Lie algebras arising from 1-cyclic perfect complexes

Let A be the path algebra of a Dynkin quiver Q over finite field, and P category projective A-modules. Denote by C1(P) 1-cyclic complexes P, n˜+ vector space spanned isomorphism classes indecomposable non-acyclic objects in C1(P). In this paper, we prove existence Hall polynomials C1(P), then establish relationship between numbers for therein those Using evaluated at 1, define Lie bracket commutators degenerate multiplication. The resulting algebras provide broad class nilpotent algebras. For example, if is bipartite, isomorphic to part corresponding semisimple algebra; linearly oriented type An, free 2-step with n-generators. Furthermore, give description root systems different n˜+. We also characterize generators relations. When A, relations are exactly defining As byproduct, construct an orthogonal exceptional pair satisfying minimal Horseshoe lemma each sincere non-projective A-module.