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 $C_2\left(P\right)$, 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 $g=n\oplus h$ at $t=-1$, where $n$ is constructed by stack functions about indecomposable radical complexes, and $h$ is by contractible complexes. For the stable category $K_2\left(P\right)$ of $C_2\left(P\right)$, we construct its moduli spaces and a $C$-Lie algebra $\tilde{g}=\tilde{n}\oplus \tilde{h}$, where $\tilde{n}$ is constructed by support-indecomposable constructible functions, and $\tilde{h}$ is by the Grothendieck group of $K_2\left(P\right)$. We prove that the natural functor $C_2\left(P\right)\to K_2\left(P\right)$ together with the natural isomorphism between Grothendieck groups of $A$ and $K_2\left(P\right)$ induces a Lie algebra isomorphism $g\cong \tilde{g}$. This makes clear that the structure constants at $t=-1$ provided by Bridgeland in [5] in terms of exact structure of $C_2\left(P\right)$ precisely equal to that given in [30] in terms of triangulated category structure of $K_2\left(P\right)$.