Exponential Formulas and Lie Algebra Type Star Products

TL;DR

This paper develops formulas and recursive methods to compute exponential expressions involving differential operators, with applications to noncommutative geometry and quantum gravity.

Contribution

It introduces new recursive formulas and differential equations for calculating exponential operator expressions related to Lie algebra star products.

Findings

Derived combinatorial recursions for $K_l$

Established differential equations for exponential expressions

Applied methods to $su(2)$ Lie algebra example

Abstract

Given formal differential operators $F_i$ on polynomial algebra in several variables $x_1,...,x_n$, 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.