Characterization of left Artinian algebras through pseudo path algebras

Journal of the Australian Mathematical Society

Fang Li¹ 

关键词: primary 16G10;   

DOI  :  10.1017/S144678870003799X

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

In this paper, using pseudo path algebras, we generalize Gabriel's Theorem on elementary algebras to left Artinian algebras over a field k when the quotient algebra can be lifted by a radical. Our particular interest is when the dimension of the quotient algebra determined by the nth Hochschild cohomology is less than 2 (for example, when k is finite or char k = 0). Using generalized path algebras, a generalization of Gabriel's Theorem is given for finite dimensional algebras with 2-nilpotent radicals which is splitting over its radical. As a tool, the so-called pseudo path algebra is introduced as a new generalization of path algebras, whose quotient by ken is a generalized path algebra (see Fact 2.6).The main result is that(i) for a left Artinian k–algebra A and r = r(A) the radical of A. if the quotient algebra A/r can be lifted then A ≅ PSEk (Δ, , ρ) with Js ⊂ (ρ) ⊂ J for some s (Theorem 3.2);(ii) If A is a finite dimensional k–algebra with 2-nilpotent radical and the quotient by radical can be lifted, then A ≅ k(Δ, , ρ) with 2 ⊂ (ρ) ⊂ 2 +∩ ker(Theorem 4.2),where Δ is the quiver of A and ρ is a set of relations.For all the cases we discuss in this paper, we prove the uniqueness of such quivers Δ and the generalized path algebras/pseudo path algebras satisfying the isomorphisms when the ideals generated by the relations are admissible (see Theorem 3.5 and 4.4).

【 授权许可】

Unknown