Exponential formulas and Lie algebra type star products. (English) Zbl 1248.81092

Meljanac, Stjepan; Škoda, Zoran; Svrtan, Dragutin

SIGMA, Symmetry Integrability Geom. Methods Appl. 8, Paper 013, 15 p. (2012).

Summary: 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.

MSC:

81R60	Noncommutative geometry in quantum theory
16S30	Universal enveloping algebras of Lie algebras
16S32	Rings of differential operators (associative algebraic aspects)
83C45	Quantization of the gravitational field
83C65	Methods of noncommutative geometry in general relativity

Keywords:

star product; exponential expression; formal differential operator