2017/05/20 by Ruan, Shiquan, Sheng, Jie, Zhang, Haicheng
#16G20 #17B20 #17B30 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1705.07307
Let A be the path algebra of a Dynkin quiver Q over a finite field, and \mathscrP be the category of projective A-modules. Denote by C1(\mathscrP) the category of 1-cyclic complexes over \mathscrP, and \mathfrakn+ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in C1(\mathscrP). In this paper, we prove the existence of Hall polynomials in C1(\mathscrP), 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 \mathfrakn+ 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, \mathfrakn+ is isomorphic to the nilpotent part of the corresponding semisimple Lie algebra; if Q is the linearly oriented quiver of type \mathbbAn, \mathfrakn+ is isomorphic to the free 2-step nilpotent Lie algebra with n-generators. Furthermore, we give a description of the root systems of different \mathfrakn+. We also characterize the Lie algebras \mathfrakn+ by generators and relations. When Q is of type \mathbbA, 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.