Exponential Formulas and Lie Algebra Type Star Products

Abstract

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 equationand 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.

Remarks on simple interpolation between Jordanian twists

“…In section 2, starting from the twist (5), we have found deformed Hopf algebra symmetry (7)-(10), star products (19), (20) and corresponding realizations for noncommutative coordinates (22). Relation between coproduct ∆p µ and star products e ik·x ⋆ e iq·x (19), (20) are given in (21). Noncommutative coordinatesx µ are also obtained from the coproduct ∆p µ in equation (23) and also from the related star product e ik·x ⋆ e iq·x = e iD(k,q)·x .…”

“…Noncommutative coordinatesx µ are also obtained from the coproduct ∆p µ in equation (23) and also from the related star product e ik·x ⋆ e iq·x = e iD(k,q)·x . In section 3, starting from realizetions of noncommutative coordinates (34), we have constructed corresponding star products (20), (40) and coproducts. Alternatively, coproduct ∆p µ is obtained from noncommutative coordinates (41).…”

“…The function D µ (k, q) from equation (40) coincides with equation (20). From the function D µ (k, q), the coproduct ∆p µ can be recovered using (21) and antipodes S(p µ ) follow analogously.…”

Interpolations between Jordanian Twists Induced by Coboundary Twists

“…. , N − 1, allows for the following generalisation [48] Proposition B.1. 10 For arbitrary realization φ(p) for the noncommutative coordinateŝ…”

“…This star product is associative (due to the fact that the twist F L,u satisfies the cocycle condition). When we choose the functions to be exponential functions e ik·x and e iq·x , then we can define new function D µ (u; k, q) [29,42,43,44,48]:…”