Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 013, 15 pages      arXiv:1006.0478      https://doi.org/10.3842/SIGMA.2012.013

Exponential Formulas and Lie Algebra Type Star Products

Stjepan Meljanac a, Zoran Škoda a and Dragutin Svrtan b
a) Division for Theoretical Physics, Institute Rudjer Bošković, Bijenička 54, P.O. Box 180, HR-10002 Zagreb, Croatia
b) Department of Mathematics, Faculty of Natural Sciences and Mathematics, University of Zagreb, HR-10000 Zagreb, Croatia

Received May 26, 2011, in final form March 01, 2012; Published online March 22, 2012

Abstract
Given formal differential operators $F_i$ on polynomial algebra in several variables $x_1,\ldots,x_n$, we discuss finding expressions $K_l$ determined by the equation $\exp(\sum_i x_i F_i)(\exp(\sum_j q_j x_j)) = \exp(\sum_l K_l x_l)$ and their applications. The expressions for $K_l$ are related to the coproducts for deformed momenta for the noncommutative space-times of Lie algebra type and also appear in the computations with a class of star products. We find combinatorial recursions and derive formal differential equations for finding $K_l$. We elaborate an example for a Lie algebra $su(2)$, related to a quantum gravity application from the literature.

Key words: star product; exponential expression; formal differential operator.

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