PBW Bases, Non–Degeneracy Conditions and Applications

We establish an explicit criteria (the vanishing of non–degeneracy conditions) for certain noncommutative algebras to have Poincaré–Birkhoff– Witt basis. We study theoretical properties of such G–algebras, concluding they are in some sense ”close to commutative”. We use the non–degeneracy conditions for practical study of certain deformations of Weyl algebras, quadratic and diffusion algebras. The famous Poincaré–Birkhoff–Witt (or, shortly, PBW) theorem, which appeared at first for universal enveloping algebras of finite dimensional Lie algebras ([7]), plays an important role in the representation theory as well as in the theory of rings and algebras. Analogous theorem for quantum groups was proved by G. Lusztig and constructively by C. M. Ringel ([6]). Many authors have proved the PBW theorem for special classes of noncommutative algebras they are dealing with ([17], [18]). Usually one uses Bergman’s Diamond Lemma ([4]), although it needs some preparations to be done before applying it. We have defined a class of algebras where the question ”Does this algebra have a PBW basis?” reduces to a direct computation involving only basic polynomial arithmetic. In this article, our approach is constructive and consists of three tasks. Firstly, we want to find the necessary and sufficient conditions for a wide class of algebras to have a PBW basis, secondly, to investigate this class for useful properties, and thirdly, to apply the results to the study of certain special types of algebras. The first part resulted in the non–degeneracy conditions (Theorem 2.3), the second one led us to the G– and GR–algebras (3.4) and their properties (Theorem 4.7, 4.8), and the third one — to the notion of G–quantization and to the description and classification of G–algebras among the quadratic and diffusion algebras. 1991 Mathematics Subject Classification. Primary 68W30, 16W70; Secondary 16Z05, 13P10. partially supported by the CRDF grant UM2–2094 and DFG. c ©0000 American Mathematical Society