Structures and Representations of Generalized Path Algebras |
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Authors: | Shouchuan Zhang Yao-Zhong Zhang |
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Institution: | (1) Department of Mathematics, Hunan University, Changsha, 410082, People’s Republic of China;(2) Department of Mathematics, University of Queensland, Brisbane, 4072, Australia |
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Abstract: | It is shown that an algebra Λ can be lifted with nilpotent Jacobson radical r = r(Λ) and has a generalized matrix unit {e
ii
}
I
with each ē
ii
in the center of if Λ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given.
As a corollary, Λ is a finite algebra with non-zero unity element over a perfect field k (e.g., a field with characteristic zero or a finite field) if Λ is isomorphic to a generalized path algebra k (D, Ω, ρ) of finite directed graph with weak relations and dim < ∞; Λ is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise
orthogonal idempotents if Λ is isomorphic to a path algebra with relations.
Presented by Idun Reiten. |
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Keywords: | generalized path algebra generalized matrix ring Jacobson radical |
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