To each abelian ideal in a simple Lie algebra we associate a family of representation of the corresponding algebra of polynomial currents. In the case, when the abelian ideals satisfy some additional conditions, the representations are generalizations of the Kirillov--Reshetikhin modules. Moreover, under a suitable equivalence of categories, these modules correspond to right indecomposable projective modules of a finite--dimensional Koszul algebra.

Furthermore, we construct an infinite dimensional Koszul algebra associated to the abelian ideal with global dimension equal to the cardinality of the ideal. This is based on joint work with Jacob Greenstein.

Lecture #12683

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