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 C 1 (P) the category of 1-cyclic complexes over P, and ñ+ 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 ñ+ 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, ñ+ is isomorphic to the nilpotent part of the corresponding semisimple Lie algebra; if Q is the linearly oriented quiver of type A n , ñ+ is isomorphic to the free 2-step nilpotent Lie algebra with n-generators. Furthermore, we give a description of the root systems of different ñ+ . We also characterize the Lie algebras ñ+ 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.

Minimal Generators of Hall Algebras of 1-cyclic Perfect Complexes

“…Let H be the subalgebra of H tw (C 1 (P)) generated by all [KP i ], 1 ≤ i ≤ n. Then H ∼ = k[x 1 , x 2 , • • • , x n ], and moreover it is contained in the center of H tw (C 1 (P)). That is, for any P ∈ P and X • ∈ C 1 (P), we have that[K P ] * [X • ] = [X • ] * [K P ](This can be easily verified by Riedtmann-Peng formula together with Proposition 2.4 in[18]). Let K 1 be the ideal of H tw (C 1 (P)) generated by all [K P i ], 1 ≤ i ≤ n. Set H tw (C 1 (P)) := H tw (C 1 (P))/K 1 .Let us reformulate Propositions 6.1 and 6.3.Proposition 7.4.…”

“…In order to state the following theorem, which is the main result of [18], we introduce the path matrix E = (a ij ) n×n of the Dynkin quiver Q: if there is a path between i and j in Q, say from i to j, then a ij = 1 and a ji = −1; otherwise, a ij = a ji = 0.…”

“…Let n + be the positive part of the simple Lie algebra associated to Q, whose canonical generators are denoted by e i , 1 ≤ i ≤ n. Since Q is bipartite, using Theorem 2.11, we can prove that there exists an isomorphism of Lie algebras γ : n + → ñ+ defined by e i → [C P i ] (cf. [18,Corollary 4.6]). Hence, it induces an isomorphism of algebras γ : U(n + ) → U(ñ + ) with e i → [C P i ].…”

“…Proof. In the whole proof, we will use the Riedtmann-Peng formula together with Proposition 2.4 in [18] to calculate Hall numbers.…”

“…Ringel and Zhang [17] have showed that the Kac theorem yields a correspondence between the isoclasses (isomorphism classes) of indecomposable objects in the stable category of C 1 (P) and the positive roots of the Kac-Moody algebra associated to A. Inspired by the above correspondence given by Ringel and Zhang, for a hereditary algebra A of Dynkin type, some research on the Lie theory of C 1 (P) has been undertaken in [18]. They gave a minimal set of generators for the Lie algebra ñ+ spanned by the isoclasses of indecomposable non-projective objects in C 1 (P), and calculated some relations in these generators.…”