Lie algebras arising from two-periodic projective complex and derived categories

Abstract

Let A be a finite-dimensional C-algebra of finite global dimension and A be the category of finitely generated right A-modules. By using of the category of two-periodic projective complexes C2(P), we construct the motivic Bridgeland's Hall algebra for A, where structure constants are given by Poincaré polynomials in t, then construct a C-Lie subalgebra g=n⊕h at t=−1, where n is constructed by stack functions about indecomposable radical complexes, and h is by contractible complexes. For the stable category K2(P) of C2(P), we construct its moduli spaces and a C-Lie algebra g˜=n˜⊕h˜, where n˜ is constructed by support-indecomposable constructible functions, and h˜ is by the Grothendieck group of K2(P). We prove that the natural functor C2(P)→K2(P) together with the natural isomorphism between Grothendieck groups of A and K2(P) induces a Lie algebra isomorphism g≅g˜. This makes clear that the structure constants at t=−1 provided by Bridgeland in [5] in terms of exact structure of C2(P) precisely equal to that given in [30] in terms of triangulated category structure of K2(P).