【作者】 王欣；

【导师】 卢涤明；

【作者基本信息】 浙江大学 ， 基础数学， 2017， 博士

【摘要】 代数扩张是利用已知代数来构造新的代数的常用方法,包括Ore扩张,双Ore扩张,正规扩张以及R-smash积等.本文主要研究非交换代数扩张的Artin-Schelter正则性,Poisson代数结构及A_∞-代数结构等代数和同调性质.文中我们考虑的是由生成元和生成关系确定的一类代数.对于这类代数的R-smash积,根据映射尺的左右可乘性,我们自然地要求线性映射只可以由生成集上的作用确定.而这一类R能够通过代数态射σ和κ-导子δ控制.此类R-smash积包含了经典Ore扩张和双Ore扩张.首先,我们利用非交换Grobner基理论给出了R-smash积的一个组合性质,这有助于我们更好地描述R-smash积.紧接着,我们讨论了R-smash积的Artin-Schelter正则性.在这一过程中,我们引入了R可逆的概念,讨论了一类带有长度意义下pure分解的代数,并证明了在这种情形下,R-smash积保持Artin-Schelter正则性.其次,我们考虑了R-smash积构造下,相应semiclassical极限的Poisson结构构造.利用上述构造,我们给出了张量积上一种带辫子的Poisson结构,并给出了这种Poisson结构的等价刻画,推广了 Poisson多项式环和双Poisson-Ore扩张的结果,并说明如此构造的Poisson结构包含很多经典的Poisson结构.最后,我们研究了分次斜扩张的Ext-代数上的结合代数结构及A∞-代数结构,证明了这类Ext-代数作为结合代数的分解定理,并且构造出了其上的一类A∞-代数结构.更多还原

【Abstract】 The method of algebra extensions is normal to construct new algebras from the known ones,including Ore extensions, double Ore extensions, normal extensions, R-smash products, and so on. In this paper, we study how the algebra properties or algebra structures extend under some algebra extensions, including Artin-Schelter regularity, Poisson structures and A∞-structures.In this paper, we consider a class of algebras which is determined by the generators and generating relations. For R-smash products of these algebras, it is natural to require the linear map R is defined only on the generating set by using its multiplicative property. In this case, R is controlled by an algebra morphism σ and a σ-derivation δ. Such a class of R-smash products contains Ore extensions and double Ore extensions.Firstly, with the theory of Grobner bases, we show a combinatorial property for the R-smash product in the above case. It helps us describe R-smash products easily. Under the given form of the linear map R, we prove the R-smash product of two Artin-Schelter regular algebras, one of which has pure resolution of length, is still Artin-Schelter regular if R is σ-invertible.Next, we construct a Poisson structure with a braiding on tensor algebras, show its equivalent describe, and apply it to R-smash products, which develops the results of Poisson polynomial extensions and double Poisson extensions, and contains many classical Poisson structures.At the end,we study the Ext-algebras of graded skew extensions. We prove a factorization theorem for such Ext-algebras as associative algebras, and construct a class of A∞-structures on such Ext-algebras.更多还原