Lie algebras arising from two-periodic projective complex and derived categories
報(bào)告人簡(jiǎn)介：肖杰，北京師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院教授、博士生導(dǎo)師。曾獲得國(guó)家杰出青年基金、教育部跨世紀(jì)人才基金，教育部自然科學(xué)一等獎(jiǎng)等。曾擔(dān)任中國(guó)科學(xué)、數(shù)學(xué)學(xué)報(bào)、數(shù)學(xué)年刊、Algebra Colloquium等編委，Pure and Applied Mathematics Quarterly 副主編，中國(guó)數(shù)學(xué)會(huì)常務(wù)理事。2006年至2017年任清華大學(xué)數(shù)學(xué)科學(xué)系主任，2014年至2017年任清華大學(xué)理學(xué)院院長(zhǎng)。主要從事代數(shù)表示論與量子群的交叉研究。在代數(shù)表示論、Ringel-Hall代數(shù)、量子群和范疇化等領(lǐng)域做出了一系列重要科研成果。相關(guān)研究成果發(fā)表于Invent. Math., Duke Math. J., Compositio Math., Adv. Math., Math. Z.等重要學(xué)術(shù)期刊。
報(bào)告內(nèi)容介紹：Let A be a ?nite-dimensional C-algebra of ?nite global dimension and consider the category of ?nitely 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). This is based on the joint work with J. Fang and Y. Lan.
