Investigation of Frobenius algebras of infinite representation type

We studied representations of finite dimensional algebars over a field. In particular，我们在研究selfinjective algebas (ie Frobenius algebras).我们有两个主要一个关于symmetric algebras的构造and the other is a characterization of symmetric algebras induced from repetitive algebras bypositive automorphism (1) Let L be a finite extension field of K. By using a given 2-cocycle of thek -‘l,我们构造a 2-cocycle of the K-algebra LQ for an arbitrary finite quiver Q without orientedcycles,我们应该对所有那些K-algebras LQ to have symmetric criterion conditionnon-splittable extension algebras defined by the 2-cocycles.(2) Let B i D4^ e D4 be the repetitivealgebra of a finite dimensional algebra B over a field K by the standard duality module over Band let ν be the Nakayama automorphism of D4 We determined the positive automorphisms ψ ofbi D4 such that the orbit algebra bi D4^ D4/(ψν) is isomorphic to a splittable extensionalgebra of B by a minimal injective cogenerator我们授权weakly symmetric algebras和symmetric algebras，of the form bi D4^ D4 /(ψν) with a positive automorphism ψ of bi D4^ D4. As an application，we characterized some class of weakly symmetric algebras with non-periodic通用标准auslander-reiten components。

We studied representations of finite dimensional algebars over a field. In particular, we concentrated on the research of selfinjective algebas (ie Frobenius algebras). We got two main results. One is about a construction of symmetric algebras, and the other is a characterization of symmetric algebras induced from repetitive algebras by positive automorphism :(1) Let L be a finite extension field of K. By using a given 2-cocycle of the K-algebra L, we constructed a 2-cocycle of the K-algebra LQ for an arbitrary finite quiver Q without oriented cycles, and we showed a criterion condition on L for all those K-algebras LQ to have symmetric non-splittable extension algebras defined by the 2-cocycles.(2) Let BィイD4^ィエD4 be the repetitive algebra of a finite dimensional algebra B over a field K by the standard duality module over B, and let ν be the Nakayama automorphism of BィイD4^ィエD4. We determined the positive automorphisms ψ of BィイD4^ィエD4 such that the orbit algebra BィイD4^ィエD4/(ψν) is isomorphic to a splittable extension algebra of B by a minimal injective cogenerator, and we characterized weakly symmetric algebras and symmetric algebras, of the form BィイD4^ィエD4 /(ψν) with a positive automorphism ψ of BィイD4^ィエD4. As an application, we characterized some class of weakly symmetric algebras with non-periodic generalized standard Auslander-Reiten components.