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).

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 2000 Mathematics Subject Classification: primary 16G10 (Metadata: XML, RSS, BibTeX)

## References

1. M. Auslander, I. Reiten and S. O. Smalø, Representation Theory of Artin Algebras (Cambridge University Press, 1995). MR1314422
2. F. U. Coelho and S. X. Liu, Generalized path algebras, volume 210 of Lecture Notes in Pure and Appl. Math. (Dekker, New York, 2000). MR1758401
3. Y. A. Drozd and V. V. Kirichenko, Finite Dimensional Algebras (Springer-Verlag, Berlin, 1994). MR1284468
4. R. S. Pierce, Associative Algebras (Springer-Verlag, New York, 1982). MR674652
5. C. M. Ringel, Tame algebras and integral quadratic forms, volume 1099 of Lecture Notes in Math. (Springer-Verlag, 1984). MR774589
 Australian Mathematical Publishing Association Inc.