J. Aust. Math. Soc. 83 (2007), no. 3, pp. 385–416.

# Characterization of left Artinian algebrasthrough pseudo path algebras

 Fang Li Department of MathematicsZhejiang UniversityHangzhouZhejiang 310027Chinafangli@zju.edu.cn
Received 20 March 2004; revised 27 May 2006
Communicated by J. Du

## Abstract

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 \ker \iota is a generalized path algebra (see Fact 2.6).

The main result is that

1. 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\cong \operatorname{PSE}_k(\Delta ,\mathcal {A},\rho ) with J^{s}\subset \langle \rho \rangle \subset J for some s (Theorem 3.2);
2. If A is a finite dimensional k-algebra with 2-nilpotent radical and the quotient by radical can be lifted, then A\cong k(\Delta ,\mathcal {A},\rho ) with \widetilde J^{2}\subset \langle \rho \rangle \subset \widetilde J^{2}+\widetilde J\cap \ker \widetilde {\varphi } (Theorem 4.2),

where \Delta is the quiver of A and \rho is a set of relations.

For all the cases we discuss in this paper, we prove the uniqueness of such quivers \Delta 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).