On the graded algebras associated with Hecke symmetries

Abstract

A Hecke symmetry R on a finite dimensional vector space V gives rise to two graded factor algebras \mathbb S (V, R) and \Lambda (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 \mathbb S (V, R) and \Lambda (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 + q + \cdots + q^{n-1} \neq 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 .