Formal Deformations of Dirac Structures.
June 2006
Wed 28 Jun 2006
Wed 21 Jun 2006
Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I.
Maxim Kontsevich, Yan Soibelman
We develop geometric approach to A-infinity algebras and A-infinitycategories based on the notion of formal scheme in the category of gradedvector spaces. Geometric approach clarifies several questions, e.g. the notionof homological unit or A-infinity structure on A-infinity functors. We discussHochschild complexes of A-infinity algebras from geometric point of view. Thepaper contains homological versions of the notions of properness and smoothnessof projective varieties as well as the non-commutative version of Hodge-to-deRham degeneration conjecture. We also discuss a generalization of Deligne’sconjecture which includes both Hochschild chains and cochains. We conclude thepaper with the description of an action of the PROP of singular chains of thetopological PROP of 2-dimensional surfaces on the Hochschild chain complex ofan A-infinity algebra with the scalar product (this action is more or lessequivalent to the structure of 2-dimensional Topological Field Theoryassociated with an “abstract” Calabi-Yau manifold).
Tue 6 Jun 2006
Deformations via Simplicial Deformation Complexes.
There has long been a philosophy that every deformation problem incharacteristic zero should be governed by a differential graded Lie algebra(DGLA). In this paper, the theory of Simplicial Deformation Complexes (SDCs) isdeveloped, as an alternative to DGLAs. These work in all characteristics, andfor many problems can be constructed canonically. In particular, SDCs areconstructed for the problems of deforming an arbitrary scheme, of deforming aHopf algebra, and of deforming a representation of the fundamental group. Incharacteristic zero, SDCs and DGLAs are equivalent.



