The Trivial Extension And Skew Group Algebra Of Selfinjective Algebras

This thesis for Ph.D degree is on the trivial extension and skew group algebra of selfinjective algebras, it mainly consists of the following three parts.1. the Koszulity of the trivial extension T(A) of the graded selfinjective algebra A. In Chapter 3, firstly we define two classes of modules over the trivial extension of selfinjective algebra and construct their projective covers. Secondly we discuss the properties of the functor F=DA(?)A-in selfinjective algebras A and obtain that the minimal projective resolution of FA0 as a A-module is…→FP2→FP2FP1→FP1FP0→FP0FA0→00, if A0 as a A-module has the minimal projective resolution…→P2→P2P1→P1P0→P0A0→0. At last by constructing the minimal projective resolutions of graded simple modules over the trivial extension of selfinjective algebra, we obtain if A is a connected selfinjective Koszul algebra, then the trivial extension algebra is also a selfinjective Koszul algebra.2. The dimension of stable module categories of skew group algebra over exterior algebra and its trivial extension and the representation dimensions of them. In chapter 4, we discuss the relations of dimensions of stable mod-ule categories between selfinjective algebra and skew group algebra over it, by the definition of dimension of triangulated category and the relations between A—modules and A*G—modules, we show both of them are equal. So we find the lower bound for the skew group algebra over exterior algebra by Rouquier's result on the dimension of stable module category of selfinjective algebra and the relations between the dimension of stable module category and the represen-tation dimension of selfinjective algebra, on the other hand, using the relations between the Loewy length and the representation dimension of selfinjective al-gebra, we obtain the representation dimension of the skew group algebra of exterior algebra over n-dimensional vector space is n+1. Lastly we investigate the representation dimension of the trivial extension algebra of skew group al-gebra over exterior algebra, and obtain the quiver and relations of its basic algebra, so we show that the representation dimension of the trivial extension algebra of skew group algebra over exterior algebra on n-dimensional vector space is n+2.3. The relations between n-cluster tilting subcategories of finite dimen-sional algebra and those of its skew group algebra and the existence of n-cluster tilting subcategories of exterior algebra of 2-dimensional vector space. In chap-ter 5, we construct the n-cluster tilting subcategories of finite-dimensional alge-bra from the n-cluster tilting subcategories of its skew group algebra. Then we construct the n-cluster tilting subcategories of skew group algebra over the finite-dimensional algebra from the n-cluster tilting subcategories of finite-dimensional algebra. And we investigate the existence of n-cluster tilting sub-categories of exterior algebra of 2-dimensional vector space finally, we find there are no n-cluster tilting subcategories in exterior algebra of 2-dimensional vector space by specific computation.