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