On Borcherds type Lie algebras

Abstract

For each even lattice \({\mathcal L}\), there is a canonical way to construct an infinite-dimensional Lie algebra via lattice vertex operator algebra theory, we call this Lie algebra and its subalgebras the Borcherds type Lie algebras associated to \({\mathcal L}\). In this paper, we apply this construction to even lattices arising from representation theory of finite-dimensional associative algebras. This is motivated by the different realizations of Kac-Moody algebras by Borcherds using lattice vertex operators and by Peng-Xiao using Ringel-Hall algebras respectively. For any finite-dimensional algebra \(A\) of finite global dimension, we associate a Borcherds type Lie algebra \(\mathfrak {BL}(A)\) to \(A\). In contrast to the Ringel-Hall Lie algebra approach, \(\mathfrak {BL}(A)\) only depends on the symmetric Euler form or Tits form but not the full representation theory of \(A\). However, our results show that for certain classes of finite-dimensional algebras whose representation theory is ’controlled’ by the Euler bilinear forms or Tits forms, their Borcherds type Lie algebras do have close relations with the representation theory of these algebras. Beyond the class of hereditary algebras, these algebras include canonical algebras, representation-directed algebras and incidence algebras of finite prinjective types.