×

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

Citations:

Zbl 0722.35001
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bongaarts, P. J. M. and Pijls, H. G. J.: Almost commutative algebra and differential calculus on the quantum hyperplane, Preprint Inst. Lorentz, 1992. · Zbl 0808.17011
[2] Borowiec, A., Marcinek, W., and Oziewicz, Z.: On multigraded differential calculus, in R. Gielerak (ed.),Quantum Groups and Related Topics, Kluwer Acad. Publ., Dordrecht, 1992, pp. 103-114. · Zbl 0834.17036
[3] Connes, A.:Géométrie non commutative, Inter. Editions, Paris, 1990. · Zbl 0707.46053
[4] Dade, E. C.: Group graded rings and modules,Math. Z. 174 (1980), 241-262. · Zbl 0439.16001
[5] Drinfeld, V. G.: Quantum groups, inProc. ICM, Amer. Math. Soc., Berkeley, 1986, pp. 798-820.
[6] Dubois-Violette, M.: Dérivations et calcul différentiel non commutatif,C.R. Acad. Sci. 307 (1988), 403-408. · Zbl 0661.17012
[7] Gurevich, D.: Hecke symmetries and braided Lie algebras, in Z. Oziewicz (ed.),Spinors, Twistors, Clifford Algebras and Quantum Deformations, Kluwer Acad. PubL, Dordrecht, 1993, pp. 317-326. · Zbl 0890.17010
[8] Jadczyk, A. and Kastler, D.: The fermionic differential calculus,Ann. Physics 179 (1987), 169-200. · Zbl 0637.17013
[9] Karoubi, M.: Homologie cyclique des groupes et des algebres,C.R. Acad. Sci. 297 (1983), 381-384. · Zbl 0528.18008
[10] Kersten, P. H. M. and Krasil’shchik, I. S.: Graded Frölicher-Nijenhuis brackets and the theory of recursion operators for super differential equations, Preprint Univ. of Twente, 1993.
[11] Krasil’shchik, I. S., Lychagin, V. V., and Vinogradov, A. M.:Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986.
[12] Lychagin, V.: Differential operators and quantizations, I, Preprint Math. Inst. Univ. Oslo 44, 1993.
[13] Lychagin, V.: Quantizations of braided differential operators, Preprint ESI 51, 1993.
[14] Lychagin, V.: Braided differential operators and quantizations in ABC-categories,C. R. Acad. Sci. Serie I 318 (1994), 857-862. · Zbl 0803.18005
[15] Lychagin, V.: Braidings and quantizations over bialgebras, Preprint ESI 61, 1993.
[16] MacLane, S.:Categories for the Working Mathematician, Springer-Verlag, Berlin, 1971. · Zbl 0705.18001
[17] Manin, Y.: Notes on quantum groups and quantum de Rham complexes, Preprint MPI 60, 1991. · Zbl 0810.17003
[18] Ogievetsky, O.: Differential operators on quantum spaces for Glq(n) and SOq(n), Preprint MPI 91-103, 1991.
[19] Sletsjøe, A. B.: Twisted Hochschild cohomologies (to appear).
[20] Vinogradov, M. M.: The main functors of differential calculus in graded algebras,Usp. Mat. Nauk 44(3) (1989), 151-152 (in Russian).
[21] Wess, J. and Zumino, B.: Covariant differential calculus on the quantum hyperplane,Nuclear Phys. 18B (1990), 303-312. · Zbl 0957.46514
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.