Quantizing Deformation Theory II

Abstract

A quantization of classical deformation theory, based on the Maurer-Cartan Equation dS + 1 2 [S, S] = 0 in dg-Lie algebras, a theory based on the Quantum Master Equation dS + ∆S + 1 2 {S, S} = 0 in dg-BV-algebras, is proposed. Representability theorems for solutions of the Quantum Master Equation are proven. Examples of "quantum" deformations are presented.

The evolution of the spectrum of a Frobenius Lie algebra under deformation

“…The structure of the paper is as follows. In Section 2, we recall the well-known infinitesimal deformation theory of Nijenhuis and Richardson (see [16,17]), wherein the deformation of a Lie algebra is controlled by the graded Chevalley-Eilenberg complex (see [29]). In Section 3, we present the classification of fourdimensional Frobenius Lie algebras by detailing in Table 1 the commutator relations, the dimension of the second and third cohomology groups of the Lie algebra with coefficients in the Lie algebra (the case of interest for deformation theory), the spectrum, and whether or not deformations of the algebra exist.…”