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 $\mathscr{P}$ be the category of projective $A$-modules. Denote by $C^1(\mathscr{P})$ the category of 1-cyclic complexes over $\mathscr{P}$, and $\tilde{\mathfrak{n}}^+$ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in $C^1(\mathscr{P})$. In this paper, we prove the existence of Hall polynomials in $C^1(\mathscr{P})$, 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 $\tilde{\mathfrak{n}}^+$ 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, $\tilde{\mathfrak{n}}^+$ is isomorphic to the nilpotent part of the corresponding semisimple Lie algebra; if $Q$ is the linearly oriented quiver of type $\mathbb{A}_{n}$, $\tilde{\mathfrak{n}}^+$ is isomorphic to the free 2-step nilpotent Lie algebra with $n$-generators. Furthermore, we give a description of the root systems of different $\tilde{\mathfrak{n}}^+$. We also characterize the Lie algebras $\tilde{\mathfrak{n}}^+$ by generators and relations. When $Q$ is of type $\mathbb{A}$, 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.