# IJPAM: Volume 58, No. 2 (2010)

**FROM LIE THEORY TO DEFORMATION THEORY**

AND QUANTIZATION

AND QUANTIZATION

Department of Mathematics

Illinois State University

Normal, IL 61790-4520, USA

e-mail: lmiones@@ilstu.edu

**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 Huebschmann-Stasheff 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.

**Received: **December 18, 2009

**AMS Subject Classification: **14Dxx, 46Lxx, 53Dxx

**Key Words and Phrases: **deformation theory, Kuranishi functor, quantization, renormalization

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2010

**Volume:** 58

**Issue:** 2