Generalized KLR Algebras and Mutation

GARCÍA IGLESIAS, AGUSTÍN

Montevideo

Congreso; Quantum 2019; 2019

Univ. de la República (Uruguay)

Khovanov-Lauda [KL] and Rouquier [R] independently constructed a graded additive monoidal category which categorifies certain quantum groups. The so-called KLR algebras are some morphism algebras in this category. As an example, the endomorphism algebra End(Ei^k), associated to a simple root object Ei, is the famous nilHecke algebra.In this talk we shall review this ideas and introduce twisted derivation algebras, built upon generalizations of the nilHecke algebra, and a mutation process which takes two such algebras and a linking polynomial Q, and produces a third twisted derivation algebra, corresponding to the commutator of two given roots. This is a step in the process of defining generalized KLR algebras, to categorify more general root vectors.This is a work in progress, joint with Ben Elias (U. of Oregon, USA).[KL] M. Khovanov, A. Lauda. A diagrammatic approach to categorification of quantum groups I. Representation Theory 13 (2009).[R] R. Rouquier. 2-kac-moody algebras. arXiv:0812.5023 (2008).