ { } be such that s i ( β ) ∈ Q + . Then H i β is not finitely generated as a K-module.Proof. Write β in the form β = γ + ni with γ ∈ L j = i Z > j of height r and n >
0. Since s i ( β ) ∈ Q + ,we have γ = n h i ∨ , γ i . We prove that under these conditions, 1 ni , γ H ini + γ ni , γ is not finitelygenerated as a K -module.Let T be a set of reduced decompositions for minimal length representatives of left cosets of S r × S n in S n + r . The two-sided ideal 1 ni , γ H ni , γ ( n − ) i + γ , i H ni , γ ni , γ of 1 ni , γ H ni , γ ni , γ is generated by the elements τ ω − τ ω ni , γ for ω ∈ T such that s ω ( r + ) =
1, where ω − denotes the reverse sequence of ω . Such ω can be written in the form ω = ω ′ [ ↑ r ] . Hence τ ω − τ ω ni , γ is an element of P ni + γ ni , γ , multipleof τ [ r ↓ ] τ [ ↑ r ] ni , γ = Q i , γ ( x r + , x , . . . , x r ) (Lemma 3.8). So 1 ni , γ H ni , γ ( n − ) i + γ , i H ni , γ ni , γ is generated by Q i , γ ( x r + , x , . . . , x r ) as a two-sided ideal of 1 ni , γ H ni , γ ni , γ .Furthermore, since γ has no component on i , we have H ni ⊗ H γ ≃ ni , γ H ni , γ ni , γ , the isomorphismbeing given by the ⋄ operation. Hence, if we denote by J the two-sided ideal of H ni ⊗ H γ generated by Q i , γ ( x r + , x , . . . , x r ) , we have 1 ni , γ H ini + γ ni , γ ≃ ( H ni ⊗ H γ ) / J . By Proposition 4.9, Q i , γ ( x r + , x , . . . , x r ) ∈ Z ( H γ )[ x r + ] . Let ˜ J be the the ( H ni ⊗ H γ ) -submodule of H γ [ x r + , . . . , x n ] generated by Q i , γ ( x r + , x . . . , x r ) .By Corollary 4.7, we have1 ni , γ H ini + γ ni , γ ≃ M ω ∈ S n ( H γ [ x r + , . . . , x n ] / ˜ J )( τ ω ⋄ γ ) Since Q i , γ ( x r + , x , . . . , x r ) does not involve the variables x r + , . . . , x n + r (and in particular, is symmetricin them), ˜ J is generated as a sub- H γ [ x r + , . . . , x n ] -module by the ∂ [ k ↓ r + ] ( Q i , γ ( x r + , x , . . . , x r )) for k ∈{ r +
1, . . . , r + n − } . However, we have ∂ [ k ↓ r + ] ( Q i , γ ( x r + , x , . . . , x r )) ∈ tx −h i ∨ , γ i− k + rk + + ∑ ℓ< −h i ∨ , γ i− k + r x ℓ k + K [ x , . . . , x k ] with t an invertible element of K × . Since n − h i ∨ , γ i , these polynomials have in particular positivedegree. Thus H γ [ x r + , . . . , x n ] / ˜ J is free of positive rank as an H γ -module. In particular, it is not finitelygenerated over K . (cid:3) Adjoint action of E i . Definition.
Consider the functor ad E i : H [ i ] → H [ i ] defined by:ad E i ( M ) = E i M / ( β , i E i M ) = π i ( E i M ) .for M ∈ H i β − mod. So ad E i is the composition of the left i -induction functor with the functor π i . Left i -induction is an exact endofunctor of H , and π i is right exact since it is a left adjoint. Hence, the functorad E i is right exact. We will prove below it is actually exact. Proposition 4.11. (1) For n > , there is a canonical isomorphism which is natural in M ∈ H [ i ] ad nE i ( M ) ≃ π i ( E ni M ) . (2) For all n > , there is an algebra morphismH op ni → End ( ad nE i ) . Hence the affine nil Hecke algebra H ni acts on ad nE i .Proof. We construct the isomorphism in (1) by induction on n . For the case n =
0, we have π i ( M ) = M for M ∈ H [ i ] , and the canonical isomorphism is just the identity. Assume that we have constructed thenatural isomorphism ad nE i ( M ) ∼ −→ π i ( E ni M ) for some n >
0. Then we have an isomorphismad n + E i ( M ) ∼ −→ ad E i ( π i ( E ni M )) = π i ( E i π i ( E ni M )) .By Lemma 4.5, there is a canonical isomorphism π i ( E i π i ( E ni M )) ∼ −→ π i ( E n + i M ) , which completes theconstruction of (1).For (2), notice that we have an action of H op ni on E ni by right multiplication, so we get an actionof H op ni on the functor M E ni M . By vertical composition, we get an action of H op ni on the functor M π i ( E ni M ) , which is canonically isomorphic to ad nE i by (1), whence the result. (cid:3) ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 19
For the rest of this paper, we fix the action of H op ni on ad nE i to be the one constructed in the proof ofProposition 4.11 via the canonical isomorphism ad nE i ( M ) ≃ π i ( E ni M ) . In particular, one can define the divided power ad ( n ) E i . Recall that the element e n = x . . . x n − n τ ω [ n ] is a primitive idempotent of H ni . Thenthe functor ad ( n ) E i is defined by ad ( n ) E i = q − n ( n − ) i e n (cid:0) ad nE i (cid:1) .From the isomorphism (3.5), we have an isomorphismad nE i ≃ [ n ] i !ad ( n ) E i .Using the definition of our chosen action of H ni on ad nE i , we see that for all M ∈ H [ i ] there is an isomor-phism ad ( n ) E i ( M ) ≃ π i (cid:0) E ( n ) i M (cid:1) which is natural in M .4.2.2. Induction.
Let us now give another description of the functor ad E i , as an induction functor. For β ∈ Q + , the right inclusion induces a (non-unital) morphism H i β → H i β + i . This endows H i β + i i , β with astructure of ( H i β + i , H i β ) -bimodule, and we get an induction functorind H i β + i H i β : ( H i β − mod → H i β + i − mod, M H i β + i i , β ⊗ H i β M . Proposition 4.12.
There is an isomorphism ad E i ≃ M β ∈ Q + ind H i β + i H i β . Proof.
Let β ∈ Q + . The canonical quotient morphism H β + α i ։ H i β + α i endows H i β + α i with a structureof ( H i β + α i , H β + α i ) -bimodule, and we have a natural isomorphism π i ( E i M ) ≃ H i β + i ⊗ H β + i ( E i M ) for M ∈ H i β − mod. Now recall that E i M = H β + i i , β ⊗ H i , β ( E i ⊗ M ) ≃ H β + i i , β ⊗ H β M ,the second isomorphism coming from the fact that E i is a free module of rank 1 over H i . Hence we havenatural isomorphisms ad E i ( M ) = π i ( E i M ) ≃ H i β + i ⊗ H β + i ( E i M ) ≃ H i β + i ⊗ H β + i H β + i i , β ⊗ H β M ≃ H i β + i i , β ⊗ H β M .In general, if A is a K -algebra, J a 2-sided ideal of A , L a left A -module, R a right A -module, such that JL = RJ =
0, we have an obvious isomorphism R ⊗ A L ≃ R ⊗ A / J L . Here, we get H i β + i β , i ⊗ H β M ≃ H i β + i β , i ⊗ H i β M ,which concludes the proof. (cid:3) The morphism τ E i , M . We now describe of the functor ad E i as the cokernel of a natural transfor-mation. For β ∈ Q + of height r and M ∈ H i β − mod, we define a morphism τ E i , M : ME i → E i M asfollows: τ E i , M : (cid:26) ME i → E i M , h β , i ⊗ H β m h τ [ ↑ r ] i , β ⊗ H β m . Proposition 4.13.
The map τ E i , M is a well-defined morphism of H β + i -modules of degree − i · β . Proof.
By adjunction between left i -induction and left i -restriction, it suffices to check that the map ( M → β , i H β + i i , β ⊗ H β M , m τ [ ↑ r ] i , β ⊗ H β m ,is a morphism of H β -modules. To check this, we need to prove that for all y ∈ H β we have (cid:16) ( y ⋄ i ) τ [ ↑ r ] − τ [ ↑ r ] ( i ⋄ y ) (cid:17) i , β ⊗ H β M = Z = (cid:16) ( y ⋄ i ) τ [ ↑ r ] − τ [ ↑ r ] ( i ⋄ y ) (cid:17) i , β . By Lemma 3.7, the element Z can be written in the form Z = r + ∑ k = τ [ k ↑ r ] z k for some z k ∈ H i , β . However, Z = β , i Z i , β and 1 β , i τ [ k ↑ r ] = τ [ k ↑ r ] β , i when k >
2. So Z = β , i Z i , β = n + ∑ k = τ [ k ↑ r ] β , i z k i , β = n + ∑ k = τ [ k ↑ r ] i , β − i , i z k i , β .Since 1 β − i , i M =
0, we deduce that Z ⊗ H β M =
0. Hence τ E i , M is well-defined. The statement about thedegree follows from the definition of the grading on the KLR algebras. (cid:3) Our morphism τ E i , M is similar to the morphism P in [KK12, Theorem 4.7]. Similar morphisms alsoappear in [BKM14], for instance in Lemma 4.9.Given the definition, it is clear that the morphism τ E i , M is natural in M , i.e. for every arrow f : M → M ′ in H [ i ] , we have a commutative diagram ME if E i (cid:15) (cid:15) τ Ei , M / / E i M E i f (cid:15) (cid:15) M ′ E i τ Ei , M ′ / / E i M ′ Hence, τ E i , ( − ) : ( − ) E i → E i ( − ) is a natural transformation of functors H [ i ] → H .We now relate τ E i , ( − ) and ad E i . Proposition 4.14.
For β ∈ Q + of height r and M ∈ H i β − mod , we have ad E i ( M ) = coker ( τ E i , M ) . Hence, wehave an exact sequence which is natural in Mq h i ∨ , β i i ME i τ Ei , M −−−→ E i M → ad E i ( M ) → Proof.
We show that im ( τ E i , M ) = H β + i β , i ( E i M ) . Let h ∈ H β + i and m ∈ M . We have τ E i , M ( h β , i ⊗ H β m ) = h τ [ ↑ r ] i , β ⊗ H β m ∈ H β + i β , i ( E i M ) .So im ( τ E i , M ) ⊆ H β + i i , β ( E i M ) . Conversely, by (3.6) we have E i M = r + M k = τ [ k ↑ r ] i , β ⊗ H i , β ( E i ⊗ M ) .Note that 1 β , i τ [ k ↑ r ] i , β = (cid:26) τ [ k ↑ r ] i , β − i , i if k > τ [ ↑ r ] β , i if k = β − i , i M =
0, we deduce that1 β , i ( ME i ) = τ [ ↑ r ] i , β ⊗ H i , β ( E i ⊗ M ) ⊆ im ( τ E i , M ) .Hence H β + i β , i ( E i M ) ⊆ im ( τ E i , M ) . So we have proved that im ( τ E i , M ) = H β + i β , i ( E i M ) , from which wededuce that ad E i ( M ) = coker ( τ E i , M ) . (cid:3) ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 21
A key property of τ E i , ( − ) is that it is compatible with the monoidal structure of H [ i ] , in the followingsense. Proposition 4.15.
For all M , N ∈ H [ i ] , we have τ E i , MN = τ E i , M N ◦ M τ E i , N .Proof. Assume that M ∈ H i β − mod with β of height r , and that N ∈ H i γ − mod with γ of height s . Let y = h β , γ ⊗ H β , γ ( a ⊗ b ) ∈ MN ,where h ∈ H β + γ , a ∈ M and b ∈ N , and z = β , γ , i ⊗ H β , γ y ∈ MNE i .Such elements z generate MNE i as an H β + γ + i -module. So it suffices to prove that τ E i , MN ( z ) = τ E i , M N ( M τ E i , N ( z )) .On the one hand, we have τ E i , MN ( z ) = τ [ ↑ r + s ] i , β + γ ⊗ H β + γ y = τ [ ↑ r + s ] ( i ⋄ h ) i , β , γ ⊗ H β , γ ( a ⊗ b ) .On the other hand, we have M τ E i , N ( z ) = M τ E i , N (cid:16) ( h ⋄ i ) β , γ , i ⊗ H β , γ ( a ⊗ b ) (cid:17) = ( h ⋄ i ) τ [ ↑ s ] β , i , γ ⊗ H β , γ ( a ⊗ b ) .So τ E i , M N ( M τ E i , N ( z )) = ( h ⋄ i ) τ [ ↑ s ] τ [ s + ↑ r + s ] i , β , γ ⊗ H β , γ ( a ⊗ b )= ( h ⋄ i ) τ [ ↑ r + s ] i , β , γ ⊗ H β , γ ( a ⊗ b ) Hence to get the result, it suffices to prove that (cid:16) ( h ⋄ i ) τ [ ↑ r + s ] − τ [ ↑ r + s ] ( i ⋄ h ) (cid:17) i , β , γ ⊗ H β , γ ( M ⊗ N ) = Z = (cid:16) ( h ⋄ i ) τ [ ↑ r + s ] − τ [ ↑ r + s ] ( i ⋄ h ) (cid:17) i , β , γ . By Lemma 3.7, we have Z ∈ r + s + ∑ k = τ [ k ↑ r + s ] H i , β + γ i , β , γ .Let S be a complete set of reduced expressions for minimal length representatives of left cosets of S r × S s in S r + s , then by (3.6) we have H β + γ β , γ = M ω ∈ S τ ω H β , γ .Hence Z can be written in the form Z = ∑ < k r + s + ω ∈ S τ [ k ↑ r + s ] ( i ⋄ τ ω ) h k , ω for some h k , ω ∈ H i , β , γ . However Z = β + γ , i Z , so Z = β + γ , i Z = ∑ < k r + s + ω ∈ S β + γ , i τ [ k ↑ r + s ] ( i ⋄ τ ω ) h k , ω = ∑ < k r + s + ω ∈ S τ [ k ↑ r + s ] β + γ , i ( i ⋄ τ ω ) h k , ω For ω ∈ S , we have s − ω ( ) ∈ { s + } . It follows that Z = ∑ < k r + s + ω ∈ S , s − ω ( )= τ [ k ↑ r + s ] ( i ⋄ τ ω ) h k , ω i , β , γ − i , i + ∑ < k r + s + ω ∈ S , s − ω ( )= s + τ [ k ↑ r + s ] ( i ⋄ τ ω ) h k , ω i , β − i , i , γ .Since both 1 β − i , i , γ and 1 β , γ − i , i act by zero on M ⊗ N , we deduce Z ⊗ H β , γ ( M ⊗ N ) =
0, which completesthe proof. (cid:3)
Example: adjoint action on Chevalley generators.
In this section, we describe the modules ad ( n ) E i ( E j ) for n > j ∈ I \ { i } . Proposition 4.16.
As left graded P n + -modules, we haveH j + ni j +( n − ) i , i ( E ni E j ) = M ω ∈ S n J n + ( τ ω ⋄ j ) ni , j ! M M ω ∈ S n k n P n + τ [ k ↓ ] ( τ ω ⋄ j ) ni , j ! , where J n + is the ideal of P n + generated by the ∂ [ k ↓ ] ( Q i , j ( x , x )) for k ∈ {
1, . . . , n } .Proof. Call L the left hand-side of the equality, and L ′ the right-hand side. Let us start by proving that L ′ is a H j + ni -submodule of H j + ni ni , j . It is clear that L ′ is stable by the left action of P n + . Furthermore,note that J n + is the (cid:0) H n ⊗ K [ x ] (cid:1) -submodule of P n + generated by Q i , j ( x , x ) . Using this and the factthat τ ni , j = Q i , j ( x , x ) , we see easily that L ′ is also stable by left multiplication by τ , . . . , τ n − . Hence L ′ is a sub- H j + ni -module.By the same argument as in the proof of Proposition 4.14, L is the H j + ni -submodule of H j + ni ni , j generated by the τ [ ↑ k ] ni , j for k ∈ {
1, . . . , n } . Since τ [ ↑ k ] ni , j ∈ L ′ for k ∈ {
1, . . . , n } , we have L ⊆ L ′ .Conversely, it is clear that M ω ∈ S n k n P n + τ [ k ↓ ] ( τ ω ⋄ j ) ni , j ⊆ L .Furthermore, Q i , j ( x , x ) ni , j = τ ni , j ∈ L . By Corollary 4.7, we deduce that J n + ni , j ⊆ L , and it followsthat L ′ ⊆ L . (cid:3) Corollary 4.17.
As left graded P n + -modules, we have ad nE i ( E j ) = M ω ∈ S n ( P n + / J n + ) ( τ ω ⋄ j ) ni , j ,ad ( n ) E i ( E j ) = q − n ( n − ) i ( P n + / J n + ) ( τ ω [ n ] ⋄ j ) ni , j .In particular, we have a simple formula for the graded ranks of ad nE i ( E j ) and ad ( n ) E i ( E j ) . They aregiven by grk ( ad nE i ( E j )) = ( − q i ) n ( − q j ) n − ∏ k = (cid:18) − q ( − c i , j − k ) i (cid:19)! ∑ ω ∈ S n q − l ( ω ) i ! ,grk ( ad ( n ) E i ( E j )) = q n ( n − ) i ( − q i ) n ( − q j ) n − ∏ k = (cid:18) − q ( − c i , j − k ) i (cid:19)! .A consequence of these formulas is the following vanishing criterion:ad nE i ( E j ) = ⇔ n > − c i , j . (4.2)We extend that result below.4.3. Main theorem and its consequences.
The first main result is the following.
Theorem 4.18.
For all β ∈ Q + and M ∈ H i β − mod , the morphism τ E i , M is injective. Hence we have a shortexact sequence which is natural in M → q h i ∨ , β i i ME i τ Ei , M −−−→ E i M → ad E i ( M ) → e i ( y ) = e i y − q h i ∨ , β i i ye i for y ∈ U + of degree β ∈ Q + . It shows that the functor ad E i is indeed a catgeorical lift of the operator ad e i . Theproof of Theorem 4.18 is done in Subsection 4.4. For the rest of this subsection, we give some corollariesof the theorem. ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 23
Exactness of ad E i . Theorem 4.18 is the analogue of [KK12, Theorem 4.7] for cyclotomic KLR alge-bras. It can be used formally in the same way to prove that the functor ad E i is exact. Corollary 4.19.
Let β ∈ Q + . The right H i β -module H i β + i i , β is projective.Proof. As in [KK12], we use the following lemma.
Lemma ([KK12, Lemma 4.18]) . Let A be a K-algebra, and N be an A [ t ] -module such that:(1) the projective dimension of N as an A [ t ] -module is at most 1,(2) there exists p ( t ) ∈ Z ( A )[ t ] with invertible leading coefficient such that p ( t ) N = .Then N is projective as an A-module. We use this lemma with A = H i β and N = H i β + i i , β . By Theorem 4.18 applied to M = H i β , we have ashort exact sequence of ( H β + i , H i ⊗ H i β ) -bimodules:0 → H β + i β , i ⊗ H β H i β → H β + i i , β ⊗ H β H i β → H i β + i i , β → (cid:0) H i ⊗ H i β (cid:1) -module structure comes from the naturality. Since H i ⊗ H i β ≃ H i β [ t ] , we can view this sequence as an exact sequence of right H i β [ t ] -modules.Since H β + i β , i is free as a right H β , i -module, H β + α i β , i ⊗ H β H i β is free as a right H i β [ t ] -module. Simi-larly, H β + i i , β ⊗ H β H i β is free as a right H i β [ t ] -module. Hence, H i β + i i , β has projective dimension at most1 as a right H i β [ t ] -module. Let p ( x n + ) = Q i , β ( x n + , x , . . . , x n ) i , β = ∑ ν ∈ I β ∏ k n ν k = i Q i , ν k ( x n + , x k ) ! i , ν ∈ P β + i .When considered as an element in P β [ x n + ] , p ( x n + ) has coefficients in Z ( H β ) by Proposition 4.9. Fur-thermore, H i β + i p ( x n + ) = (cid:3) Corollary 4.20.
The functor ad E i is exact.Proof. This follows immediately from Proposition 4.12 and Corollary 4.19. (cid:3)
Compatibility with products.
We now state and prove the compatibility between the monoidal struc-ture of H [ i ] and the action of the functor ad E i . We can think of the following result as expressing the factthat ad E i is a “categorical derivation”. This categorifies the compatibility of the adjoint action of a Hopfalgebra with its product. Corollary 4.21.
For M ∈ H i β − mod and N ∈ H i γ − mod with β , γ ∈ Q + , there is a short exact sequence whichis natural in M , N → q h i ∨ , β i i M ad E i ( N ) → ad E i ( MN ) → ad E i ( M ) N → Proof.
This a consequence of the following elementary statement: given arrows f = f ◦ f in an abeliancategory, we have an exact sequencecoker ( f ) → coker ( f ) → coker ( f ) → f , the second map is the canonical quotient map. Furthermore, if f isinjective, the map coker ( f ) → coker ( f ) is injective as well. By Proposition 4.15 and Theorem 4.18, thestatement applies to f = τ E i , MN , f = M τ E i , N and f = τ E i , M N , yielding the result. (cid:3) Remark . In general, the short exact sequence in Corollary 4.21 does not split. Let us give an example.Let j ∈ I \ { i } and c = − c i , j . Assume that c >
0. Let S j be the simple H j -module equal to K as a graded K -module, with x acting by 0. We havead E i ( S j ) = K [ x ] / x c ij ,where x acts by multiplication and x , τ and τ act by zero. In particular, the element x c ij of H i , j actsby zero. From this we deduce that the central element χ = x c jji + x c jij + x c ijj ∈ H i + j acts by zero on both S j ad E i ( S j ) and ad E i ( S j ) S j . We can also compute ad E i ( S j ) , where S j = S j S j , andfind that it decomposes as a P j + i -module asad E i ( S j ) = i ,2 j (cid:16)(cid:16) K [ x ] / x c (cid:17) ⊗ S j (cid:17) ⊕ (cid:16) τ i ,2 j ⊗ H i ,2 j (cid:16) ( K [ x ] / x c ) ⊗ S j (cid:17)(cid:17) .Since c > c > c , so χ does not act by zero on the first summand. Hence the short exactsequence 0 → q − ci S j ad E i ( S j ) → ad E i ( S j ) → ad E i ( S j ) S j → nE i ( MN ) . To describe it, let us set up some notation. Given an integer k >
0, and its binary expansion k = ∑ ℓ > k ℓ ℓ where k ℓ ∈ {
0, 1 } are almost all 0, we define its binary weight ζ ( k ) as the number of k ℓ that are equal to1. There are inductive relations for the binary weight function: for all k > (cid:26) ζ ( k ) = ζ ( k ) , ζ ( k + ) = ζ ( k ) +
1. (4.3)We also define a function σ by σ ( k ) = ∑ ℓ > ℓ k ℓ − ζ ( k )( ζ ( k ) − ) σ : for all k > (cid:26) σ ( k ) = σ ( k ) + ζ ( k ) , σ ( k + ) = σ ( k ) . (4.4)With these notations we can now state and prove the result. Proposition 4.23.
Let M ∈ H i β − mod , N ∈ H i γ − mod with β , γ ∈ Q + and n > . Then there is a filtration of ad nE i ( MN ) = V − ⊆ V ⊆ · · · ⊆ V n − = ad nE i ( MN ) , with quotients given by V k / V k − ≃ q ( n − ζ ( k )) h i ∨ , β i + σ ( k ) i ad ζ ( k ) E i ( M ) ad n − ζ ( k ) E i ( N ) . Proof.
We proceed inductively on n . The case n = n we have a filtrationof ad nE i ( MN ) = V − ⊆ V ⊆ · · · ⊆ V n − = ad nE i ( MN ) ,with quotients given by V k / V k − ≃ q ( n − ζ ( k )) h i ∨ , β i + σ ( k ) i ad ζ ( k ) E i ( M ) ad n − ζ ( k ) E i ( N ) .Since ad E i is exact by Corollary 4.20, we can apply ad E i to the filtration above to get a filtration ofad n + E i ( MN ) = ad E i ( V − ) ⊆ ad E i ( V ) ⊆ · · · ⊆ ad E i ( V n − ) = ad n + E i ( MN ) ,with quotients given byad E i ( V k ) /ad E i ( V k − ) ∼ → p k q ( n − ζ ( k )) h i ∨ , β i + σ ( k ) i ad E i (cid:16) ad ζ ( k ) E i ( M ) ad n − ζ ( k ) E i ( N ) (cid:17) .By Corollary 4.21, there are short exact sequences0 → q h i ∨ , β i + ζ ( k ) i ad ζ ( k ) E i ( M ) ad n + − ζ ( k ) E i ( N ) f k −→ ad E i (cid:16) ad ζ ( k ) E i ( M ) ad n − ζ ( k ) E i ( N ) (cid:17) → ad ζ ( k )+ E i ( M ) ad ζ ( k ) E i ( N ) → n + E i ( MN ) . More precisely, for k ∈ {−
1, . . . 2 n + − } let W k = ( ad E i ( V ( k − ) /2 ) if k is odd, p − k /2 ( im ( f k /2 )) if k is even. ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 25
We get a filtration of ad n + E i ( MN ) of the form0 = W − ⊆ W ⊆ · · · ⊆ W n + − = ad n + E i ( XY ) ,with quotients W k / W k − ≃ q ( n − ζ (( k − ) /2 )) h i ∨ , β i + σ (( k − ) /2 ) i ad ζ (( k − ) /2 )+ E i ( M ) ad n − ζ (( k − ) /2 ) E i ( N ) if k is odd, q ( n − ζ ( k /2 )) h i ∨ , β i + σ ( k /2 )+ h i ∨ , β i + ζ ( k /2 ) i ad ζ ( k /2 ) E i ( M ) ad n + − ζ ( k /2 ) E i ( N ) if k is even.The conclusion follows from the recursive relations (4.3) for ζ and (4.4) for σ . (cid:3) A vanishing criterion.
We determine for which β ∈ Q + the algebra H i β is zero, extending the resultsof Subsection 4.2.4. Let us introduce some notation first. For β ∈ Q + and ν ∈ I β , we let E ν = E ν | β | . . . E ν = H β ν .Note that H i β ν = π i ( E ν ) . Proposition 4.24.
Let β ∈ Q + . The algebra H i β is zero if and only if s i ( β ) / ∈ Q + .Proof. We proved in Proposition 4.10 that if s i ( β ) ∈ Q + , then H i β is not finitely generated as a K -module(in particular non zero).Conversely, assume s i ( β ) / ∈ Q + . Write β in the form β = γ + ni , with γ ∈ L j = i Z > j and n > s i ( β ) / ∈ Q + , we have n > − h i ∨ , γ i . Consider an idempotent of H i β of the form 1 n i , ν ,..., n k i , ν k with n + . . . + n k = n , and ν . . . ν k ∈ I γ . Then H i β n i , ν ,..., n k i , ν k = π i ( E n i E ν . . . E n k i E ν k ) ≃ π i (cid:0) E n i π i ( E ν . . . E n k i E ν k ) (cid:1) by Proposition 4.5. Furthermore by Proposition 4.11, we have isomorphisms π i (cid:0) E n i π i ( E ν . . . E n k i E ν k ) (cid:1) ≃ ad n E i (cid:0) π i (cid:0) E ν . . . E n k i E ν k (cid:1)(cid:1) ≃ ad n E i (cid:0) E ν π i (cid:0) E n i E ν . . . E n k i E ν k (cid:1)(cid:1) where the second isomorphism is obtained by applying Proposition 4.5 to take the E ν outside of the π i .Hence there is an isomorphism H i β n i , ν ,..., n k i , ν k ≃ ad n E i (cid:0) E ν π i (cid:0) E n i E ν . . . E n k i E ν k (cid:1)(cid:1) .This procedure can be repeated inductively and we obtain an isomorphism H i β n i , ν ,..., n k i , ν k ≃ ad n E i (cid:16) E ν ad n E i (cid:16) E ν . . . ad n k E i ( E ν k ) . . . (cid:17)(cid:17) .Using Proposition 4.23 repeatedly, we see that H i β n i , ν ,..., n k i , ν k has a filtration with quotients being shiftsof modules of the form ad m r E i ( E j r ) . . . ad m E i ( E j ) for some m , . . . , m r such that m + . . . + m r = n and ( j r , . . . , j ) ∈ I γ . We now prove that such quotients are zero. Indeed, for such a quotient, since m + . . . + m r = n > − (cid:10) i ∨ , γ (cid:11) = − (cid:10) i ∨ , j (cid:11) − . . . − − (cid:10) i ∨ , j r (cid:11) there exists an index ℓ such that m ℓ > − h i ∨ , j ℓ i . But then by the vanishing criterion (4.2), we havead m ℓ E i ( E j ℓ ) =
0, and ad m r E i ( E j r ) . . . ad m E i ( E j ) =
0. Hence all the quotients in the filtration of H i β n i , ν ,..., n k i , ν k are zero. So H i β n i , ν ,..., n k i , ν k =
0. Since 1 β is the sum of all idempotents of the form 1 n i , ν ,..., n k i , ν k , weconclude that H i β = (cid:3) In particular, we deduce from Proposition 4.24 that H i β + ni is zero for n large enough. An immediateconsequence is the following corollary. Corollary 4.25.
The functor ad E i is locally nilpotent. Propositions 4.10 and 4.24 imply in particular that H i β is either zero, or not finitely generated as a K -module. This contrasts with the case of cyclotomic KLR algebras, which are always finitely generatedas K -modules.4.4. Proof of the main theorem.
Preliminaries.
We start by proving weaker versions of Corollary 4.20 and Proposition 4.23 that willbe useful in the proof.
Proposition 4.26.
Let M ∈ H [ i ] together with a filtration = M − ⊆ M ⊆ · · · ⊆ M r = M . For all k ∈ {
0, . . . , r } , put V k = M k / M k − and assume that τ E i , V k is injective. Then τ E i , M is injective, and wehave a filtration of ad E i ( M ) = ad E i ( M − ) ⊆ ad E i ( M ) ⊆ · · · ⊆ ad E i ( M r ) = ad E i ( M ) , such that for all k ∈ {
0, . . . , r } , ad E i ( M k ) /ad E i ( M k − ) ≃ ad E i ( V k ) .Proof. We proceed by induction on r . The result is clear if r =
0. In general, the short exact sequence0 → M r − → M → M r → / / M r − E i / / τ Ei , Mr − (cid:15) (cid:15) ME i / / τ Ei , M (cid:15) (cid:15) V r E i / / τ Ei , Vr (cid:15) (cid:15) / / E i M r − / / E i M / / E i V r / / τ E i , V r is injective, and by induction τ E i , M r − is injective as well. Hence τ E i , M is injective.Furthermore, we have a commutative diagram0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / M r − E i / / τ Ei , Mr − (cid:15) (cid:15) ME i / / τ Ei , M (cid:15) (cid:15) V r E i / / τ Ei , Vr (cid:15) (cid:15) / / E i M r − / / (cid:15) (cid:15) E i M / / (cid:15) (cid:15) E i V r / / (cid:15) (cid:15) / / ad E i ( M r − ) (cid:15) (cid:15) / / ad E i ( M ) (cid:15) (cid:15) / / ad E i ( V r ) (cid:15) (cid:15) / /
00 0 0in which the first two rows are exact, and the three columns are exact. By the nine lemma, the last rowis also exact. By induction, we have a filtration of ad E i ( M r − ) of the form0 = ad E i ( M − ) ⊆ ad E i ( M ) ⊆ · · · ⊆ ad E i ( M r − ) ,and such that for all k , ad E i ( M k ) /ad E i ( M k − ) ≃ ad E i ( V k ) . This filtration together with the third exactrow of the diagram above gives the desired filtration of ad E i ( M ) . (cid:3) Proposition 4.27.
Let M ∈ H i β − mod and let N ∈ H i γ − mod with β , γ ∈ Q + . Assume that for all k > , τ E i ,ad kEi ( M ) and τ E i ,ad kEi ( N ) are injective. Then there is a filtration = V − ⊆ V ⊆ V ⊆ · · · ⊆ V n − = ad nE i ( MN ) , such that V k / V k − ≃ q ( n − ζ ( k )) h i ∨ , β i + σ ( k ) i ad ζ ( k ) E i ( M ) ad n − ζ ( k ) E i ( N ) . ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 27
Proof.
The proof is the same inductive proof as that of Proposition 4.23, with one difference. Instead ofusing the exactness of ad E i to find a filtration of ad n + E i ( MN ) from that of ad nE i ( MN ) , we use Proposition4.26. The assumptions of Proposition 4.26 are satisfied since for all k > τ E i ,ad kEi ( M ) and τ E i ,ad kEi ( N ) areinjective by assumption. (cid:3) Proof of Theorem 4.18.
The strategy of the proof is as follows. We start by proving that Theorem4.18 holds for modules of the form ad ( n ) E i ( E j r . . . E j ) , where j , . . . , j r ∈ I \ { i } and n >
0, and theirsubquotients. This part of the proof is similar to the argument of [KK12]: it is done by constructing aquasi left inverse to τ E i , ( − ) . Then, we prove that as a Serre subcategory of H , the subcategory H [ i ] isgenerated by such modules. This is done using the compatibility with the monoidal structure proved inProposition 4.27. Once these two points are proved, the conclusion follows from Proposition 4.26. Theorem 4.28.
Let j , . . . , j r ∈ I \ { i } and let n > . Let N be a subquotient of M = ad ( n ) E i ( E j r . . . E j ) . Then τ E i , N is injective.Proof. Let β = ni + j + . . . + j r , so that M is an H i β -module. We construct a map τ M , E i : E i M → ME i such that the composition τ M , E i ◦ τ E i , M is injective. The map τ M , E i is defined by τ M , E i : (cid:26) E i M → ME i h i , β ⊗ H β m hP τ [ n + r ↓ ] β , i ⊗ H β m where P = Q i , β ( x n + r + , x , . . . , x n + r ) = ∑ ν ∈ I β ∏ k n + r ν k = i Q i , ν k ( x n + r + , x k ) ! i , ν ∈ P β + i .To check that this is well-defined, we need to prove that for all y ∈ H β , we have (cid:16) ( i ⋄ y ) P τ [ n + r ↓ ] − P τ [ n + r ↓ ] ( y ⋄ i ) (cid:17) β , i ⊗ H β M = Z = (cid:16) ( i ⋄ y ) P τ [ n + r ↓ ] − P τ [ n + r ↓ ] ( y ⋄ i ) (cid:17) β , i . The element P is central in H i , β by Proposition 4.9, sowe have Z = P (cid:16) ( i ⋄ y ) τ [ n + r ↓ ] − τ [ n + r ↓ ] ( y ⋄ i ) (cid:17) β , i . By Proposition 3.7, there are elements z k ∈ H β , i such that Z = P n + r − ∑ k = τ [ k ↓ ] z k β , i = n + r − ∑ k = τ [ k ↓ ] Pz k β , i .However P β , i = ∑ ν ∈ I β ∏ k n + r ν k = i Q i , ν k ( x n + r + , x k ) ! i , ν β , i = ∑ µ ∈ I β − i ∏ k n + r − µ k = i Q i , µ k ( x n + r + , x k + ) ! i , µ , i = ∑ ρ ∈ I β ρ n + r = i Q i , ρ ( x n + r + , x , . . . , x n + r + ) ρ , i = ∑ ρ ∈ I β ρ n + r = i Q i , ρ ( x n + r , x , . . . , x n + r ) ρ ! ⋄ i .By Proposition 4.8, Q i , ρ ( x n + r , x , . . . , x n + r ) ρ acts by zero on M for all ρ ∈ I β such that ρ n + r = i . So Z ⊗ H β M =
0, and the morphism τ M , E i is well defined.Next, we compute the composition τ M , E i ◦ τ E i , M . It is given by τ M , E i ◦ τ E i , M : (cid:26) ME i → ME i h β , i ⊗ H β m h τ [ ↑ n + r ] P τ [ n + r ↓ ] β , i ⊗ H β m Note that ME i is cyclic, generated by the class c of τ ω [ r + n + r + ] ni , j r ,..., j , i in ME i . We start by computingthe image of c under τ M , E i ◦ τ E i , M . In the following computations, we do not write the idempotent1 ni , j r ,..., j , i on the right to help readability, but all the elements are considered in H β + i ni , j r ,..., j , i . Thedetailed explanations are below the computation. We have τ [ ↑ n + r ] P τ [ n + r ↓ ] τ ω [ r + n + r + ] = τ [ ↑ n + r ] P τ ω [ r + n + r + ] τ [ r ↓ ] = τ [ ↑ r ] ∂ [ r + ↑ n + r ] ( P ) τ ω [ r + n + r + ] τ [ r ↓ ] = s [ ↑ r ] (cid:16) ∂ [ r + ↑ n + r ] ( P ) (cid:17) τ [ ↑ r ] τ [ n + r ↓ ] τ ω [ r + n + r + ] = s [ ↑ r ] (cid:16) ∂ [ r + ↑ n + r ] ( P ) (cid:17) τ [ n + r ↓ r + ] τ [ ↑ r ] τ r + τ [ r ↓ ] τ ω [ r + n + r + ] .For the first equality, we have commuted τ [ r ↓ ] to the right of τ ω [ r + n + r + ] , and the factor τ [ n + r ↓ r + ] τ ω [ r + n + r + ] that appears is equal to τ ω [ r + n + r + ] . The second equality comes from applying part (2) of Lemma 3.6.For the third equality, we have commuted ∂ [ r + ↑ n + r ] ( P ) to the left of τ [ ↑ r ] using relation (6) from Defini-tion 3.2, and we have undone the first step. Finally, the fourth equality is obtained by commuting τ [ ↑ r ] to the right of τ [ n + r ↓ r + ] . Now by relation (3) in Lemma 3.8 we have τ [ ↑ r ] τ r + τ [ r ↓ ] = (cid:16) τ [ r + ↓ ] τ τ [ ↑ r + ] + ˜ P (cid:17) = ˜ P mod ( β − i , i , i ) where ˜ P = ∂ r + (cid:16) Q i , ( j ,..., j r ) ( x r + , x , . . . , x r + ) (cid:17) . By part (2) of Lemma 3.6 we have τ [ n + r ↓ r + ] ˜ P τ ω [ r + n + r + ] = ∂ [ n + r ↓ r + ] ( ˜ P ) τ ω [ r + n + r + ] .Hence if we let θ = s [ ↑ r ] (cid:16) ∂ [ r + ↑ n + r ] ( P ) (cid:17) ∂ [ n + r ↓ r + ] ( ˜ P ) ni , j r ,..., j , i ∈ P β + i we have proved that τ [ ↑ n + r ] P τ [ n + r ↓ ] τ ω [ r + n + r + ] = θτ ω [ r + n + r + ] mod (cid:0) β − i , i , i (cid:1) .This proves that τ M , E i ◦ τ E i , M ( c ) = θ c . Now let Θ = ∑ ω ∈ S n + r + /Stab ( ni , j r ,..., j , i ) ω ( θ ) ∈ P β + i ,where Stab ( ni , j r , . . . , j , i ) denotes the stabilizer of the sequence ( ni , j r , . . . , j , i ) in S n + r + . Let us checkthat Θ is in the center of H β + i . By Proposition 3.4, this amounts to checking that Θ is fixed by S n + r + .Given the definition of Θ , it suffices to check that for ω ∈ Stab ( ni , j r , . . . , j , i ) we have ω ( θ ) = θ . ByProposition 4.9, we already know that P , ˜ P are fixed under those ω who permute the variables labeledby j , . . . , j r , thus so is θ . But then the Demazure operators in ∂ [ r + ↑ n + r ] ( P ) , ∂ [ n + r ↓ r + ] ( ˜ P ) make thesepolynomials symmetric in the variables labeled by i . Hence θ is indeed fixed by Stab ( ni , j r , . . . , j , i ) , and Θ is central. Furthermore, we have Θ c = Θ ni , j r ,..., j , i c = θ c = τ M , E i ◦ τ E i , M ( c ) .Hence τ M , E i ◦ τ E i , M and multiplication by Θ (which is a morphism of H β + i -modules since Θ is central)agree on c . Since c generates ME i , we conclude that τ M , E i ◦ τ E i , M ( x ) = Θ x for all x ∈ ME i . Now remarkthat Θ β , i has the form Θ β , i ∈ tx ℓ β , i + ∑ k <ℓ x k ( P β ⋄ i ) β , i for some integer ℓ and t ∈ K × . It follows that multiplication by Θ β , i is an injective endomorphism of M ⊗ E i . But since ME i = n + r M k = τ [ k ↓ ] β , i ⊗ H β , i ( M ⊗ E i ) ,multiplication by Θ acts diagonally along this decomposition, by multiplication by Θ β , i on each sum-mand M ⊗ E i . We conclude that multiplication by Θ is an injective endomorphism of ME i . Hence τ E i , M is injective.We now prove that this implies the result for the subquotient N of M . We can fit N in a diagram N ֒ → N ′ f և M ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 29 for some H β -module N ′ . Given its definition, the map τ M , E i restricts to a map E i ker ( f ) → ker ( f ) E i . Sowe have an induced map τ N ′ , E i : E i N ′ → N ′ E i such that τ N ′ , E i ◦ τ E i , N ′ is multiplication by Θ . By the sameargument as above, multiplication by Θ is injective on N ′ E i . Hence τ E i , N ′ is injective. But by naturality τ E i , N is the restriction of τ E i , N ′ to NE i , so it is injective as well. (cid:3) Corollary 4.29.
Let N be in the Serre subcategory of H [ i ] generated by the modules ad nE i ( E j r . . . E j ) for n > ,r > and j , . . . , j r ∈ I \ { i } . Then τ E i , N is injective.Proof. Since ad nE i ≃ [ n ] i !ad ( n ) E i , we know that N is in the Serre subcategory of H [ i ] generated by themodules ad ( n ) E i ( E j r . . . E j ) for n > r > j , . . . , j r ∈ I \ { i } . There is a filtration of N ⊆ N ⊆ · · · ⊆ N ℓ = N such that each N k / N k − is a subquotient of some ad ( n ) E i ( E j r . . . E j ) . Hence τ E i , N k / N k − is injective byTheorem 4.28. By Proposition 4.26, we conclude that τ E i , N is injective. (cid:3) Theorem 4.30.
As a Serre subcategory of H , the subcategory H [ i ] is generated by the modules ad nE i ( E j r . . . E j ) for n > , r > and j , . . . , j r ∈ I \ { i } .Proof. Let H ′ [ i ] be the Serre full subcategory of H generated by the modules ad nE i ( E j r . . . E j ) for n > r > j , . . . , j r ∈ I \ { i } . It is clear that H ′ [ i ] ⊆ H [ i ] . We prove the converse inclusion.Let M ∈ H [ i ] . There exists a projective module P ∈ H that surjects onto M . Since π i is rightexact, π i ( P ) surjects onto π i ( M ) = M . Hence it suffices to show that π i ( P ) ∈ H ′ [ i ] for every projectivemodule P . Since every projective module is a direct sum of summands of modules of the form E j m . . . E j ,it suffices to show π i ( E j m . . . E j ) ∈ H ′ [ i ] for all m and j , . . . , j m ∈ I . We proceed inductively on m .For m =
1, we have π i ( E j ) = E j if j = i and π i ( E i ) =
0. In all cases, we have π i ( E j ) ∈ H ′ [ i ] .Assume now π i ( E j m − . . . E j ) ∈ H ′ [ i ] , we consider two cases: • if j m = j = i , then π i ( E j m . . . E j ) ≃ E j π i ( E j m − . . . E j ) by Lemma 4.5. So it suffices to show that E j H ′ [ i ] ⊆ H ′ [ i ] . If M ∈ H ′ [ i ] , then we have a filtration0 = M − ⊆ M ⊆ · · · ⊆ M s = M ,such that M k / M k − is a subquotient of ad n k E i ( N k ) for some n k > N k a product of E a , a ∈ I \ { i } . By exactness of multiplication by E j , we have a filtration of E j ME = E j M − ⊆ E j M ⊆ · · · ⊆ E j M s = E j M ,such that E j M k / E j M k − is a subquotient of E j ad n k E i ( N k ) . By Theorem 4.28, N k and E j satisfy theconditions of Proposition 4.27, so we have an injection (up to a grading shift) E j ad n k E i ( N k ) ֒ → ad n k E i ( E j N k ) .Hence E j M is also in H ′ [ i ] . • if j m = i , we have π i ( E j m . . . E j ) ≃ ad E i ( π i ( E j m − . . . E j )) by Lemma 4.5. So it suffices to provethat H ′ [ i ] is stable under ad E i . If M ∈ H ′ [ i ] , then we have a filtration0 = M − ⊆ M ⊆ · · · ⊆ M s = M such that M k / M k − is a subquotient of some ad n k E i ( N k ) for N k a product of E a , a ∈ I \ { i } . ByCorollary 4.29, τ E i , M k / M k − is injective for all k . So we can apply Proposition 4.26 to get a filtrationof ad E i ( M ) = ad E i ( M − ) ⊆ ad E i ( M ) ⊆ · · · ⊆ ad E i ( M s ) = ad E i ( M ) ,such that ad E i ( M k ) /ad E i ( M k − ) ≃ ad E i ( M k / M k − ) .By assumption, M k / M k − fits into a diagram M k / M k − ֒ → N և ad n k E i ( N k ) . Since τ E i , ( − ) is injective for the three modules in this diagram, by Proposition 4.26 we can applyad E i to this diagram to getad E i ( M k / M k − ) ֒ → ad E i ( N ) և ad n k + E i ( N k ) .So ad E i ( M k ) /ad E i ( M k − ) is a subquotient of ad n k + E i ( N k ) . Hence ad E i ( M ) ∈ H ′ [ i ] , and the proofis complete. (cid:3) Now Corollary 4.29 and Theorem 4.30 imply Theorem 4.18.4.5.
Generating objects.
As an immediate consequence of Theorem 4.30 and Proposition 4.23, we getthe following generating result for H [ i ] . Corollary 4.31.
As a Serre and monoidal full subcategory of H , H [ i ] is generated by the objects ad ( n ) E i ( E j ) forn > and j ∈ I \ { i } . Corollary 4.31 is a categorification of the definition of U + [ i ] as the subalgebra generated by thead ( n ) e i ( e j ) for n > j ∈ I \ { i } . Lusztig also proves that U + = ∑ n > U + [ i ] e ( n ) i [Lus10, Lemma38.1.2]. We finish this section by proving a similar result for H . Proposition 4.32.
The Serre subcategory of H generated by the subcategories H [ i ] E ( n ) i for n > is equal to H .Proof. Let H ′ be the Serre subcategory of H generated by the subcategories H [ i ] E ni for n >
0. We have E j ∈ H ′ for all j ∈ I . So to prove that H = H ′ , it suffices to prove that H ′ is also a monoidal subcategory.We start by proving that if M ∈ H ′ , then E i M ∈ H ′ . (4.5)Given M ∈ H ′ , there is a filtration 0 = M ⊆ M ⊆ · · · ⊆ M r = M such that each quotient M k / M k − isa subquotient of an object of H [ i ] E n k i . We want to prove E i M ∈ H ′ . By induction on r , it suffices to provethe result for M = M ′ E ni , with M ′ ∈ H [ i ] . But then by Theorem 4.18 we have a short exact sequence0 → M ′ E n + i → E i M → ad E i ( M ′ ) E ni → M ′ E n + i , ad E i ( M ′ ) E ni ∈ H ′ , we conclude that E i M ∈ H ′ . Hence statement (4.5) is established.Since H [ i ] is monoidal, it is also clear thatif M ∈ H ′ and N ∈ H [ i ] , then MN ∈ H ′ . (4.6)Now let M , N ∈ H ′ , we want to prove that MN ∈ H ′ . We have filtrations0 = M ⊆ M ⊆ · · · ⊆ M r − ⊆ M r = M ,0 = N ⊆ N ⊆ · · · ⊆ N s − ⊆ N s = N ,such that M k / M k − is a subquotient of an object of H [ i ] E m k i and N k / N k − is a subquotient of an objectof H [ i ] E n k i . There is an exact sequence0 → MN s − → MN → M ( N / N s − ) → s , it suffices to show the result for s =
1. Similarly there is an exact sequence0 → M r − N → MN → ( M / M r − ) N → r , it suffices to show the result for r =
1. In that case, M is a subquotient of M ′ E mi and N is a subquotient of N ′ E ni with M ′ , N ′ ∈ H [ i ] . Then by exactness of the tensor product, MN is asubquotient of M ′ E mi N ′ E ni . But since N ′ E ni ∈ H ′ , applying statement (4.5) m times gives E mi N ′ E ni ∈ H ′ .Then statement (4.6) gives M ′ E mi N ′ E ni ∈ H ′ . Thus MN ∈ H ′ . (cid:3) ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 31
5. C
ATEGORICAL sl ACTION
In this section we finish proving that there is a categorical action of sl on H [ i ] . We start by recallingthe definition of categorical sl actions. With what we have already proved, the only axiom left to checkis the categorical [ e i , f i ] = h i relation. The proof mirrors that of [KK12] for cyclotomic KLR algebras, andis based on two ingredients: the short exact sequence of Theorem 4.18 and the Mackey decompositionsfor KLR algebras. One difference, as in the proof of Theorem 4.18, is that we only do explicit com-putations for modules of the form ad nE i ( E j r . . . E j ) , which is enough as they generate H [ i ] by Theorem4.30.5.1. There are various setups for 2-representations (on additive, abelian or triangu-lated categories). In our framework, we use a version for abelian categories. Since it is not important forour purpose, we will not introduce the 2-category attached to sl , but rather just define what a represen-tation of it is, as in the original approach of [CR08]. Definition 5.1. A of sl is the data of • a graded, abelian, K -linear category V together with a decomposition into weight subcategories V = ⊕ w ∈ Z V w , • endofunctors E , F of V restricting to E w : V w → V w + and F w : V w → V w − for all w ∈ Z , • natural transformations x : E → E of degree 2, τ : E → E of degree -2, η : 1 w → F E w ofdegree 1 + w and ε : E F w → w of degree 1 − w for all w ∈ Z ,subject to the following conditions(1) the functors E , F are exact,(2) the natural transformation ε , η are the counit and unit respectively of adjunctions ( E w , q w + F w + ) for all w ∈ Z ,(3) the natural transformations x and τ induce actions of affine nil Hecke algebras on powers of E ;which means that the following equalities hold: τ = τ ◦ x E − E x ◦ τ = x E ◦ τ − τ ◦ E x = E , τ E ◦ E τ ◦ τ E − E τ ◦ τ E ◦ E τ = w ∈ Z , there are isomorphisms if w > E F w ρ w −→ ∼ F E w ⊕ [ w ] w ,if w E F w ⊕ [ − w ] w ρ w −→ ∼ F E w ,where the ρ w are some explicit natural transformations defined below.To complete the definition, we need to define the natural transformations ρ w . We start by defining amap ϕ : EF → FE as the composition ϕ = (cid:18) E F η E F −−→ F E E F F τ F −−−→ F E E F FE ε −−→ F E (cid:19) . (5.1)If w >
0, we define the natural transformation ρ w as a column vector as follows ρ w = ϕεε ◦ x F ... ε ◦ x w − F .If w >
0, we define the natural transformation ρ w as a row vector as follows ρ w = h ϕ η F x ◦ η . . . F x w − ◦ η i .This completes the definition of a categorical sl -action. Remark . The degrees of the natural transformations can also be dilated by some factor. In our case, the sl categorical action arises from an action of the subalgebra U i , hence all the degrees will be multipliedby d i . We want to prove that there is a structure of 2-representation of sl on the abelian category H [ i ] ,induced by the functor ad E i . Let us start by giving all the necessary pieces of data. The weight subcate-gories of H [ i ] are the subcategories H [ i ] w , w ∈ Z , defined by H [ i ] w = M β ∈ Q + h i ∨ , β i = w H i β − mod.The adjoint pair of endofunctors of H [ i ] that we consider is ( ad E i , ad F i ) . The functor ad E i is the one westudied in Section 4, the functor ad F i is defined to be its right adjoint, up to a grading shift chosen sothat the unit η and counit ε of adjunction have the desired degree. Explicitly, ad F i is given byad F i = M β ∈ Q + q −h i ∨ , β i i res H i β H i β − i .More precisely, if M ∈ H i β − mod, we have ad F i ( M ) = q −h i ∨ , β i i i , β − i M , viewed as an H i β − i -modulevia the right inclusion H i β − i → i , β − i H i β i , β − i . The fact that ad E i and ad F i restrict as desired on theweight subcategories follows simply from h i ∨ , i i =
2. Finally, we have to define x ∈ End ( ad E i ) and τ ∈ End ( ad E i ) . These natural transformations are defined by Proposition 4.11. Theorem 5.3.
This data defines a 2-representation of sl on H [ i ] .Proof. We must prove that the conditions in Definition 5.1 are satisfied. The fact that ad E i is exact wasproved in Corollary 4.20, and the exactness of ad F i follows similarly from Corollary 4.19. The action ofthe affine nil Hecke algebra on powers of ad E i was proved in Proposition 4.11. Hence only the invert-ibility of the maps ρ w , w ∈ Z , remains to be proved, which we do in Theorem 5.6 below. (cid:3) Mackey formulas.
The KLR algebra case.
We follow [KK12, Section 3]. Let β ∈ Q + and i ∈ I . Recall the left i -inductionfunctor (cid:26) H β − mod → H β + i − mod, M E i M = H β + i i , β ⊗ H i , β ( E i ⊗ M ) .Since E i is a free module of rank 1 over H i , we have E i M ≃ H β + i i , β ⊗ H β M . We also defined the left i -restriction functor F i as the right adjoint of the left i -induction functor. It is given by (cid:26) H β + i − mod → H β , N F i ( N ) = i , β N .There is a Mackey type result for left i -induction and left i -restriction. Proposition 5.4 ([KK12, Theorem 3.6]) . There is an isomorphism of graded H β -bimodules ( q i (cid:16) H β i , β − i ⊗ H β − i i , β − i H β (cid:17) ⊕ H i , β → i , β H β + i i , β , (( y ⊗ y ′ ) , z ) ( i ⋄ y ) τ n ( i ⋄ y ′ ) + z . It induces a natural isomorphism in M ∈ H β − mod q i E i F i ( M ) ⊕ M [ X i ] ∼ −→ F i ( E i M ) where X i is a variable of degree d i . More explicitly,M [ X i ] = M ⊗ K [ X i ] ≃ M k > q − ki M .Right i -induction and right i -restriction functors are defined similarly, and satisfy an analogousMackey type decomposition. Finally, there is a Mackey type result for right i -induction and left i -restriction. Proposition 5.5 ([KK12, Theorem 3.9]) . For all β ∈ Q + , there is a short exact sequence of graded H β -bimodules → H β β − i , i ⊗ H β − i i , β − i H β R −→ i , β H β + i β , i S −→ q h i ∨ , β i i H β , i → ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 33
It induces a short exact sequence which is natural in M ∈ H β − mod0 → F i ( M ) E i R M −−→ F i ( ME i ) S M −→ q h i ∨ , β i i M [ X i ] → R and S are. The morphism R is given by (cid:26) H β β − i , i ⊗ H β − i i , β − i H β → i , β H β + i β , i , y ⊗ y ′ ( i ⋄ y )( y ′ ⋄ i ) .To describe the morphism S , we use decomposition (3.6) to write H β + i β , i = n M k = τ [ k ↓ ] H β , i .If z ∈ i , β H β + i β , i , we can thus decompose z as z = n ∑ k = τ [ k ↓ ] z k for some unique z , . . . , z n ∈ H β , i Then S ( z ) = z n . Equivalently, we can also compute S ( z ) by decom-posing along 1 i , β H β + i = n + M k = H i , β τ [ n ↓ k ] and taking the coefficient of τ [ n ↓ ] (see [KK12, Corollary 3.8]).5.2.2. Mackey formulas in H [ i ] . We can now prove the categorical [ e i , f i ] = h i relation. Theorem 5.6.
Let M ∈ H i β − mod with β ∈ Q + , and let w = h i ∨ , β i . Then we have isomorphisms if w >
0, ad E i ad F i ( M ) ρ w −→ ∼ ad F i ad E i ( M ) ⊕ [ w ] i M , if w
0, ad E i ad F i ( M ) ⊕ [ − w ] i M ρ w −→ ∼ ad F i ad E i ( M ) .We follow the approach of [KK12, Theorem 5.2], and using jointly Theorem 4.18 and the Mackeydecompositions for KLR algebras. Lemma 5.7.
Let M ∈ H i β − mod with β ∈ Q + , and let w = h i ∨ , β i . There is a commutative diagram with exactrows which is natural in M / / q − i F i ( M ) E i R M / / τ Ei , Fi ( M ) (cid:15) (cid:15) q − i F i ( ME i ) S M / / F i ( τ Ei , M ) (cid:15) (cid:15) q w − i M [ X i ] Φ M (cid:15) (cid:15) / / / / q − wi E i F i ( M ) T M / / q − − wi F i ( E i M ) W M / / q − − wi M [ X i ] / / Proof.
The maps R M and S M are those of Proposition 5.5, and the maps T M and W M are those inducedby the isomorphism of Proposition 5.4. The exactness of the two rows also follows from Propositions 5.4and 5.5. The map Φ M is defined by commutativity of the diagram. So the only thing to prove is that theleftmost square of the diagram commutes.Let y = β − i , i ⊗ H β − i i , β − i m ∈ F i ( M ) E i , with m ∈ M . We have τ E i , F i ( M ) ( y ) = τ [ ↑| β |− ] i , β − i ⊗ H β − i i , β − i m , T M ( τ E i , F i ( M ) ( y )) = τ [ ↑| β | ] i , β − i ⊗ H β m ,and R M ( y ) = i , β − i , i ⊗ H β m , F i ( τ E i , M )( f M ( y )) = τ [ ↑| β | ] i , β ⊗ H β m .Hence T M ( τ E i , F i ( M ) ( y )) = F i ( τ E i , M )( R M ( y )) . Since elements y ∈ F i ( M ) E i of this form generate F i ( M ) E i as an H β -module, we deduce that T M ◦ τ E i , F i ( M ) = F i ( τ E i , M ) ◦ R M , and the proof is complete. (cid:3) For M ∈ H i β − mod, the morphism τ E i , F i ( M ) is injective by Theorem 4.18, and its cokernel is ad E i ( F i ( M )) = q h i ∨ , β i− i ad E i ( ad F i ( M )) . Since the left i -restriction functor F i is exact, we also know that F i ( τ E i , M ) is in-jective, and its cokernel is F i ( ad E i ( M )) = q h i ∨ , β i + i ad F i ( ad E i ( M )) . Hence by the snake lemma there is ashort exact sequence which is natural in M ∈ H i β − mod0 → ker ( Φ M ) → ad E i ad F i ( M ) T M −−→ ad F i ad E i ( M ) → coker ( Φ M ) →
0. (5.3)
Lemma 5.8.
For all M ∈ H i β − mod with β ∈ Q + , the map T M : ad E i ad F i ( M ) → ad F i ad E i ( M ) in the sequence(5.3) is equal to the map ϕ defined in equation (5.1).Proof. We prove this by chasing the diagram giving rise to the exact sequence (5.3). Let y be an elementof ad E i ad F i ( M ) of the form y = i , β − i ⊗ H i β − i i , β − i m , for m ∈ M . We can write y as the image of theelement z = i , β − i ⊗ H β − i i , β − i m ∈ E i F i ( M ) by the canonical quotient morphism. Then we have T M ( z ) = τ | β | i , β − i ⊗ H β m .Hence T M ( y ) = τ | β | i , β − i ⊗ H i β m = ϕ ( y ) .So ϕ and T M coincide on elements y of the form given above. Since these generate ad E i ad F i ( M ) as an H i β -module, we deduce that ϕ = T M . (cid:3) We now want to compute explicitly the kernel and cokernel of the map Φ M . An element y of M [ X i ] will be written formally in the form y = ℓ ∑ k = m k X ki where ℓ >
0, and m , . . . , m ℓ ∈ M . Our discussion up to this point is valid for any module in H [ i ] , butwe now restrict to the generators of H [ i ] to simplify the computations. Proposition 5.9.
Let M = ad ( n ) E i ( E j r . . . E j ) for some n > and j , . . . , j r ∈ I \ { i } . Let β = ni + j + . . . + j r and let w = h i ∨ , β i . Then there exists an invertible element b of K such that for all m ∈ M and k > we have Φ M ( mX ki ) ∈ bmX k − wi + ∑ ℓ< k − w MX ℓ i , where we put X ℓ i = when ℓ < .Proof. The module M is cyclic, generated by the class c of τ ω [ r + n + r ] ni , j r ,..., j in π i ( E ( n ) i E j r . . . E j ) . So itsuffices to prove the result for m = c .Let c k = x kn + r + τ [ n + r ↓ ] β , i ⊗ H β c ∈ F i ( ME i ) . Then cX ki = S M ( c k ) by definition of S . Since therightmost square in the commutative diagram (5.2) commutes, Φ M ( cX ki ) = W M ( τ E i , M ( c k )) . We have τ E i , M ( c k ) = x kn + r + τ [ n + r ↓ ] τ [ ↑ n + r ] i , β ⊗ H β c = x kn + r + τ [ n + r ↓ ] τ [ ↑ n + r ] τ ω [ r + n + r ] ( n + ) i , j r ... j ⊗ H β x r + . . . x n − n + r c the second equality coming from the fact that c = τ ω [ r + n + r ] x r + . . . x n − n + r c . Let c ′ k = x kn + r + τ [ n + r ↓ ] τ [ ↑ n + r ] τ ω [ r + n + r ] ( n + ) i , j r ... j .To compute W M ( τ E i , M ( c k )) , we must find the component of c ′ k on H i , β in the direct sum decomposition1 i , β H β + i i , β = H i , β ⊕ q i ( H β i , β − i ⊗ H β − i i , β − i H β ) given by Proposition 5.4. The following computations are done in H β + i ( n + ) i , j r ,..., j , but we omit theidempotent 1 ( n + ) i , j r ,..., j on the right to help readability. The explanation of each equality is writtenbelow the computation. Let P = r ∏ ℓ = Q i , j ℓ ( x r + , x ℓ ) ! ( n + ) i , j r ,..., j ∈ P β + i . ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 35
We have c ′ k = x kn + r + τ [ n + r ↓ ] τ [ ↑ n + r ] τ ω [ r + n + r ] = x kn + r + τ [ n + r ↓ r + ] P τ [ r + ↑ n + r ] τ ω [ r + n + r ] = x kn + r + τ [ n + r ↓ r + ] P τ ω [ r + n + r + ] = x kn + r + ∂ [ n + r ↓ r + ] ( P ) τ ω [ r + n + r + ] = x kn + r + ∂ [ n + r ↓ r + ] ( P ) τ [ r + ↑ n + r ] τ ω [ r + n + r ] .In the second equality, we have applied Lemma 3.8 to replace τ [ r ↓ ] τ [ ↑ r ] by P . For the third equality, wehave used the fact that ω [ r + n + r + ] = s [ r + ↑ n + r ] ω [ r + n + r ] to replace τ [ r + ↑ n + r ] τ ω [ r + n + r ] by τ ω [ r + n + r + ] . The fourth equality comes from the relations of the affine nil Hecke algebra proved inLemma 3.6. Finally in the fifth equality, we have just reversed the third step.Before we continue the computation, we write ∂ [ n + r ↓ r + ] ( P ) in a more explicit way. First, given theassumptions on the form of the polynomials Q i , j , we can write P in the form P ∈ tx n − wr + + ∑ ℓ< n − w x ℓ r + K [ x , . . . , x r ] .where t is an invertible element of K . Hence ∂ [ n + r ↓ r + ] ( P ) = ∑ ℓ n − w x ℓ n + r + p ℓ for some p ℓ ∈ K [ x , . . . , x n + r ] , with p n − w = ( − ) n t . We can now resume the computation of c ′ k . Withthis form for ∂ [ n + r ↓ r + ] ( P ) , we have c ′ k = ∑ ℓ n − w x k + ℓ n + r + p ℓ τ [ r + ↑ n + r ] τ ω [ r + n + r ] = ∑ ℓ n − w p ℓ τ [ r + ↑ n + r − ] x k + ln + r + τ n + r τ ω [ r + n + r ] = ∑ ℓ n − w p l τ [ r + ↑ n + r ] x k + ℓ n + r τ ω [ r + n + r ] + ∑ ℓ n − wm k + ℓ − p ℓ τ [ r + ↑ n + r − ] x mn + r + x k + ℓ − − mn + r τ ω [ r + n + r ] .For the second equality, we have used relation (6) of Definition 3.2 from to move the factor x k + ℓ n + r + allthe way to the left of τ n + r . The third equality is what we get from the commutation relation of x k + ℓ n + r + and τ n + r from Lemma 3.6. However ∑ ℓ n − w p ℓ τ [ r + ↑ n + r ] x k + ℓ n + r τ ω [ r + n + r ] ∈ H β i , β − i ⊗ H β − i i , β − i H β .Hence, modulo H β i , β − i ⊗ H β − i i , β − i H β we have c ′ k = ∑ ℓ n − wm k + ℓ − p ℓ τ [ r + ↑ n + r − ] x mn + r + x k + ℓ − − mn + r τ ω [ r + n + r ] = ∑ ℓ n − wm k + ℓ − p ℓ x mn + r + ∂ [ r + ↑ n + r − ] ( x k + ℓ − − mn + r ) τ ω [ r + n + r ] .We have ∂ [ r + ↑ n + r − ] ( x k + ℓ − − mn + r ) = (cid:26) k + ℓ − − m < n − k + ℓ − − m = n − Thus the previous equation can be written c ′ k = ∑ ℓ n − wm k + ℓ − n p ℓ x mn + r + ∂ [ r + ↑ n + r − ] ( x ℓ − − mn + r ) τ ω [ r + n + r ] ∈ ( − ) n tx k − wn + r + τ ω [ r + n + r ] + ∑ ℓ< k − w x ℓ n + r + ( i ⋄ H β ) .We can now conclude that Φ M ( cX ki ) = W M ( c ′ k ⊗ β x r + . . . x n − r + n c ) ∈ ( − ) n tcX k − wi + ∑ ℓ< k − w MX ℓ i .If we let b = ( − ) n t ∈ K × , then the proof is complete. (cid:3) Remark . Since the map Φ M is natural in M , Proposition 5.9 actually holds for M any subquotient ofa module of the form ad ( n ) E i ( E j r . . . E j ) .As an immediate consequence of Proposition 5.9, we get an explicit description of the kernel andcokernel of Φ M . Corollary 5.11.
Let M be a subquotient of ad ( n ) E i ( E j r . . . E j ) for n > and j , . . . , j r ∈ I \ { i } and w defined asabove. Then we have if w >
0, ker ( Φ M ) = q w − i w − M k = MX ki ! = [ w ] i M and coker ( Φ M ) = if w
0, ker ( Φ M ) = and coker ( Φ M ) = q − w − i − w − M k = MX ki ! = [ − w ] i M .With this proved, we can now show that the maps ρ w are isomorphisms. We separate two casesdepending on whether the weight is positive or negative. Proposition 5.12.
Let M be a subquotient of ad ( n ) E i ( E j r . . . E j ) for some n > and j , . . . , j r ∈ I \ { i } . Let β = ni + j + . . . + j r and let w = h i ∨ , β i . Assume that w . Then ρ w is an isomorphism.Proof. Given Lemma 5.8 and Corollary 5.11, the short exact sequence (5.3) becomes0 → ad E i ad F i ( M ) ϕ −→ ad F i ad E i ( M ) → [ − w ] i M → q − − wi F i ( E i M ) W M / / (cid:15) (cid:15) q − − wi M [ X i ] (cid:15) (cid:15) [ − w ] i M ⊇ o o ∼ x x qqqqqqqqqqq ad F i ad E i ( M ) / / coker ( Φ M ) The map W M admits a right inverse A M given by the direct sum decomposition1 i , β H i + β i , β = H i , β ⊕ q i ( H β i , β − i ⊗ H β − i i , β − i H β ) .This right inverse provides a map A M : [ − w ] i M → ad F i ad E i ( M ) that splits the short exact sequenceabove. Let us compute the map A M . Let y ∈ [ − w ] i M , we view y as an element of q − − wi M [ X i ] that wewrite in the form y = − w − ∑ k = m k X ki where m , . . . , m − w − ∈ M . Then we have A M ( y ) = − w − ∑ k = x kn + r + i , β ⊗ H β m k ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 37 So A M ( y ) = − w − ∑ k = x kn + r + i , β ⊗ H i β m k = − w − ∑ k = x kn + r + η M ( m k ) .Hence A M is exactly the component of ρ w on [ − w ] i M , so ρ w is an isomorphism. (cid:3) Proposition 5.13.
Let M be a subquotient of ad ( n ) E i ( E j r . . . E j ) for some n > and j , . . . , j r ∈ I \ { i } . Let β = ni + j + . . . + j r and let w = h i ∨ , β i . Assume that w > . Then ρ w is an isomorphism.Proof. Given Lemma 5.8 and Corollary 5.11, the short exact sequence (5.3) becomes0 → [ w ] i M δ −→ ad E i ad F i ( M ) ϕ −→ ad F i ad E i ( M ) → δ is the connecting map given by the snake lemma. Let Γ be the component of ρ w which maps in [ w ] i M . To prove that ρ w is an isomorphism, it suffices to prove that Γ δ is an isomorphism. To this end,we prove that there exists an invertible element b ∈ K such that for all m ∈ M and k ∈ {
0, . . . , w − } ,we have Γ δ ( mX ki ) = bmX w − − ki + ∑ w − − k <ℓ w − MX ℓ i . (5.4)By naturality, it suffices to prove this for M = ad ( n ) E i ( E j r . . . E j ) . In this case, M is cyclic, generated bythe class c of τ ω [ r + n + r ] ni , j r ..., j in π i ( E ( n ) i E j r . . . E j ) , and it suffices to treat the case m = c . We start bycomputing δ ( cX ki ) , which we do by chasing the diagram (5.2). We resume the computations done in theproof of Proposition 5.9. Recall that cX ki = S M ( c k ) with c k = x kn + r + τ [ n + r ↓ ] β , i ⊗ H β x r + . . . x n − r + n c . Weproved that τ E i , M ( c k ) = c ′ k ⊗ H β x r + . . . x n − r + n c , where c ′ k is an element proved to be equal to c ′ k = ∑ ℓ n − w p ℓ τ [ r + ↑ n + r ] x k + ℓ n + r τ ω [ r + n + r ] + ∑ ℓ n − wm k + ℓ − p ℓ x mn + r + ∂ [ r + ↑ n + r − ] ( x k + ℓ − − mn + r ) τ ω [ r + n + r ] where p ℓ ∈ K [ x , . . . , x n + r ] , with p n − w ∈ K × . Since cX ki ∈ ker ( Φ M ) , the second sum vanishes (this canalso be checked directly from the fact that k w − τ E i , M ( c k ) = T M ∑ ℓ n − w p ℓ τ [ r + ↑ n + r − ] i , β − i ⊗ H β − i x k + ℓ n + r c ! ,and δ ( cX ki ) = ∑ ℓ n − w p ℓ τ [ r + ↑ n + r − ] i , β − i ⊗ H i β − i x k + ℓ n + r c .So Γ δ ( cX ki ) = ∑ ℓ n − wu w − p ℓ ∂ [ r + ↑ n + r − ] ( x k + ℓ + un + r ) cX ui .We have ∂ [ r + ↑ n + r − ] ( x k + ℓ + un + r ) = u > w − − k , and we get Γ δ ( cX ki ) = p n − w cX w − − ki + ∑ w − − k <ℓ w − MX ℓ i .Since p n − w ∈ K × , we have established equation (5.4). So Γ δ is an isomorphism and the proof is complete. (cid:3) Hence, we have proved that Theorem 5.6 holds for all the subquotients of modules of the formad ( n ) E i ( E j r . . . E j ) . To conclude that Theorem 5.6 holds for every module in H [ i ] , we use the followingobvious lemma. Lemma 5.14.
Let C , D be abelian categories, G , G ′ : C → D be exact functors, and α : G → G ′ be a naturaltransformation. Assume that we have a collection M of objects of C such that C is generated by M as a Serresubcategory, and α M is an isomorphism for any subquotient M of an object of M . Then α is an isomorphism. By Theorem 4.30, modules of the form ad ( n ) E i ( E j r . . . E j ) generate H [ i ] as a Serre subcategory. SinceTheorem 5.6 holds for their subquotients, we conclude that it holds for every module in H [ i ] .6. P ROJECTIVE RESOLUTIONS In U , there are simple formulas for ad ne i and ad ( n ) e i that can be obtained by induction. For all y ∈ U ofweight β ∈ Q + we have ad ne i ( y ) = n ∑ k = ( − ) k q k ( n − + h i ∨ , β i ) i (cid:20) nk (cid:21) i e n − ki ye ki ,ad ( n ) e i ( y ) = n ∑ k = ( − ) k q k ( n − + h i ∨ , β i ) i e ( n − k ) i ye ( k ) i , (6.1)where (cid:20) nk (cid:21) i = [ n ] i ! [ k ] i ! [ n − k ] i !is the q i -binomial coefficient. In this section, we introduce complexes Ad nE i ( M ) and Ad ( n ) E i ( M ) for M ∈ H [ i ] . These complexes categorify the alternating sums (6.1). We prove that their cohomologyis concentrated in top degree, and is equal to ad nE i ( M ) and ad ( n ) E i ( M ) respectively. From this we obtainprojective resolutions for the generators of H [ i ] .The higher order quantum Serre relations in U + state that for j ∈ I \ { i } , m > n > − mc i , j wehave ad ( n ) e i ( e mj ) = ( n ) E i ( E mj ) is zero when n > − mc i , j ,as follows from Proposition 4.24. We also prove that the complex Ad ( n ) E i ( E mj ) is null-homotopic when n > − mc i , j , which categorifies the vanishing of the alternating sum. Convention.
All the complexes we write are cochain complexes (so they have degree +1 differential).Given a complex M = ( . . . → M r d r −→ M r + → . . . ) we denote by H r ( M ) = ker ( d r ) /Im ( d r − ) its r th cohomology group. Given complexes M , N with respective differentials d M , d N , and a morphism ofcomplexes f : M → N , its cone is the complex Cone ( f ) defined byCone ( f ) r = M r + ⊕ N r ,with differential (cid:20) − d r + M f r d rN (cid:21) : M r + ⊕ N r → M r + ⊕ N r + .Then there is a long exact sequence in cohomology · · · → H k ( M ) H k ( f ) −−−→ H k ( N ) → H k ( Cone ( f )) → H k + ( M ) → · · · .6.1. Adjoint action as a complex.
We start by constructing the complex Ad nE i ( M ) for M ∈ H [ i ] . To dothis, we need to express the q i -binomial coefficients in a combinatorial way. Let P kn be the set of subsetsof {
1, . . . , n } containing k elements. For S ∈ P kn , we let Σ ( S ) be the sum of the elements of S . Then wehave (see [KC02, Theorem 6.1]) (cid:20) nk (cid:21) i = q − k ( n + ) i ∑ S ∈P kn q Σ ( S ) i . ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 39
With this expression, we can rewrite ad ne i ( y ) asad ne i ( y ) = n ∑ k = ( − ) k q k h i ∨ , β i i ∑ S ∈P kn q ( Σ ( S ) − k ) i e n − ki ye ki . (6.2)We construct a complex which categorifies this alternating sum. The construction is based on the fol-lowing elementary lemma. Lemma 6.1.
Let β ∈ Q + of height n. In H n + , the following relations hold τ [ ↑ n + ] τ [ ↑ n ] τ n + i , β = τ τ [ ↑ n + ] τ [ ↑ n ] i , β , τ n + τ [ n ↓ ] τ [ n + ↓ ] β ,2 i = τ [ n ↓ ] τ [ n + ↓ ] τ β ,2 i . Proof.
This just follows from applying the braid relation (8) of Definition 3.2 repeatedly, noticing that weare always in the case where the polynomial error term is zero. (cid:3)
Definition 6.2.
Let M ∈ H i β − mod, n > w = h i ∨ , β i . We define a complex Ad nE i ( M ) of H β + ni -modules of the form0 → q n ( w + n − ) i ME ni → · · · → M S ∈P kn q kw + ( Σ ( S ) − k ) i E n − ki ME ki → · · · → E ni M → E ni M is in cohomological degree 0. To define the differential of Ad nE i ( M ) , we first define a map d S , S ′ : q kw + ( Σ ( S ) − k ) i E n − ki ME ki → q ( k − ) w + ( Σ ( S ′ ) − k + ) i E n + − ki ME k − i for any S ∈ P kn and S ′ ∈ P k − n asfollows: • if S ′ is not a subset of S , then d S , S ′ = • if S ′ = S \ { ℓ } , let a be the number of elements of S which are > ℓ . Then we define d S , S ′ = ( − ) a (cid:16) τ [ ↑ ℓ − − a ] ME k − i (cid:17) ◦ (cid:16) E n − ki τ E i , M E k − i (cid:17) ◦ (cid:16) E n − ki M τ [ k − a ↑ k − ] (cid:17) = ( E n − ki ME ki → E n − k + i ME k − i ,1 ( n − k ) i , β , i ⊗ H β m ( − ) a τ [ k − a + ↑ k − a + ℓ + | β | ] ( n − k + ) i , β , ( k − ) i ⊗ H β m .Notice that d S , S ′ has indeed the required degree − d i ( w + ( l − )) .Then, the differential of Ad nE i ( M ) is defined to be the direct sum of all maps d S , S ′ . This ends the definitionof Ad nE i ( M ) .It can be checked directly using Lemma 6.1 that the differential we defined on Ad nE i ( M ) indeedsquares to 0. Alternatively, we can also notice that Ad nE i ( M ) can be constructed inductively, as thefollowing proposition explains. Proposition 6.3.
Let M ∈ H i β − mod and let n > . There is a morphism of complexes τ E i ,Ad nEi ( M ) : q h i ∨ , β i + ni Ad nE i ( M ) E i → E i Ad nE i ( M ) defined on the component of cohomological degree − k as the direct sum over S ∈ P kn of the maps τ S = (cid:16) τ [ ↑ n − k ] ME ki (cid:17) ◦ (cid:16) E n − ki τ E i , M E ki (cid:17) ◦ (cid:16) E n − ki M τ [ ↑ k ] (cid:17) = ( E n − ki ME k + i → E n − k + i ME ki ,1 ( n − k ) i , β , ( k + ) i ⊗ H β m τ [ ↑ n + | β | ] ( n − k + ) i , β , ki ⊗ H β m . The cone of this morphism is Ad n + E i ( M ) . Proof.
We will denote the differential of Ad nE i ( M ) by d n . Let us check that the defined map is indeed amorphism of complexes. We need to prove that for S ∈ P kn and S ′ ∈ P k − n the diagram E n − ki ME k + i d nS , S ′ E i / / τ S (cid:15) (cid:15) E n − k + i ME ki τ S ′ (cid:15) (cid:15) E n − k + i ME ki E i d nS , S ′ / / E n − k + i ME k − i commutes. It is clear if S ′ is not a subset of S , because then d nS , S ′ =
0. If S ′ = S \ { ℓ } , let a be the numberof elements in S that are > ℓ . We have τ S ′ ◦ d nS , S ′ E i = ( E n − ki ME k + i → E n − k + i ME k − i ( n − k ) i , β , ( k + ) i ⊗ H β m ( − ) a τ [ k − a + ↑ k − a + ℓ + | β | + ] τ [ ↑ n + β ] ( n − k + ) i , β , ki ⊗ H β mE i d nS , S ′ ◦ τ S = ( E n − ki ME k + i → E n − k + i ME k − i ( n − k ) i , β , ( k + ) i ⊗ H β m ( − ) a τ [ ↑ n + β ] τ [ k − a + ↑ k − a + ℓ + | β | ] ( n − k + ) i , β , ki ⊗ H β m By Lemma 6.1, we have τ [ k − a + ↑ k − a + ℓ + | β | + ] τ [ ↑ n + β ] ( n − k + ) i , β , ki = τ [ ↑ n + β ] τ [ ↑ n + β ] τ [ k − a + ↑ k − a + ℓ + | β | ] ( n − k + ) i , β , ki .Hence the diagram commutes, and the morphism of complexes τ E i ,Ad nEi ( M ) is well-defined.We now need to check that the cone of this morphism is Ad n + E i ( M ) . Let w = h i ∨ , β i . The term of thecone in cohomological degree − k is given by q w + ni M S ∈P k − n q ( k − ) w + ( Σ ( S ) − k + ) i E n + − ki ME ki M M S ∈P kn q kw + ( Σ ( S ) − k ) i E n − ki ME ki = M S ∈P k − n q kw + ( Σ ( S )+ n + − k ) i E n + − ki ME ki M M S ∈P kn q kw + ( Σ ( S ) − k ) i E n − ki ME ki .The terms of the first sum can be indexed by the S ∈ P kn + such that n + ∈ S , and the terms ofthe second sum by the S ∈ P kn + such that n + ∈ S , and we see that we indeed get the term ofcohomological degree − k of Ad n + E i ( M ) . Finally, we check that the differential of the cone matches thatof Ad n + E i ( M ) . Let S ∈ P kn + and S ′ ∈ P k − n + . With the parametrization of the terms of the cone givenabove, we let d ′ S , S ′ be the differential of the cone from the term indexed by S to the term indexed by S ′ .By definition of the cone, we have d ′ S , S ′ = τ S = d n + S , S \{ n + } if n + ∈ S and S ′ = S \ { n + } ,0 if n + ∈ S and S ′ = S \ { n + } , E i d nS , S ′ if n + ∈ S , S ′ , − d nS \{ n + } , S ′ \{ n + } E i if n + ∈ S , S ′ .From this, we see easily that d ′ S , S ′ = d n + S , S ′ . (cid:3) We can now prove the main result regarding the cohomology of Ad nE i . Theorem 6.4.
Let M ∈ H i β − mod and let n > . ThenH k ( Ad nE i ( M )) = (cid:26) ad nE i ( M ) if k = otherwise.Proof. We proceed by induction on n . The result is clear if n =
0. Assume that the theorem is proved forAd nE i ( M ) . By Proposition 6.3 there is a distinguished triangle q h i ∨ , β i + ni Ad nE i ( M ) E i → E i Ad nE i ( M ) → Ad n + E i ( M ) [ ] −→ . ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 41
It gives rise to a long exact sequence in cohomology · · · → H k − ( Ad n + E i ( M )) → q h i ∨ , β i + ni H k ( Ad nE i ( M )) E i → E i H k ( Ad nE i ( M )) → H k ( Ad n + E i ( M )) → · · · We deduce from this exact sequence that H k ( Ad n + E i ( M )) = k = −
1, and that there is an exactsequence 0 → H − ( Ad n + E i ( M )) → q h i ∨ , β i + ni ad nE i ( M ) E i → E i ad nE i ( M ) → H ( Ad n + E i ( M )) → q h i ∨ , β i + ni Ad nE i ( M ) E i → E i Ad nE i ( M ) in Proposition 6.3, we see that theinduced map q h i ∨ , β i + ni ad nE i ( M ) E i → E i ad nE i ( M ) is simply τ E i ,ad nEi ( M ) . By Theorem 4.18, this map isinjective with cokernel ad n + E i ( M ) . The result follows. (cid:3) Divided powers complex.
We now introduce a similar complex for the divided powers.
Definition 6.5.
Let M ∈ H i β − mod with β of height r , and n >
0. We define a complex Ad ( n ) E i ( M ) of theform 0 → q n ( n + w − ) i ME ( n ) i → · · · → q k ( n + w − ) i E ( n − k ) i ME ( k ) i → · · · → E ( n ) i M → E ( n ) i M is in cohomological degree 0 and w = h i ∨ , β i . The differential of Ad ( n ) E i ( M ) is defined by d k = ( − ) k − (cid:16) τ [ ↑ n − k ] ME k − i (cid:17) ◦ (cid:16) E n − ki τ E i , M E k − i (cid:17) = ( q k ( n + w − ) i E ( n − k ) i ME ( k ) i → q ( k − )( n + w − ) i E ( n − k + ) i ME ( k − ) i τ ω [ k + r , n + r ] τ ω [ k ] ( n − k ) i , β , ki ⊗ H β m ( − ) k − τ ω [ k + r , n + r ] τ ω [ k ] τ [ k ↑ n + r ] ( n − k + ) i , β , ( k − ) i ⊗ H β m Let us explain why this is well-defined. First, by Lemma 6.1, we have τ ω [ k + r , n + r ] τ ω [ k ] τ [ k ↑ n + r ] ( n − k + ) i , β , ( k − ) i = τ [ ↑ k + r − ] τ ω [ k − + r , n + n ] τ ω [ k − ] ( n − k + ) i , β , ( k − ) i which proves that d k indeed takes values in the summand E ( n − k + ) i ME ( k − ) i of E n − k + i ME k − i . Further-more, in H β + ni ( n − k + ) i , β , ( k − ) i we have τ ω [ k + r , n + r ] τ ω [ k ] τ [ k ↑ n + r ] τ [ k − ↑ n + r ] = τ ω [ k + r , n + r ] τ ω [ k ] τ k − τ [ k ↑ n + r ] τ [ k − ↑ n + r − ] = d k − d k =
0, and we have indeed defineda complex.We now show, as in Theorem 6.4, that the cohomology of Ad ( n ) E i ( M ) is concentrated in degree zero. Theorem 6.6.
Let M ∈ H [ i ] and let n > . ThenH k ( Ad ( n ) E i ( M )) = ( ad ( n ) E i ( M ) if k = otherwise.Proof. Assume that M ∈ H β − mod with β ∈ Q + of height r . The proof is by induction on n . The resultis clear if n =
0. Assume the result proved for some n >
0. There is a morphism τ E i ,Ad ( n ) Ei ( M ) : q h i ∨ , β i + ni Ad ( n ) E i ( M ) → E i Ad ( n ) E i ( M ) defined on the component of cohomological degree − k to be the map τ S = (cid:16) τ [ ↑ n − k ] ME ki (cid:17) ◦ (cid:16) E n − ki τ E i , M E ki (cid:17) ◦ (cid:16) E n − ki M τ [ ↑ k ] (cid:17) = ( E ( n − k ) i ME ( k ) i E i → E i E ( n − k ) i ME ( k ) i , τ ω [ k + + r , n + r + ] τ ω [ k + ] ( n − k ) i , β , ( k + ) i ⊗ H β m τ ω [ k + + r , n + r + ] τ ω [ k + ] τ [ ↑ n + r ] ( n − k + ) i , β , ki ⊗ H β m .Note that this just the restriction of the morphism defined in Proposition 6.3, and we prove similarlythat this is well-defined. Let Υ be the cone of τ E i ,Ad ( n ) Ei ( M ) . By the same argument as the proof of Theorem6.4, Υ has its cohomology concentrated in degree zero and equal to ad E i ( ad ( n ) E i ( M )) . We now prove that Ad ( n + ) E i ( M ) is a direct summand of Υ , up to a shift. First, we define a map G : Ad ( n + ) E i ( M ) → Υ as follows: on the component of cohomological degree − k , G is the canonicalinclusion of E ( n + − k ) i ME ( k ) i in the first summand of E ( n + − k ) i ME ( k − ) i E i ⊕ E i E ( n − k ) i ME ( k ) i . It followseasily from the definitions that G is a morphism of complexes.We need to define a left inverse to G . We will denote by h d the complete symmetric polynomial ofdegree d . Recall the idempotent e n = x . . . x n − n τ ω [ n ] ∈ H ni . We define new idempotents f k , f ′ k in H β +( n + ) i by f k = ( e n + − k ⋄ β ⋄ e k ) , f ′ k = ( − ) k − h k − ( x , x k + r + , . . . , x n + r + )( e n + − k ⋄ β ⋄ (( e k − ⋄ i ) τ [ ↑ k − ] )) .Note that f k f ′ k = f k and f ′ k f k = f ′ k . We have a morphism ( E n + − ki ME ki → E ( n + − k ) i ME ( k ) i ( n + − k ) i , β , ki ⊗ H β m f k ⊗ H β m that we will just denote f k , and similarly for f ′ k . Then we define a morphism F : Υ → Ad ( n + ) E i ( M ) asfollows: on the component of cohomological degree − k , is defined by the row matrix h f ′ k x kn + r f k i : E ( n + − k ) i ME ( k − ) i E i M E i E ( n − k ) i ME ( k ) i → E ( n + − k ) i ME ( k ) i .It is clear that FG =
1, we just need to check that F is indeed a morphism of complexes. This amountsto checking the following relations in H β +( n + ) i ( n + − k ) i , β , ( k − ) i ⊗ H β H i β : x kn + r + f k τ [ k ↑ n + r ] = ( i ◦ e n − k ⋄ β ◦ e k ) τ [ k ↑ n + r − ] x k − n + r + f k − , f ′ k τ [ k ↑ n + r ] = ( e n + − k ⋄ β ⋄ e k − ◦ i )( τ [ ↑ n + r ] x k − n + r + f k − − τ [ k ↑ n + r ] f ′ k − ) .The first relation follows from the following computation ( i ⋄ e n − k ⋄ β ⋄ e k ) τ [ k ↑ n + r − ] x k − n + r + f k − = x k − n + r + ( x k + r + . . . x n − k − n + r )( ( n + − k ) i ⋄ β ⋄ e k ) τ ω [ k + r + n + r ] τ [ k ↑ n + r − ] ( e n + − k ⋄ β ⋄ e k − )= x k − n + r + ( x k + r + . . . x n − k − n + r )( ( n + − k ) i ⋄ β ⋄ e k ) τ [ k ↑ k + r − ] τ ω [ k + r , n + r ] ( e n + − k ⋄ β ⋄ e k − )= x k − n + r + ( x k + r + . . . x n − k − n + r )( ( n + − k ) i ⋄ β ⋄ e k ) τ [ k ↑ k + r − ] x n + − kn + r + ( τ ω [ n + − k ] ⋄ β ⋄ e k − )= x nn + r + ( x k + r + . . . x n − k − n + r )( τ ω [ n + − k ] ⋄ β ⋄ e k ) τ [ k ↑ n + r ] = x kn + r + ( e n + − k ⋄ β ⋄ e k ) τ [ k ↑ n + r ] .Note that this computation actually holds in H β +( n + ) i ( n + − k ) i , β , ( k − ) i , without needing to tensor by H i β on the right.For the second equality, we start by computing ( e n + − k ⋄ β ⋄ e k − ◦ i ) τ [ ↑ n + r ] x k − n + r + f k − and ( e n + − k ⋄ β ⋄ e k − ⋄ i ) τ [ k ↑ n + r ] f ′ k − separately. We have ( e n + − k ⋄ β ⋄ e k − ⋄ i ) τ [ ↑ n + r ] x k − n + r + f k − = ( e n + − k ⋄ β ⋄ e k − ⋄ i ) τ [ ↑ n + r ] x k − n + r + ( e n + − k ⋄ β ⋄ e k − )= (( x . . . x n − kn + − k ) ⋄ β ⋄ e k − ⋄ i ) τ [ ↑ k + r − ] h k − ( x k + r , . . . , x r + n + )( τ ω [ n + − k ] ⋄ β ⋄ e k − )= ( e n + − k ⋄ β ⋄ e k ◦ i ) τ [ ↑ k + r − ] h k − ( x k + r , . . . , x r + n + ) τ [ k + r ↑ n + r ] .Now in H β +( n + ) i ( n + − k ) i , β , ( k − ) i ⊗ H β H i β we have τ [ k ↑ k + r − ] x k + r = x k τ [ k ↑ k + r − ] because ( τ [ k ↑ k + r − ] x k + r − x k τ [ k ↑ k + r − ] ) i , β , ( k − ) i ∈ ( i , β − i , i , ( k − ) i ) (this is the same argument that we usedto prove that τ E i , M is well-defined in Proposition 4.13). Hence we conclude that ( e n + − k ⋄ β ⋄ e k − ⋄ i ) τ [ ↑ n + r ] x k − n + r + f k − = ( e n + − k ⋄ β ⋄ e k ⋄ i ) τ [ ↑ k − ] h k − ( x k , x k + r + . . . , x n + r + ) τ [ k ↑ n + r ] . ATEGORIFICATION OF THE ADJOINT ACTION OF QUANTUM GROUPS 43
Similarly, we have ( e n + − k ⋄ β ⋄ e k − ⋄ i ) τ [ k ↑ n + r ] f ′ k − = ( e n + − k ⋄ β ⋄ e k − ⋄ i ) h k − ( x , x k , x k + r + , . . . , x n + r + ) τ [ ↑ k − ] τ [ k ↑ n + r ] To conclude, we use the following formula in the affine nil Hecke algebra H k : τ ω [ k ] τ [ ↑ k − ] x ak = τ ω [ k ] x a τ [ ↑ k − ] + τ ω [ k ] f a − ( x , x k ) τ [ ↑ k − ] .This is easily derived by induction on a . Then we have ( e n + − k ⋄ β ⋄ e k − ⋄ i ) (cid:16) τ [ ↑ n + r ] x k − n + r + f k − − τ [ k ↑ n + r ] f ′ k − (cid:17) = ( e n + − k ⋄ β ⋄ e k − ⋄ i ) (cid:16) τ [ ↑ k − ] h k − ( x k , x k + r + . . . , x n + r + ) − h k − ( x , x k , x k + r + , . . . , x n + r + (cid:17) τ [ ↑ k − ] ) τ [ k ↑ n + r ] = ( e n + − k ⋄ β ⋄ e k − ⋄ i ) h k − ( x , x k + r + . . . , x n + r + ) τ [ ↑ n + r ] = f ′ k τ [ k ↑ n + r ] .This completes the proof that FG = ( n + ) E i ( M ) is a direct factor of Υ , hence has cohomology only in degree 0. Furthermore, ondegree zero cohomology, the morphism GF of Υ induces the idempotent projecting on the summandad ( n + ) E i ( M ) of ad E i ( ad ( n ) E i ( M )) . Hence H ( Ad ( n + ) E i ( M )) = ad ( n + ) E i ( M ) . (cid:3) Projective resolutions and Serre relations.
Using Theorems 6.4 and 6.6, we can construct projec-tive resolutions.
Proposition 6.7.
Let M be a module in H [ i ] which is projective as a module in H . Then Ad nE i ( M ) (resp. Ad ( n ) E i ( M ) ) is a projective resolution of ad nE i ( M ) (resp. ad ( n ) E i ( M ) ) in H . Typically, this applies to M = E j . . . E j r for j , . . . , j r = i . In particular, we have obtained projectiveresolutions for the generators of H [ i ] .We now state the categorical version of the quantum higher Serre relations. Theorem 6.8.
Let j ∈ I \ { i } and let m > . If n > − mc i , j , then Ad ( n ) E i ( E mj ) is null-homotopic.Proof. By Proposition 6.7, Ad ( n ) E i ( E mj ) is a projective resolution of ad ( n ) E i ( E mj ) . Assume n > − mc i , j . ByProposition 4.24, H ini + mj =
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