Finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids
Finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids
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由齐次二次单项式关系及其底层幺半群定义的有限生成代数
DOI：
10.1016/j.jalgebra.2015.05.017
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发表时间：
2015
影响因子：
0.9
通讯作者：
M. V. Campenhout
中科院分区：
文献类型：
--
作者：
E. Jespers;J. Okniński;M. V. Campenhout
我们考虑域K上的代数，其生成元x 1，x 2，.，xn服从（n 2）二次关系x i x j= x k x l与（i，j）<$（k，l），而且每个单项式x i x j在其中一个定义关系中至多出现一次。如果这些关系是非退化的，那么它表明，代数是左和右诺特，满足多项式单位元，并有Gelfand-Kirillov维数至多n。如果定义关系是无平方的，这已经由Gateva-Ivanova，Jespers和Okniovski建立。为了证明这些结果，我们调查的基础幺半群S的结构，定义相同的介绍。它被称为二次幺半群。我们发现，有一个强有力的联系，整除幺半群和幺半群的I型（也称为YB-幺半群）。I型幺半群是非退化二次幺半群的例子。它们是Yang-Baxter方程的非退化对合集合论解的么半群解释，近年来Gateva-Ivanova和货车den Bergh、Etingof、Schedler和Soloviev、Jespers和Okniovski等学者对它们进行了深入的研究。可除性幺半群是由Kuske引入的。这些是包含I型幺半群类的可消幺半群。我们表明，他们有一个介绍，最多（n 2）的关系，如果他们有精确的（n 2）定义的关系，那么他们是幺半群的I型。
We consider algebras over a field K with generators x 1, x 2,…, x n subject to (n 2) quadratic relations of the form x i x j= x k x l with (i, j)≠(k, l) and, moreover, every monomial x i x j appears at most once in one of the defining relations. If these relations are non-degenerate then it is shown that the algebra is left and right Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension at most n. In case the defining relations are square-free this was already established by Gateva-Ivanova, Jespers and Okniński. To prove these results we investigate the structure of the underlying monoid S, defined by the same presentation. It is called a quadratic monoid. We show that there is a strong link with the divisibility monoids and monoids of I-type (also referred to as YB-monoids). Monoids of I-type are examples of non-degenerate quadratic monoids. They are a monoid interpretation of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation and have been studied quite intensively in recent years, by Gateva-Ivanova and Van den Bergh, Etingof, Schedler and Soloviev, Jespers and Okniński. Divisibility monoids have been introduced by Kuske. These are cancellative monoids that include the class of monoids of I-type. We show that they have a presentation with at most (n 2) relations and if they have precisely (n 2) defining relations then they are monoids of I-type.