Infinity algebras and their deformations

Abstract

Infinity algebras are generalizations of associative and Lie algebras. They play a role in both mathematics and mathematical physics. We study low dimensional examples of these algebras, and classify the nonisomorphic structures. Deformation theory is concerned with how one structure smoothly changes into another structure, and the object of studying the deformations is to understand how the space of all such structures is glued together. In physics, deformations arise because the algebra of quantum mechanics is a deformation of the algebra of the phase space of classical physics. In mathematics, one is interested in the structure of the space of algebras, which is called a moduli space. We present some examples of low dimensional moduli spaces of algebras, and show how the deformations give a picture of these moduli spaces.