Structures and Representations of Generalized Path Algebras · arXivDeskmath/0402188Feb 12, 200419 pages. To appear in Algebras and Representation Theory
Structures and Representations of Generalized Path Algebras
Shouchuan Zhang, Yao-Zhong Zhang
Abstract
It is shown that an algebra Λ can be lifted with nilpotent Jacobson radical r=r(Λ) and has a generalized matrix unit {eii}I
with each
in the center of
iff
is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary,
is a finite algebra with non-zero unity element over perfect field
(e.g. a field with characteristic zero or a finite field) iff
is isomorphic to a generalized path algebra
k(D,Ω,ρ) of finite directed graph with weak relations and
dim } Ω< ∞
;
is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise orthogonal idempotents iff
is isomorphic to a path algebra with relations.