微分分次代数的Koszul对偶与Hochschild上同调

本项目为代数表示论、同调代数、非交换几何的交叉领域，研究微分分次（DG）代数的Koszul对偶与Hochschild上同调方面的若干前沿问题。将建立DG代数的Calabi-Yau消解理论，作为应用建立整体维数无限代数的广义丛范畴理论；将利用循环同调给出代数的Hochschild扩张代数为对称代数的判别准则，并通过Koszul对偶揭示代数的平凡扩张的形变与形变Calabi-Yau完备之间的关系；顺应同调代数向同伦代数发展的趋势，将证明（强对称）DG代数及其Koszul对偶的Hochschild上链复形作为同伦Gerstenhaber（Batalin-Vilkovisky）代数拟同构；将证明（对称）代数的特征同态为同伦Gerstenhaber（Batalin-Vilkovisky）代数同态。

This project is the crossing field of representation theory of algebras, homological algebra and non-commutative geometry. We will study some problems on the Hochschild cohomology and Koszul duals of differential graded (DG) algebras. We will set up the Calabi-Yau resolution theory of DG algebras, and apply it to build the generalized cluster category theory of algebras of infinite global dimension. We will provide a criterion for a Hochschild extension algebra to be symmetric via cyclic homology, and clarify the relationship between the deformations of trivial extensions of algebras and the deformed Calabi-Yau completion of algebras by Koszul duality. Conforming with the trend from homological algebra to homotopic algebra, we will prove the Hochschild cochain complexes of a DG algebra and its Koszul dual are quasi-isomorphic as homotopy Gerstenhaber algebras, and further the Hochschild cochain complexes of a strong symmetric DG algebra and its Koszul dual are quasi-isomorphic as homotopy Batalin-Vilkovisky algebras. We will show that the characteristic homomorphism of an algebra is a homomorphism of homotopy Gerstenhaber algebras, and further the characteristic homomorphism of a symmetric algebra is a homomorphism of homotopy Batalin-Vilkovisky algebras.