Some Research On Algebra Extensions

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 ? and a ?-derivation ?. 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 ?-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.