一类Gelfand-Kirilov维数为3的广义Weyl代数的同调光滑性

发布时间：2018-04-20 11:57

本文选题：广义 + Weyl代数　；参考：《扬州大学》2017年硕士论文

【摘要】：同调光滑性是关于结合代数的一种同调性质.作为交换意义下光滑性的非交换版本,同调光滑性在非交换代数几何,量子群,算子代数,数学物理等数学领域都扮演着重要角色.许多同调光滑代数的Hochschild同调与上同调具有对偶性,亦即Van den Bergh对偶.一般而言,判断一个代数是否同调光滑是困难的.本论文研究了一类Gelfand-Kirilov维数为3的广义Weyl代数的同调光滑性.广义Weyl代数的概念是Bavula于1992年引入的,目的是研究一些类似于经典Weyl代数的代数.本文所讨论的广义Weyl代数取,以二元多项式环k[z1,z2]为子代数,由两个参数σ和φ决定,其中σ是代数k[z1,z2]的一个仿射型自同构,φ=φ(z_1,z_2)是一个非零的二元多项式.通过构造W的同伦双复形,我们首先得到了 W作为We-模的一个投射分解,然后找出了 W具有同调光滑性的一个充分条件.更准确地说,我们证明了:若φ,αφ/αz1,αφ/αz2生成的理想就等于k[z1,z2]本身,则W是同调光滑的.作为上述结论的应用,我们证明了量子群Oq(SL2)和U(sl2)都是同调光滑代数,这与K.A.Brown,J.J.Zhang在2008年取得的一个结果相吻合;还研究了陈惠香于1999年定义的代数M(1,q),证明M(1,q)模去一个正规正则元生成的主理想所得的商代数也是同调光滑的.文章的最后部分给出了与上述结论相关的一些前景和展望.
[Abstract]:Homology smoothness is a homology property of associative algebra. As a non-commutative version of smoothness in the sense of commutative homology smoothness plays an important role in the fields of noncommutative algebraic geometry quantum group operator algebra mathematical physics and so on. The Hochschild homology and cohomology of many homology smooth algebras are duality, that is, Van den Bergh duality. Generally speaking, it is difficult to judge whether an algebra is homology smooth. In this paper, we study the homology smoothness of a class of generalized Weyl algebras with Gelfand-Kirilov dimension 3. The concept of generalized Weyl algebra was introduced by Bavula in 1992 in order to study some algebras similar to classical Weyl algebras. The generalized Weyl algebra discussed in this paper takes the bivariate polynomial ring k [z1z2] as a subalgebra and is determined by two parameters 蟽 and 蠁, where 蟽 is an affine automorphism of the algebra k [z1z2], 蠁 = 蠁 z1zst2) is a nonzero binary polynomial. By constructing a homotopy bicomplex of W, we first obtain a projective decomposition of W as a We-module, and then find a sufficient condition for W to have homological smoothness. More accurately, we prove that if the ideal generated by 蠁, 伪蠁 / 伪 z 1, 伪蠁 / 伪 z 2 is equal to k [z 1z2] itself, then W is homologically smooth. As an application of the above conclusions, we prove that both the quantum group Oqn SL2) and Uttril 2) are homology smooth algebras, which is in agreement with a result obtained by K.A. Brownberg J. J. Zhang in 2008. In this paper, we also study the algebra M ~ (1) Q ~ (1) defined by Chen Huixiang in 1999, and prove that the quotient number of the principal ideal generated by M _ (1) Q _ () module is also homologically smooth. The last part of the paper gives some prospects and prospects related to the above conclusions.
【学位授予单位】：扬州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O153

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