Lie algebras arising from 1-cyclic perfect complexes

Abstract

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