2007/04/17 by Lucian M. Ionescu, Ionescu, Lucian M.
Mathematics · #14Dxx #46Lxx #53Dxx #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.0704.2213
openalex publication_date 2007/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.