J. Aust. Math. Soc. 83 (2007), no. 3, pp. 385–416. |
Characterization of left Artinian algebras through pseudo path algebras |
Fang Li |
Department of Mathematics Zhejiang University Hangzhou Zhejiang 310027 China fangli@zju.edu.cn |
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
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 | ||
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