Vector Field Twisting of Lie-Algebras

In quantum groups coproducts of Lie-algebras are twisted in terms of generators of the corresponding universal enveloping algebra. If representations are considered, twists also serve as starproducts that accordingly quantize representation spaces. In physics, requirements turn out to be the other way around. Physics comes up with noncommutative spaces in terms of starproducts that miss a suiting quantum symmetry. In general the classical limit is known, i.e. there exists a representation of the Lie-algebra on a corresponding finitely generated commutative space. In this setup quantization can be considered independently from any representation theoretic issue. We construct an algebra of vector fields from a left cross-product algebra of the representation space and its Hopf-algebra of momenta. The latter can always be defined. The suitingly devided cross-product algebra is then lifted to a Hopf-algebra that carries the required genuine structure to accomodate a matrix representation of the universal enveloping algebra as a subalgebra. We twist the Hopf-algebra of vector fields and thereby obtain the desired twisting of the Lie-algebra. Since we twist with vector fields and not with generators of the Lie-algebra, this is the most general twisting that can possibly be obtained. In other words, we push starproducts to twists of the desired symmetry algebra and to this purpose solve the problem of turning vector fields into a Hopf-algebra. We give some genuine example.