ABSTRACT
Let k denote an algebraically closed field. We say that a finite dimensional k-algebra ∧ is quasi-schurian, if it satisfies the following two conditions:
QS1) dimkHom ∧(P,Q) ≤ 1 if P, Q are not isomorphic indecomposable projective A-modules.
QS2) dimkEnd ∧(P) = 2 for each indecomposable projective A-module P.
An important class of quasi-schurian algebras is the trivial extensions of finite representation type.
In this paper, we give necessary and sufficient conditions for a given quasischurian algebra ∧ to be weakly-symmetric or symmetric. These conditions are given in a combinatorial approach using a graph GS (∧) associated to ∧, and a function ϕ ∧: Ch(GS(∧)) k where Ch(GS(∧)) is the set of chains of the graph GS (∧). Finally we give some connections between symmetric quasi-schurian algebras and trivial extensions of algebras.
