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A diagrammatic approach to categorification of quantum groups. I. (English) Zbl 1188.81117

From the introduction: The goal of this paper is to categorify \(U^- = U^-_q (\mathfrak g)\), for an arbitrary simply laced Kac-Moody algebra \(\mathfrak g\). Here \(U^-\) stands for the quantum deformation of the universal enveloping algebra of the “lower-triangular” subalgebra of \(\mathfrak g\).
Following the discovery of quantum groups \(U_q (\mathfrak g)\) by Drinfeld and Jimbo, Ringel found a Hall algebra interpretation of the negative half \(U^-\) of the quantum group in the simply-laced Dynkin case. Lusztig gave a geometric interpretation of \(U^-\) and produced a canonical basis there via a sophisticated approach which required the full strength of the theory of \(l\)-adic perverse sheaves. Kashiwara defined a crystal basis of \(U^-\) at 0, a graph equipped with extra data, and constructed the so-called global crystal basis of \(U^-\). Grojnowski and Lusztig proved that the global crystal basis and the canonical basis are the same. The canonical basis \(\mathbf B\) of \(U^-\) gives rise to bases in all irreducible integrable \(U\)-representations. Lusztig also produced an idempotent version \(\dot U\) of \(U\) and defined a basis there.
The work of Ariki can be viewed as a categorification of the restricted dual of \(U^-(\mathfrak g)\) for \(\mathfrak g = \mathfrak{sl}_N\) and \(\mathfrak g = \widehat{\mathfrak{sl}}_N\) and a categorification of all irreducible integrable representations of these Lie algebras. An integral version of the restricted dual of \(U^-(\mathfrak g)\) becomes the sum of Grothendieck groups of suitable blocks of affine Hecke algebra representations. An earlier work of Zelevinsky can be understood in this context as a parametrization of basis elements of \(U^-(\mathfrak g)^*\) via certain irreducible representations of affine Hecke algebras. Irreducible integrable representations of \(U(\mathfrak g)\) become Grothendieck groups of Ariki-Koike cyclotomic Hecke algebras, which are certain finite-dimensional quotient algebras of affine Hecke algebras.
Grojnowski found a purely algebraic way to understand these categorifications via a generalization of Kleshchev’s methods for studying modular representations of the symmetric group. This approach was further developed by Grojnowski and Vazirani, Vazirani, Brundan and Kleshchev and others. It is explained by Kleshchev in the context of degenerate affine Hecke algebras.
In this paper we introduce graded algebras categorifying \(U_q (\mathfrak g)\), for an arbitrary simply-laced \(\mathfrak g\). We start with an unoriented graph \(\Gamma\) without loops and multiple edges. Let \(I\) be the set of vertices of \(\Gamma\). The bilinear Cartan form on \(N[I]\) is given on the basis elements \(i, j \in I\) by \[ i \cdot j =\begin{cases} 2 \text{ if } i = j,\\ -1 \text{ if }i \text{ and } j \text{ are joined by an edge},\\ 0 \text{ otherwise}.\end{cases} \]
The algebra \(U^-\) over \(Q(q)\), the negative (or positive) half of the quantum universal enveloping algebra, has generators \(\theta_i\), \(i\in I\), and defining relations \[ \theta_i\theta_j = \theta_j\theta_i\qquad \text{if } i \cdot j = 0, \]
\[ (q + q^{-1})\theta_i\theta_j\theta_i = \theta_i^2\theta_j + \theta_j^2\theta_i\qquad \text{if } i \cdot j = -1. \]
The algebra \(U^-\) contains a subring \({}_A\mathbf f\), which is the \(\mathbb Z[q, q^{-1}]\)-lattice generated by all products of quantum divided powers \(\theta_i^{(a)}\). The canonical basis \(\mathbf B\) is a basis of \({}_A\mathbf f\) viewed as a free \(\mathbb Z[q, q^{-1}]\)-module.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16T20 Ring-theoretic aspects of quantum groups
18D35 Structured objects in a category (MSC2010)
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