Colour calculus and colour quantizations (Q1903563)

Let \(A\) be an associative algebra over a commutative field \(k\), let \(\lambda \in \Aut A\). 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 \textit{A. M. Vinogradov}, \textit{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.