A diagrammatic approach to categorification of quantum groups II

We categorify one-half of the quantum group associated to an arbitrary Cartan datum.

We set d ij = −2 i•j i•i ∈ AE.To a Cartan datum assign a graph Γ with the set of vertices I and an edge between i and j if and only if i • j = 0. We recall the definition of the negative half of the quantum group associated to a Cartan datum, following [6].Let [1] i .Let ′ f be the free associative algebra over É(q) with generators θ i , i ∈ I and denote θ (n) i = θ n i /[n] i !.We equip ′ f with an AE[I]-grading by assigning to θ i grading i.The tensor square ′ f ⊗ ′ f is an associative algebra with the multiplication There is a unique algebra homomorphism r : ′ f−→ ′ f ⊗ ′ f given on generators by r(θ i ) = θ i ⊗ 1 + 1 ⊗ θ i . 1 Proposition 1. Algebra ′ f carries a unique É(v)-bilinear form such that (1, 1) = 1 and • (θ i , θ j ) = δ i,j (1 − q 2 i ) −1 for all i, j ∈ I, • (x, yy ′ ) = (r(x), y ⊗ y ′ ) for x, y, y ′ ∈ ′ f, • (xx ′ , y) = (x ⊗ x ′ , r(y)) for x, x ′ , y ∈ ′ f.This bilinear form is symmetric.
The radical I of (, ) is a two-sided ideal of ′ f.The bilinear form descends to a non-degenerate bilinear form on the associative É(q)-algebra f = ′ f/I.The AE[I]-grading also descends: The quantum version of the Gabber-Kac theorem says the following.
Proposition 2. The ideal I is generated by the elements Thus, f is the quotient of ′ f by the so-called quantum Serre relations Denote by A f the [q, q −1 ]-subalgebra of f generated by the divided powers θ (a) i , over all i ∈ I and a ∈ AE.
Algebras R(ν).As in [4], we consider braid-like planar diagrams, each strand labelled by an element of I, and impose the following relations (2) For each ν ∈ AE[I] define the graded ring where j R(ν) i is the abelian group of all linear combinations of diagrams with bot(D) = i and top(D) = j modulo the relations ( 2)-( 7) and Seq(ν) is the set of weight ν sequences of elements of I.The multiplication is given by concatenation.Degrees of the generators are The rest of [4, Section 2.1] generalizes without difficulty to an arbitrary Cartan datum.To define the analogue of the module Poℓ ν over R(ν), we choose an orientation of each edge of Γ, then faithfully follow the exposition in Section 2.3 of [4], only changing the action of δ k,i in the last of the four cases to Computations in the nilHecke ring.In this section we slightly enhance the graphical calculus for computations in the nilHecke ring and record several lemmas to be used in the proof of categorified quantum Serre relations below.We use notations from Section 2.2 of [4].
A box with n incoming and n outgoing edges and ∂(n) written inside denotes the longest divided difference ∂ w 0 , the nonzero product of n(n−1) A box labelled e i,n denotes the corresponding idempotent in R(ν): Remark 4. Similar-looking diagrams are used in the graphical calculus of Jones-Wenzl projectors, see [3], but the latter has no direct relation to the graphical calculus in our paper.
Lemma 5. We have Proof is by induction on n: The first equality uses that x 1 x 2 . . .x n−1 is central in the nilHecke ring NH n−1 , allowing us to move these dots across ∂(n − 1).The second equality is the induction hypothesis.
The lemma implies the following graphical identities: The following also hold: For each i ∈ I the ring R(mi) is isomorphic to the nilHecke ring.The grading of a dot is now i•i, while that of a crossing is −i•i.For this reason one needs to generalize the grading convention described in [4, Section 2.2] and define i,m P to be the right graded projective module e i,m R(mi The Grothendieck group, bilinear form and projectives.We retain all notations and assumptions from [4], working over a field k, denoting by K 0 (R(ν)) the Grothendieck group of the category R(ν)−pmod of graded finitely-generated projective left R(ν)-modules and forming the direct sum where and The character ch(M) of an R(ν)-module M, the divided power sequences Seqd(ν), and idempotents 1 i for i = i Define graded left, respectively right, projective module and also write α + a,b when i and j are fixed.To prove the categorified quantum Serre relations, we assume that a By adding vertical lines on the left and on the right of the diagram, α + a,b can be viewed, more generally, as an element of for any sequences i ′ , i ′′ and the corresponding ν.We can replace sequences i ′ and i ′′ by dots to simplify notation.
Left multiplication by α + a,b is a homomorphism of projective modules terminating on the left at and on the right at Furthermore, Therefore, as elements of R(ν), see below: (−1) a−1 e i,a e i,b i i i i j i i i , as endomorphisms of the projective module (a,b) P , since e i,a ⊗ 1 j ⊗ e i,b acts by the identity on (a,b) P . Likewise, e i,d e i,d+1 e i,d+1 , where d ′ = d ji , and α − (1,d) α + (0,d+1) = Id, as endomorphisms of (0,d+1) P .A similar computation shows that as elements of R(ν), and as endomorphisms of (d+1,0) P .Proposition 6.For each i, j ∈ I, i = j there are isomorphisms of graded right projective modules Proof.When i • j = 0 the isomorphism reads ...ji... P ∼ = ...ij... P and is given by left multiplication by the ij intersection.When i • j < 0, earlier computations show that the maps given by are mutually-inverse isomorphisms, implying the proposition.Maps α ′ , α ′′ together are given by summing over all arrows in the diagram (24), with every fourth arrow appearing with the minus sign.

Corollary 7.
For each i, j ∈ I, i = j there are isomorphisms of graded left projective modules Proposition 6 and Corollary 7 generalize Proposition 2.13 in [4] and can be considered a categorification of the quantum Serre relations.Corollaries 2.14 and 2.15 of [4], establishing quantum Serre relations for the characters of any M ∈ R(ν)−mod, generalize to an arbitrary Cartan datum in the same way.
Due to the quantum Gabber-Kac theorem, this homomorphism is injective.Surjectivity of γ follows from the arguments identical to those given in [4, Section 3.2], which, in turn, were adopted from [5, Section 5].Alternatively, the arguments could be adopted from [2] and [9]; we settled on using a single source.We obtain This theorem holds without any restrictions on the Cartan datum and on the ground field k over which R(ν) is defined.All other results and observations of Sections 3.2 and 3.3 of [4] extend to the general case as well.The cyclotomic quotients of R(ν) described in [4,Section 3.4] generalize to an arbitrary Cartan datum.
It would be interesting to relate our construction to Lusztig's geometric realization of U − in the non-simply laced case [6] and to Brundan-Kleshchev's categorification [1], [5] of U − q=1 in the affine Dynkin case A A multi-grading.For every pair (i, j) of vertices of Γ, algebras R(ν) can be equipped with an additional grading, by assigning degrees −1 and 1 to the ij and ji crossings, respectively, deg and degree 0 to all other diagrammatic generators of R(ν).These gradings are independent, and together with the principal grading, introduced above, make R(ν) into a multi-graded ring (with n(n−1) 2 + 1 independent gradings where n = |Supp(ν)|).The direct sum of the categories of multi-graded finitely-generated projective left R(ν)-modules, over all ν ∈ AE[I], categorifies a multi-parameter deformation [7], [8] of the quantum universal enveloping algebra U − , the quotient of the free associative algebra on θ i , i ∈ I, by the relations where q ij are formal variables subject to conditions q ij q ji = 1.
Modifications in the simply-laced case.This section explains how to deform algebras R(ν) in the simply-laced case so that the main results of [4] will hold for the modified algebras.These deformations can be nontrivial only when the graph has cycles.As in [4], we start with an unoriented graph Γ without loops and multiple edges.Next, fix an orientation of each edge of Γ, work over a base field k, and, for each oriented edge i −→ j, choose two invertible elements τ ij and τ ji in k.Denote such a datum {orientations, invertible elements} by τ .
For each ν ∈ AE[I] consider k-vector space Poℓ ν defined as in [4].This space is the sum of polynomial rings in |ν| variables, over all sequences in Seq(ν).Define R τ (ν) to be the endomorphism algebra of Poℓ ν generated by the endomorphisms 1 i , x k,i , δ k,i , over all possible k and i , with the action as in [4, Section 2.3], with the only difference being the action of δ k,i in the last of the four cases: The algebra R τ (ν) has a diagrammatic description similar to that of R(ν), with the following defining relations Reverse the orientation of a single edge i − j and change τ ij to −τ ij and τ ji to −τ ji .Denote the new datum by τ ′ .Algebras R τ (ν) and R τ ′ (ν) are isomorphic via a map which is the identity on diagrams.This way, the study of R τ (ν) reduces to the case of any preferred orientation of Γ. Rescaling one of the two possible types of the ij crossing by λ ∈ k changes τ ij to λτ ij and τ ji to λτ ji while keeping the rest of the data fixed.We see that R τ (ν) depends only on products τ ij τ −1 ji , over all edges of Γ, via non-canonical isomorphisms.Rescalings of ii crossings and dots further reduce the number of parameters to the rank of the first homology group of Γ.When graph Γ is a forest (has no cycles), algebras R τ (ν) are all isomorphic to R(ν) via rescaling of generators.When Γ has a single cycle, rescaling of generators reduces this family of algebras to a one-parameter family, with the parameter taking values in k * .It is likely that R τ (ν) has a description via equivariant convolution algebras in Lusztig's geometrization [6] of U − when all τ ij = 1 (compare with Conjecture 1.2 in [4]). Form R τ (ν).
The Grothendieck group K 0 (R τ ) of the category of finitely-generated graded left projective modules can be naturally identified with the integral version A f of U − .All other essential constructions and results of [4] generalize from R(ν) to algebras R τ (ν) in a straightforward fashion.
Modifications in the general case.Rings R(ν) associated to an arbitrary Cartan datum admit similar modifications that depend on choosing an orientation of Γ and invertible elements τ ij , τ ji of the ground field k for each oriented edge i −→ j.The key point is the change in the definition of the endomorphism algebra, making δ k,i act by in the last of the four cases, with d = d i k+1 i k and d ′ = d i k i k+1 .Our proof of categorified quantum Serre relations for R(ν) requires only minor changes in the general case of R τ (ν).Everything else generalizes as well.