tailieunhanh - Đề tài " On deformations of associative algebras "

In a classic paper, Gerstenhaber showed that first order deformations of an associative k-algebra a are controlled by the second Hochschild cohomology group of a. More generally, any n-parameter first order deformation of a gives, due to commutativity of the cup-product on Hochschild cohomology, a graded algebra morphism Sym• (kn ) → Ext2•bimod (a, a). We prove that any extension aof the n-parameter first order deformation of a to an infinite order formal deformation provides a canonical ‘lift’ of the graded algebra morphism above to a dg-algebra morphism Sym• (kn ) → RHom• . | Annals of Mathematics On deformations of associative algebras By Roman Bezrukavnikov and Victor Ginzburg Annals of Mathematics 166 2007 533 548 On deformations of associative algebras By Roman Bezrukavnikov and Victor Ginzburg Abstract In a classic paper Gerstenhaber showed that first order deformations of an associative k-algebra a are controlled by the second Hochschild cohomology group of a. More generally any n-parameter first order deformation of a gives due to commutativity of the cup-product on Hochschild cohomology a graded algebra morphism Sym kra Ext2 bimod a a . We prove that any extension of the n-parameter first order deformation of a to an infinite order formal deformation provides a canonical lift of the graded algebra morphism above to a dg-algebra morphism Sym kra RHom a a where the symmetric algebra Sym kra is viewed as a dg-algebra generated by the vector space kra placed in degree 2 equipped with zero differential. 1. Main result . Let k be a field of characteristic zero and write G Gk Hom Homk etc. Given a k-vector space V let V Hom V k denote the dual space. We will work with unital associative k-algebras to be referred to as algebras . Given such an algebra B we write mB B G B - B for the corresponding multiplication map and put QB Ker mB c B G B. This is a B-bimodule which is free as a right B-module in effect Qb B k G B is a free right B-module generated by the subspace B k c Qb formed by the elements b G 1 1 G b b E B. Fix a finite dimensional vector space T and let O k T be the commutative local k-algebra with unit 1 E k and with maximal ideal T c O such that T 2 0. Thus O T k. The algebra O is Koszul and one has a canonical isomorphism Torf k k T . We are interested in multi-parameter first order deformations of a given algebra a. Specifically by an O-deformation of a we mean a free O-algebra A Both authors are partially supported by the NSF. 534 ROMAN BEZRUKAVNIKOV AND VICTOR GINZBURG that is O is a central subalgebra in A and A is a

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