Algebraic Deformation Quantization

In this note we give an algebraic proof of “deformation quantization” by making use of the theory of Unital Gröbner bases over a valuation ring. MSC: 16Z05,13P10 In this note we give an algebraic proof of deformation quantization (c.f. [7]). We do this be developing in (Sec. 1) the theory of unital Gröbner bases over a valuation ring. We then in (Sec. 2) obtain, almost immediately, our desired deformation quantization in a completely analogous manner to the Gröbner basis proof of the Poincaré-Birkhoff-Witt[PBW] Theorem for the universal enveloping algebra of a Lie algebra ([12]). Proceeding in this manner, we obtain, essentially for free, that our “quantization map” is defined over Q, rather than R, and the Casimir functions are central in the quantized algebra. We present here only the “local” picture. The global picture in the analytic case is described [3], and in the algebraic case in [8] and [15]. The holomorphic case is discussed in [13], while the positive characteristic setting is described in [1]. 1 Gröbner Bases over Valued Rings In this section, we develop the theory of unital Gröbner bases over a valuation ring. The case where R is an arbitrary ring, without a valuation, is discussed in [10]. The essential difference is that in this paper, we are allowing the base ring to affect the term ordering, while this is not the case in [10]. One may recover the results of [10] by taking val to be the discrete or trivial valuation val(r) = 0 for r ∈ R. In fact, all proofs are carried directly over from [10] once one emphasizes the role of a leading term rather than of a leading monomial and once one takes care of topological issues. We repeat the proofs here for completeness and clarity. Partially supported by a VATAT fellowship