Lie algebras arising from two-periodic projective complex and derived categories (Q6611887)

This paper formulates two types of Lie algebras from the exact category of two-periodic projective complexes and its stable triangulated category respectively, associated to any finite-dimensional algebra \(A\) of finite global dimension. The constructions follow from \textit{T. Bridgeland} [Ann. Math. (2) 177, No. 2, 739--759 (2013; Zbl 1268.16017)] and \textit{L. Peng} and \textit{J. Xiao} [Invent. Math. 140, No. 3, 563--603 (2000; Zbl 0966.16006)] respectively, however, this time over the complex field.\N\NFirstly, for the category of 2-periodic projective complexes \(\mathcal{C}_2(\mathcal{P})\) of two-periodic projective complexes over \(A\), the authors define a moduli stack to parametrize isomorphism classes of objects from \(\mathcal{C}_2(\mathcal{P})\), and construct a motivic Bridgeland Hall algebra in the sense of Joyce, where the structure constants are basing on Bridgeland's Hall algebra construction, however, using Poincaré polynomials to instead of Hall numbers. Finally, they obtain a Lie algebra \(\mathfrak{g}\) by evaluating the variable \(t=-1\) in the Poincaré polynomials. This construction is a generalization of Bridgeland's Hall Lie algebra, since \(A\) is not necessary a hereditary algebra.\N\NThe authors are clever to use (virtual) Poincaré polynomials as the structure constants, which gives an interpretation for the artificial twisting by \(\sqrt{q}^{\langle X^1,Y^1\rangle+\langle X^0,Y^0\rangle}\) in the original construction of Bridgeland's Hall algebra. Moreover, they overcome the difficulty of realizing the Cartan elements via constructible functions.\N\NSecondly, for the triangulated category \(\mathcal{K}_2(\mathcal{P})\) of the stable category \(\mathcal{C}_2(\mathcal{P})\) of two-periodic projective complexes, the authors define an ind-constructible stack to parametrize the homotopy equivalence classes of objects in \(\mathcal{K}_2(\mathcal{P})\). Then they define a Lie algebra \(\tilde{\mathfrak{g}}\) following Peng-Xiao's construction. The basis of \(\tilde{\mathfrak{g}}\) come from constructible functions together with the Grothendieck group. This construction is a reformulation and a refinement of Xiao-Xu-Zhang's original unpublished work, but with a self-contained proof. One of the main difficulties is to prove the Jacobi identity by using of the octahedral axiom.\N\NIt is well known that there is a natural functor \(\Phi: \mathcal{C}_2(\mathcal{P})\to \mathcal{K}_2(\mathcal{P})\). The main result of this paper states that the functor \(\Phi\) induces an isomorphism between the two algebras \(\mathfrak{g}\) and \(\tilde{\mathfrak{g}}\), which are constructed from the exact category \(\mathcal{C}_2(\mathcal{P})\) and the triangulated catagory \(\mathcal{K}_2(\mathcal{P})\) respectively. The key step to prove the theorem is by comparing the structure constants in both constructions, which are congruence modulo \(q-1\), where \(q\) is the cardinality of the based field.\N\NThere are several applications of the main result:\N\NSince the two algebras \(\mathfrak{g}\) and \(\tilde{\mathfrak{g}}\) are isomorphic, and \(\tilde{\mathfrak{g}}\) is completely determined by \(\mathcal{K}_2(\mathcal{P})\) which is derived invariant, one obtains that \(\mathfrak{g}\) is also derived invariant.\N\NThe highlight of this paper is to construct the motivic Bridgeland's Hall algebra in the framework of motive Hall algebra in the sense of Joyce, which has not been studied before. The authors also interpret Riedtmann-Peng formula and the twisted/localization process in Bridgeland's original construction by cartesian diagrams of stacks, which should be helpful for further research in this field. Moreover, both proofs for \(\mathfrak{g}\) and \(\tilde{\mathfrak{g}}\) being Lie algebras require detailed and complicated calculations, which need deep geometric knowledge. Furthermore, the proof of \(\mathfrak{g}\cong\tilde{\mathfrak{g}}\) are isomorphic is highly nontrivial. In my opinion, the main results in this paper are important, interesting and necessary for the readers to understand the deep relation between two different kinds of realization of Lie algebras from exact category and triangulated category respectively.