alexa A canonical semi-classical star product
ISSN: 1736-4337

Journal of Generalized Lie Theory and Applications
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Research Article

A canonical semi-classical star product

Lucian M. IONESCU* and Papa A. SISSOKHO

Illinois State University, Normal, IL 61790-4520, USA

*Corresponding Author:
Lucian M. IONESCU
Illinois State University,
Normal, IL 61790-4520,
USA
E-mail:
[email protected] and [email protected]

Received date: December 21, 2006; Revised date: May 29, 2007

Abstract

We study the Maurer-Cartan equation of the pre-Lie algebra of graphs controlling the deformation theory of associative algebras. We prove that there is a canonical solution (choice independent) within the class of graphs without circuits, i.e. at the level of the free operad, without imposing the Jacobi identity. The proof is a consequence of the unique factorization property of the pre-Lie algebra of graphs (tree operad), where composition is the insertion of graphs. The restriction to graphs without circuits, i.e. at “tree level”, accounts for the interpretation as a semi-classical solution. The fact that this solution is canonical should not be surprising, in view of the Hausdorff series, which lies at the core of almost all quantization prescriptions.

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