An approach to categorification of Verma modules

Abstract

We give a geometric categorification of the Verma modules M (λ) for quantum sl2. ContentsFrom the work of Lauda in [34,35] adjusted to our context it follows that for each n 0 there is an action of the nilHecke algebra NH n on F n and on E n . As a matter of fact, there is an enlargement of NH n , which we denote as A n , acting on F n and E n , also admitting a nice diagrammatic description (see § 1.1.6 for a sketch).We define M as the direct sum of all the categories M k and functors F, E and Q in the obvious way. One of the main results in this paper is the following. † All our functors are, in fact, superfunctors which we tend to see as functors between categories endowed with a Z/2Z-action, whence the use of the terminology functor. ‡ We thank Aaron Lauda for explaining this to us.

A survey on categorification of Verma modules

“…In this series of lectures I give an introduction to my joint work with Grégoire Naisse on categorification of Verma modules for quantum Kac-Moody algebras [16,17,19].…”

A DG-extension of symmetric functions arising from higher representation theory

“…The definition. The extended nilHecke algebra NH ext n , first defined in [NV16], is a graded superalgebra with even generators x 1 , · · · , x n and ∂ 1 , · · · , ∂ n−1 , and odd generators ω 1 , · · · , ω n satisfying equations (1.1) and the following relations…”

“…For each w ∈ S n fix a reduced expression. A basis for the superalgebra NH ext n (m) is given in [NV16] by the set of elements…”

“…Recently, Naisse and Vaz [NV16] have introduced an extension of the nilHecke algebra NH ext n that we refer to as the extended nilHecke algebra. This algebra arose in the study of a fundamental issue in higher representation theory.…”

“…The cohomology rings of Grassmannian Gr(k, N ) can be thought of as the sl N homology of the k-colored unknot [Wu14,Yon11], while the ring of extended symmetric functions Λ ext k can be thought of as the HOMFLY-PT homology of the k-colored unknot [WW09]. The Koszul differential d N considered here and in [NV16] is then a special case of Rasmussen's sl N differential [Ras15]. We expect that the trimodules Ω n+k F (n) Ω k and Ω k−n E (n) Ω k appear in this setting as the homologies of certain MOY diagrams, namely the colored theta graphs.…”