Quantized Symmetrizations and ⋆-products

The purpose of this article is to make precise some of the results we have reported in talks in Bedlevo (Sep. 2000) and Krakow (July 2001). We have pointed out there that for any algebra having a PBW-type basis, it is possible to construct a ⋆-product. At the conference this has also been made pertinent by J. Wess in his talk and R. Twarock also uses Bergman’s Diamond Lemma . On the other hand, in our article 7 we have shown, in a work that falls naturally in the line of investigations of Procesi and de Concini , and also of the Diamond Lemma, that it is possible to construct (apparently) another ⋆-product for a class of quadratic algebras. In this connection we wish to stress, in view of Kontsevich’s result , the importance of determining the gauge group. For all of the known cases, these algebras are generated by certain elements of the quantized enveloping algebra corresponding to a hermitian symmetric space, and actually, a close inspection and fine tuning of the construction reveals that the results extend to both the quantized Borel subalgebra as well as the full quantized enveloping algebra. Here, the result of Levendorskii 12 , is also needed. Even an arbitrary (classical) Lie algebra falls under this and we rediscover here the Kontsevich ⋆-product . The ⋆-products have direct implications for the quantized Dirac-operator an operator we want to act on “classical” functions on a “classical” (indeed, linearly flat) space.