From Lie Theory to Deformation Theory and Quantization

TL;DR

This paper explores the deep connections between Lie Theory and Deformation Theory, focusing on key constructions like Maurer-Cartan solutions and Kuranishi functors, to enhance understanding of quantization and renormalization processes.

Contribution

It introduces a unified perspective on deformation constructions, linking Lie Theory concepts with advanced deformation and quantization techniques.

Findings

Maurer-Cartan solutions act as exponentials in deformation theory

Kuranishi functor serves as the logarithm of deformation functors

Connections between quantization, renormalization, and quantum groups are clarified

Abstract

Deformation Theory is a natural generalization of Lie Theory, from Lie groups and their linearization, Lie algebras, to differential graded Lie algebras and their higher order deformations, quantum groups. The article focuses on two basic constructions of deformation theory: the universal solution of Maurer-Cartan Equation (MCE), which plays the role of the exponential of Lie Theory, and its inverse, the Kuranishi functor, as the logarithm. The deformation functor is the gauge reduction of MCE, corresponding to a Hodge decomposition associated to the strong deformation retract data. The above comparison with Lie Theory leads to a better understanding of Deformation Theory and its applications, e.g. the relation between quantization and Connes-Kreimer renormalization, quantum doubles and Birkhoff decomposition.