Connes-Kreimer quantizations and PBW theorems for pre-Lie algebras
Abstract
The Connes-Kreimer renormalization Hopf algebras are examples of a canonical quantization procedure for pre-Lie algebras. We give a simple construction of this quantization using the universal enveloping algebra for so-called twisted Lie algebras (Lie algebras in the category of symmetric sequences of k-modules). As an application, we obtain a simple proof of the (quantized) PBW theorem for Lie algebras which come from a pre-Lie product (over an arbitrary commutative ring). More generally, we observe that the quantization and the PBW theorem extend to pre-Lie algebras in arbitrary abelian symmetric monoidal categories with limits. We also extend a PBW theorem of Stover for connected twisted Lie algebras to this categorical setting.
Cite
@article{arxiv.0907.1717,
title = {Connes-Kreimer quantizations and PBW theorems for pre-Lie algebras},
author = {Travis Schedler},
journal= {arXiv preprint arXiv:0907.1717},
year = {2010}
}
Comments
final version, 24 pages; main results generalized to categorical setting, appendix added, and new references included. To appear in SMF proceedings of "Operads 2009" (Luminy).