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Quiver Hecke algebras and 2-Lie algebras. (English) Zbl 1247.20002
This important paper provides an introduction to some basic constructions and results related to quiver Hecke algebras and $$2$$-representation theory of Kac-Moody algebras. Section 2 is an introduction to nil (affine) Hecke algebras of type $$A$$. The author recalls basic properties of Hecke algebras of the symmetric group, provides a construction for the nil Hecke algebras via BGG-Demazure operators and constructs symmetrizing forms.
Section 3 is devoted to quiver Hecke algebras. The author proposes a new simpler definition for these algebras by showing that the more complicated relation from the original definition follows from the other ones. Further, the author constructs a faithful polynomial representation and explains the relation with affine Hecke algebras (for type $$A$$ quivers). Finally, it is explained how to put together all quiver Hecke algebras associated with a quiver to obtain a monoidal category that categorifies a half Kac-Moody algebra and its quantum version.
Section 4 introduces $$2$$-Kac-Moody algebras (which categorify the usual Kac-Moody algebras) and discusses their $$2$$-representations (which categorify integrable representations of the corresponding Kac-Moody algebras). The authors provides some results which allow one to check that a category is equipped with a structure of an integrable $$2$$-representation and explains the universal construction of “simple” $$2$$-representations with a detailed description for $$\mathfrak{sl}_2$$. At the end of the section the author explains construction of Fock spaces from representations of symmetric groups in this framework.
The last section describes geometric methods available in the case of symmetrizable Kac-Moody algebras. The author shows that Lusztig’s category of perverse sheaves on the moduli space of representations of a quiver is equivalent to the monoidal category of quiver Hecke algebras. This provides a relation to Lusztig’s canonical basis. Finally, the author shows that microlocalization categories of sheaves can be endowed with the structure of a $$2$$-representation isomorphic to the universal simple $$2$$-representation, which again is applied to establish a connection with Lusztig’s canonical basis.

##### MSC:
 20C08 Hecke algebras and their representations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 16G20 Representations of quivers and partially ordered sets 20C30 Representations of finite symmetric groups
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