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 $V$ .

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关于与Hecke对称有关的梯度代数

有限维向量空间$ V $上的Hecke对称$ R $产生张量代数$ V $的两个梯度因子代数$ \ mathbb S（V，R）$和$ \ Lambda（V，R）$被认为是对称代数和外部代数的量子类似物。与$ R $相关的另一个分级代数是Faddeev–Reshetikhin–Takhtajan双代数$ A（R）$，它与$ \ mathbb S（V，R）$和$ \ Lambda（V，R）$共同作用。还存在关于成对的Hecke对称性定义的更一般的渐变代数，并根据量子同空间进行了解释。在Hecke关系的参数$ q $使得所有$ n> 0 $都为$ 1 + q + \ cdots + q ^ {n-1} \ neq 0 $的假设下，已知它们的良好行为。本文试图研究在$ q $上没有这种条件的几个问题。尤其是我们对这些渐变代数的Koszulness和Gorensteinness感兴趣。对于$ q $，要得到1个正结果的根，必须限制$ A $类型的Hecke代数的不可分解模块，这些模块可以作为张量幂$ V $的表示的直接加和出现。