本文選題：廣義 + Weyl代數(shù)��； 參考：《揚(yáng)州大學(xué)》2017年碩士論文

【摘要】：同調(diào)光滑性是關(guān)于結(jié)合代數(shù)的一種同調(diào)性質(zhì).作為交換意義下光滑性的非交換版本,同調(diào)光滑性在非交換代數(shù)幾何,量子群,算子代數(shù),數(shù)學(xué)物理等數(shù)學(xué)領(lǐng)域都扮演著重要角色.許多同調(diào)光滑代數(shù)的Hochschild同調(diào)與上同調(diào)具有對(duì)偶性,亦即Van den Bergh對(duì)偶.一般而言,判斷一個(gè)代數(shù)是否同調(diào)光滑是困難的.本論文研究了一類Gelfand-Kirilov維數(shù)為3的廣義Weyl代數(shù)的同調(diào)光滑性.廣義Weyl代數(shù)的概念是Bavula于1992年引入的,目的是研究一些類似于經(jīng)典Weyl代數(shù)的代數(shù).本文所討論的廣義Weyl代數(shù)取,以二元多項(xiàng)式環(huán)k[z1,z2]為子代數(shù),由兩個(gè)參數(shù)σ和φ決定,其中σ是代數(shù)k[z1,z2]的一個(gè)仿射型自同構(gòu),φ=φ(z_1,z_2)是一個(gè)非零的二元多項(xiàng)式.通過(guò)構(gòu)造W的同倫雙復(fù)形,我們首先得到了 W作為We-模的一個(gè)投射分解,然后找出了 W具有同調(diào)光滑性的一個(gè)充分條件.更準(zhǔn)確地說(shuō),我們證明了:若φ,αφ/αz1,αφ/αz2生成的理想就等于k[z1,z2]本身,則W是同調(diào)光滑的.作為上述結(jié)論的應(yīng)用,我們證明了量子群Oq(SL2)和U(sl2)都是同調(diào)光滑代數(shù),這與K.A.Brown,J.J.Zhang在2008年取得的一個(gè)結(jié)果相吻合;還研究了陳惠香于1999年定義的代數(shù)M(1,q),證明M(1,q)模去一個(gè)正規(guī)正則元生成的主理想所得的商代數(shù)也是同調(diào)光滑的.文章的最后部分給出了與上述結(jié)論相關(guān)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, 偽 蠁 / 偽 z 1, 偽 蠁 / 偽 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.
【學(xué)位授予單位】：揚(yáng)州大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O153

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