Given an n-dimensional Lie algebra g over a field k ⊃ Q, together with its vector space basis X 0 1 , . . . , X 0 n , we give a formula, depending only on the structure constants, representing the infinitesimal generators,, where t is a formal variable, as a formal power series in t with coefficients in the Weyl algebra A n . Actually, the theorem is proved for Lie algebras over arbitrary rings k ⊃ Q.We provide three different proofs, each of which is expected to be useful for generalizations. The first proof is obtained by direct calculations with tensors. This involves a number of interesting combinatorial formulas in structure constants. The final step in calculation is a new formula involving Bernoulli numbers and arbitrary derivatives of coth(x/2). The dimensions of certain spaces of tensors are also calculated. The second method of proof is geometric and reduces to a calculation of formal right-invariant vector fields in specific coordinates, in a (new) variant of formal group scheme theory. The third proof uses coderivations and Hopf algebras.

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For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way, therefore obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.Comment: uses kluwer.cls; v. 2: 25 pages, significant corrections, 3 authors; version 3: significant revision, 32 pages, corrections and added geometrical viewpoint and preliminaries on formal differential operators; version 4: final corrections and slightly improved readability; accepted in Letters in Mathematical Physic

In our earlier article [Lett. Math. Phys. 107 (2017), 475-503], we explicitly described a topological Hopf algebroid playing the role of the noncommutative phase space of Lie algebra type. Ping Xu has shown that every deformation quantization leads to a Drinfeld twist of the associative bialgebroid of h-adic series of differential operators on a fixed Poisson manifold. In the case of linear Poisson structures, the twisted bialgebroid essentially coincides with our construction. Using our explicit description of the Hopf algebroid, we compute the corresponding Drinfeld twist explicitly as a product of two exponential expressions.

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.