On the graded algebras associated with Hecke symmetries

Abstract

© 2020 European Mathematical Society Publishing House. All rights reserved. A Hecke symmetry R on a finite dimensional vector space V gives rise to two graded factor algebras S(V;R) and Λ(V;R) of the tensor algebra of V which are regarded as quantum analogs of the symmetric and the exterior algebras. Another graded algebra associated with R is the Faddeev-Reshetikhin-Takhtajan bialgebra A.R/ which coacts on S(V;R) and .Λ(V;R). There are also more general graded algebras defined with respect to pairs of Hecke symmetries and interpreted in terms of quantum hom-spaces. Their nice behaviour has been known under the assumption that the parameter q of the Hecke relation is such that 1 C q C C qn-1 ≠ 0 for all n > 0. The present paper makes an attempt to investigate several questions without this condition on q. Particularly we are interested in Koszulness and Gorensteinness of those graded algebras. For q a root of 1 positive results require a restriction on the indecomposable modules for the Hecke algebras of type A that can occur as direct summands of representations in the tensor powers of V.