##
**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.

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 |

### Keywords:

colour commutative algebra; colour differential form; colour differential operator; skew-symmetry; Jacobi identities; derivations; colours; group algebras; crossed products; quantizations; monoidal categories; colour calculus### Citations:

Zbl 0722.35001
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\textit{V. Lychagin}, Acta Appl. Math. 41, No. 1--3, 193--226 (1995; Zbl 0846.18006)

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### References:

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