2023/05/11 by Jiepeng Fang, Fang, Jiepeng, Yixin Lan +3
Mathematics · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2305.06664
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 \mathfrakg=\mathfrakn⊕ \mathfrakh at t=-1, where \mathfrakn is constructed by stack functions about indecomposable radical complexes, and \mathfrakh is by contractible complexes. For the stable category K2(P) of C2(P), we construct its moduli spaces and a ℂ-Lie algebra \mathfrakg=\mathfrakn⊕ \mathfrakh, where \mathfrakn is constructed by support-indecomposable constructible functions, and \mathfrakh 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 \mathfrakg≅\mathfrakg. 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).