Connes-Kreimer quantizations and PBW theorems for pre-Lie algebras

Details Connes-Kreimer quantizations and PBW theorems for pre-Lie algebras Connes-Kreimer quantizations and PBW theorems for pre-Lie algebras

2009-07-10

arxiv.org/abs/0907.1717v2

Mathematics

Rings and Algebras

final version, 24 pages; main results generalized to categorical setting, appendix added, and new references included. To appe

Scientific paper

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.

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