A diagrammatic approach to categorification of quantum groups II

Abstract

We categorify one-half of the quantum group associated to an arbitrary Cartan datum. Cartan data and algebras Af . A Cartan datum (I, ·) consists of a finite set I and a symmetric bilinear form on Z[I] taking values in Z, subject to conditions • i · i ∈ {2, 4, 6, . . .} for any i ∈ I, • 2 i·j i·i ∈ {0,−1,−2, . . .} for any i 6= j in I. We set dij = −2 i·j i·i ∈ N. 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 6= 0. We recall the definition of the negative half of the quantum group associated to a Cartan datum, following [6]. Let qi = q i·i 2 , [n]i = q n−1 i + q n−3 i + · · · + q i , [n]i! = [n]i[n − 1]i . . . [1]i. Let f be the free associative algebra over Q(q) with generators θi, i ∈ I and denote θ i = θ i /[n]i!. We equip f with an N[I]-grading by assigning to θi grading i. The tensor square f ⊗ f is an associative algebra with the multiplication (x1 ⊗ x2)(x ′ 1 ⊗ x ′ 2) = q ′1x1x ′ 1 ⊗ x2x ′ 2 for homogeneous x1, x2, x ′ 1, x ′ 2. There is a unique algebra homomorphism r : f−→f ⊗ f given on generators by r(θi) = θi ⊗ 1 + 1⊗ θi. 1 2 Proposition 1. Algebra f carries a unique Q(v)-bilinear form such that (1, 1) = 1 and • (θi, θj) = δi,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 Q(q)-algebra f = f/I. The N[I]-grading also descends: f = ⊕ ν∈N[I] fν . The quantum version of the Gabber-Kac theorem says the following. Proposition 2. The ideal I is generated by the elements ∑ a+b=dij+1 (−1)θ (a) i θjθ (b) i over all i, j ∈ I, i 6= j. Thus, f is the quotient of f by the so-called quantum Serre relations ∑ a+b=dij+1 (−1)θ (a) i θjθ (b) i = 0. (1) Denote by Af the Z[q, q]-subalgebra of f generated by the divided powers θ (a) i , over all i ∈ I and a ∈ N. 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 i j =    0 if i = j,

Cite this paper

@inproceedings{Khovanov2008ADA, title={A diagrammatic approach to categorification of quantum groups II}, author={Mikhail Khovanov and Aaron D. Lauda}, year={2008} }