Fedosov quantization in positive characteristic

By R. Bezrukavnikov, D. Kaledin. 2005

65 citations recorded by the source corpus.

The so-called deformation quantization of symplectic manifolds originally ap peared in C?? symplectic geometry; the crucial breakthrough was independently made in the early 1980s by B. Fedosov and M. De Wilde-P. Lecomte. The input of a deformation quantization problem is a symplectic (or, more generally, a Poisson) manifold M; the output is a non-commutative one-parameter deformation of the algebra of functions on M. Both Fedosov and De Wilde-Lecomte provided general procedures which solve the deformation quantization problem for C?? manifolds. Recently, motivated in part by M. Kontsevich [K], there has been much interest in generalizing the deformation quantization procedures to the case of algebraic manifolds equipped with an algebraic symplectic form (see, e.g., [BK1], [Y], and an earlier paper [NT] for the holomorphic situation). In particular, in [BK1] it has been shown that under some mild assumptions on the manifold, the Fedosov quantization can be made to work in the algebraic setting. The present paper is a continuation of [BK1] (to which we refer the reader for a more complete bibliography and historical discussion). Namely, one of the most important assumptions in [BK1] (as well as in other papers on the subject) was that the field of definition for all algebraic manifolds has characteristic 0. In this paper, we study what happens in the case of positive characteristic. The most obvious new feature of the theory in positive characteristic is the presence of a large Poisson center in the sheaf Ox of functions on a Poisson manifold X: since for any two local functions f,g Ox and any Poisson bracket { ?,?} on Ox we have {fp,g} = 0, the image ?px C Ox of the Frobenius map lies in the center of any Poisson structure. This phenomenon, already observed in [BMR], allows for interesting applications (see, e.g., [BK2]) but makes the quantization procedures more involved. In this paper, we were not able to prove any meaningful results for general quantizations in positive characteristic, and we had to restrict our attention to a special class of them: the so-called Frobenius-constant quantization. Roughly speaking the precise definition is Definition 1.4 below a quantization is Frobenius-constant if the Poisson center Opx C Ox stays central in the quantized algebra Oh For quantization of this type, we were able to achieve, under mild assumptions on the manifold X, a reasonably complete classification theorem. In particular, Frobenius-constant quantizations do exist.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a9c68f1d0ef8d8ab3a1d48113da8fa1877aedc95

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Citations in the corpus

Completions of symplectic reflection algebras (2010, methodology): “ommutative Poisson algebras and their Poisson bimodules. These are algebras and bimodules equipped with an additional structure: a Poisson bracket. A similar notion already appeared, for instance, in [BeKa]. After giving all necessary deﬁnitions we study some simple properties of Poisson bimodules. ”
Derived Equivalences by Quantization (2005, background): “m. Now, by [BK3, Proposition 1.22], there is a family of one-parameter deformations O h of the structure sheaf O X which are numbered by elements of the cohomology group Q(X) = H1 et(X,O∗ X/O ∗p); in [BK3], they are called Frobenius-constant quantizations on X. In this language, the deformation O a ”
On global deformation quantization in the algebraic case (2006, methodology): “ sheaves over a generalized complex manifold. In the case of a symplectic manifold this should be some variant of the Fukaya category. Other papers relevant for algebraic deformation quantization are [7, 6, 31]. [6] is especially interesting as it discusses Fedosov quantization in positive character”