QUANTIZATION OF REDUCTIVE LIE ALGEBRAS: CONSTRUCTION AND UNIVERSALITY

Abstract

The present paper addresses the question of universality of the quantization of reductive Lie algebras. Quantization is viewed as a torsion free deformation depending upon several parameters which are treated formally and not as complex numbers. The coalgebra and algebra structures are shown to restrict very sharply the possibilities for the infinite series in the generators of the Cartan subalgebra. Under an Ansatz which can be viewed as requiring that the two Borel subalgebras are deformed as Hopf algebras we construct a multi-parameter quantization which has the required property of universality. We also show that such a quantization can be defined so that the algebra structure is the same as that of the standard one-parameter quantization, the remaining parameters being relegated to the coalgebra structure. We discuss an example in which only the latter parameters appear in the deformation. We then complete the study of the universal deformations by developing some aspects of the representation theory of the deformed algebras. Using this theory, especially the freeness of the irreducible modules, we prove the analogue of the Poincaré-Birkhoff-Witt theorem, and, as a consequence, the torsion freeness of the universal deformations.