Colour calculus and colour quantizations.(English)Zbl 0846.18006

Let $$A$$ be an associative algebra over a commutative field $$k$$, let $$\lambda \in \operatorname{Aut} A$$. Then a $$\lambda$$-derivation of the algebra $$A$$ is defined to be a $$k$$-linear map $$X_\lambda : A \to A$$ such that the Leibniz rule holds in the following version: $$X_\lambda (ab) = X_\lambda a \cdot b + \lambda (a) \cdot X_\lambda b$$. In section 1, the aim is to set up appropriate definitions so that the $$\lambda$$-derivations admit a Lie structure and an $$A$$-module structure. Formulations obtained are: Let $$G$$ be a group, $$A = \sum_{g \in G} A_g$$ a $$G$$-graded algebra. A colour on $$G$$ is defined to be a mapping $$s : G \times G \to A$$ such that each $$s_{\alpha, \beta} \in A$$ is an invertible element and appropriate identities hold to ensure, basically, that we have versions of skew-symmetry and Jacobi identities for the bracket $$[X_\alpha, X_\beta] = X_\alpha \circ X_\beta - s_{\alpha, \beta} \cdot X_\beta \circ X_\alpha$$, and that by the formula $$\alpha (b) = s_{\alpha, \beta} \cdot b$$, $$b \in A_\beta$$, a $$G$$-action is defined on $$A$$. Under further natural requirements (e.g., that derivations preserve the graded structure) the $$\lambda$$-derivations are shown to form a $$G$$-graded $$A$$-module $$\text{Der}_* (A) = \sum_{\lambda \in G} \text{Der}_\lambda (A)$$ with nice properties.
Examples of colours are abundant, among them all group algebras and their generalizations, named crossed products. Simplest examples for $$G = \mathbb{Z}$$ are $$s_{\alpha, \beta} = 1$$ and $$s_{\alpha, \beta} = (-1)^{\alpha \beta}$$, the latter being a basis for standard supercalculus. Section 1 ends with definitions underlying extensions of the above concepts from the algebra $$A$$ to its modules (colour symmetric bimodules). Section 2 starts with an inductive definition of differential operators between colour symmetric bimodules. On this basis, and along the lines of “Geometry of jet spaces and nonlinear partial differential equations” by A. M. Vinogradov, I. S. Krasil’shchik and the author (1986; Zbl 0722.35001), a colour calculus is built. In particular, colour symbol modules, colour Poisson brackets, colour de Rham complexes, colour jet modules and colour Spencer complexes are introduced. Finally, Section 3 is devoted to the description of symmetries and quantizations in two monoidal categories related to the colour calculus.
Reviewer: M.Marvan (Opava)

MSC:

 18D20 Enriched categories (over closed or monoidal categories) 58A50 Supermanifolds and graded manifolds 35A99 General topics in partial differential equations 58J99 Partial differential equations on manifolds; differential operators 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 35Q53 KdV equations (Korteweg-de Vries equations) 53D50 Geometric quantization 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 16W25 Derivations, actions of Lie algebras 17B99 Lie algebras and Lie superalgebras

Zbl 0722.35001
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