Exploring Noncommutative Algebras via Deformation Theory

In this lecture 1 I would like to address the following question: given an associative algebra A 0 , what are the possible ways to deform it? Consideration of this question for concrete algebras often leads to interesting mathematical discoveries. I will discuss several approaches to this question, and examples of applying them. 1. Deformation theory 1.1. Formal deformations. The most general approach to the question " how to deform A 0 ? " is the theory of formal deformations. Let k be a field and K := k[[ 1 , ..., ℓ ]] the ring of formal power series in variables i. Let m be the maximal ideal in K. A K-module M is said to be topologically free if it is isomorphic to M 0 [[ 1 , ..., ℓ ]] for some vector space M 0. Let A 0 be an algebra over k. 2 Definition 1.1. An ℓ-parameter flat formal deformation of A 0 is an algebra A over K which is topologically free as a K-module, together with an isomorphism of algebras φ : A/m → A 0. 3 For simplicity we will mostly consider 1-parameter deformations. If A is a 1-parameter flat formal deformation of A 0 then we can choose an identification A → A 0 [[]] as K-modules, which reduces to φ modulo. Then the algebra structure on A transforms into a new K-linear multiplication law µ on A 0 [[]]. Such a multiplication law is determined by the product µ(a, b), a, b ∈ A 0 ⊂ A 0 [[]], which is given by the formula µ(a, b) = µ 0 (a, b) + µ 1 (a, b) + 2 µ 2 (a, b) + ..., a, b ∈ A 0 , where µ i : A 0 ⊗ A 0 → A 0 are linear maps, and µ 0 (a, b) is the undeformed product ab in A 0. Thus, to find formal deformations of A 0 means to find all such series µ which satisfy the associativity equation, modulo the automor-phisms of the K-module A 0 [[]] which are the identity modulo. 4 1 This lecture was delivered at " Giornata IndAM " , Naples, June 7, 2005. I would like to thank the organizers, in particular Prof. Corrado De Concini and Paolo Piazza for this wonderful opportunity. I am also grateful to J. Stasheff for useful comments. 2 …