4,099 research outputs found
Cohomology theories for homotopy algebras and noncommutative geometry
This paper builds a general framework in which to study cohomology theories
of strongly homotopy algebras, namely and
-algebras. This framework is based on noncommutative geometry as
expounded by Connes and Kontsevich. The developed machinery is then used to
establish a general form of Hodge decomposition of Hochschild and cyclic
cohomology of -algebras. This generalizes and puts in a conceptual
framework previous work by Loday and Gerstenhaber-Schack.Comment: This 54 pages paper is a substantial revision of the part of
math.QA/0410621 dealing with algebraic Hodge decompositions of Hochschild and
cyclic cohomology theories. The main addition is the treatment of cohomology
theories corresponding to unital infinity-structure
Chiral Koszul duality
We extend the theory of chiral and factorization algebras, developed for
curves by Beilinson and Drinfeld in \cite{bd}, to higher-dimensional varieties.
This extension entails the development of the homotopy theory of chiral and
factorization structures, in a sense analogous to Quillen's homotopy theory of
differential graded Lie algebras. We prove the equivalence of
higher-dimensional chiral and factorization algebras by embedding factorization
algebras into a larger category of chiral commutative coalgebras, then
realizing this interrelation as a chiral form of Koszul duality. We apply these
techniques to rederive some fundamental results of \cite{bd} on chiral
enveloping algebras of -Lie algebras
Poincare-Birkhoff-Witt Theorems
We sample some Poincare-Birkhoff-Witt theorems appearing in mathematics.
Along the way, we compare modern techniques used to establish such results, for
example, the Composition-Diamond Lemma, Groebner basis theory, and the
homological approaches of Braverman and Gaitsgory and of Polishchuk and
Positselski. We discuss several contexts for PBW theorems and their
applications, such as Drinfeld-Jimbo quantum groups, graded Hecke algebras, and
symplectic reflection and related algebras.Comment: 30 pages; survey article to appear in Mathematical Sciences Research
Institute Proceeding
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