Lie algebras arising from 1-cyclic perfect complexes

[摘要] 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 C-1(P) the category of 1-cyclic complexes over P, and (n) over tilde (+ )the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in C-1 (P). In this paper, we prove the existence of Hall polynomials in C-1(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) over tilde (+) 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) over tilde (+)is isomorphic to the nilpotent part of the corresponding semisimple Lie algebra; if Q is the linearly oriented quiver of type A(n), (n) over tilde (+ )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) over tilde (+). We also characterize the Lie algebras (n) over tilde (+) 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. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-11-15

[关键词] 1-cyclic complex;Hall polynomial;Simple Lie algebra;Nilpotent Lie algebra;Minimal Horseshoe lemma