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 $1+q+\cdots +q^{n-1}\ne 0$ 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 $V$ .