Covariant Realization of Quantum Spaces as Star Products by Drinfeld Twists

Abstract

Covariance of a quantum space with respect to a quantum enveloping algebra ties the deformation of the multiplication of the space algebra to the deformation of the coproduct of the enveloping algebra. Since the deformation of the coproduct is governed by a Drinfeld twist, the same twist naturally defines a covariant star product on the commutative space. However, this product is in general not associative and does not yield the quantum space. It is shown that there are certain Drinfeld twists which realize the associative product of the quantum plane, quantum Euclidean 4-space, and quantum Minkowski space. These twists are unique up to a central 2-coboundary. The appropriate formal deformation of real structures of the quantum spaces is also expressed by these twists.