Colour calculus and colour quantizations
type: Item
instance of: scholarly article
title: Colour calculus and colour quantizations
zbMATH Open document ID: 0846.18006
DOI: 10.1007/BF00996113
published in: Acta Applicandae Mathematicae
publication date: 1996-09-16T00:00:00Z
review text: 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. …
reviewed by: Michal Marvan
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 58A50
Mathematics Subject Classification ID: 35A99
Mathematics Subject Classification ID: 58J99
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 17B99
zbMATH DE Number: 824529
zbMATH Keywords: colour commutative algebra
zbMATH Keywords: colour differential form
zbMATH Keywords: colour differential operator
zbMATH Keywords: skew-symmetry
zbMATH Keywords: Jacobi identities
zbMATH Keywords: derivations
zbMATH Keywords: colours
zbMATH Keywords: group algebras
zbMATH Keywords: crossed products
zbMATH Keywords: quantizations
zbMATH Keywords: monoidal categories
zbMATH Keywords: colour calculus
author: Valentin Lychagin
MaRDI profile type: Publication
cites work: Almost commutative algebra and differential calculus on the quantum hyperplane
cites work: Q4859191
cites work: Q3488733
cites work: Q3809971
cites work: Q4356241
cites work: Graded Lie-Cartan pairs. II: The fermionic differential calculus
cites work: Q3310034
cites work: Q4303683
cites work: Q4693151
cites work: Notes on quantum groups and quantum de Rham complexes
cites work: Covariant differential calculus on the quantum hyperplane
full work available at URL: https://doi.org/10.1007/bf00996113
OpenAlex ID: W4241685582
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description: scientific article; zbMATH DE number 824529