Abstract
We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra \({\text{sl}}^{c}_{2} \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Artin, M., Schelter, W., Tate, J.: Quantum deformations of \(GL_{\,n}\). Comm. Pure Appl. Math. XLIV, 879–895 (1990)
Drozd, Yu. A., Kirichenko, V.V.: Finite Dimensional Algebras. Springer, Berlin, Heidelberg, New York (1993)
Gorodnii, M. F., Podkolzin, G. B.: Irreducible representations of a graded Lie algebra, in Spectral theory of Operators and infinite-dimensional Analysis, Inst. Math. Acad. Sci. UkrSSR, Kiev, pp. 66–76. (1984)
Hu, N.H.: Quantum group structure associated to the quantum affine space, preprint (2004)
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory, Graduate Texts in Math., vol. 9. Springer, Berlin Heidelberg New York (1972)
Karpilovsky, G.: Projective Representations of Finite Groups, Monographs in Pure and Applied Math., vol. 94. Marcel Dekker, New York (1985)
Kharchenko, V.K.: An algebra of skew primitive elements, math.QA/0006077
Kirkman, E.E., Small, L.W.; Examples of FCR-algebras. Comm. Algebra 30(7), 3311–3326 (2002)
Kraft, H., Small, L.W.: Invariant algebras and completely reducible representations. Math. Res. Lett. 1, 297–307 (1994)
Kraft, H., Small, L.W., Wallach, N.R.: Properties and examples of FCR-algebras. Manuscripta Math. 104, 443–450 (2001)
McConnell, J.C., Robson, J.C.: Noncommuative Noetherian Rings. Chichester, Wiley (1987)
Nǎstǎsescu, C., van Oystaeyen, F.: Dimension of Ring Theory, Mathematics and Its Applications. Reidel, Dordrecht, Holland (1987)
Nǎstǎsescu, C., van Oystaeyen, F.: Methods of Graded Rings, Lecture Notes in Math., vol. 1836. Springer, Berlin Heidelberg New York (2004)
Ostrovskyi, V.L., Silvestrov, S.D.: Representations of the real forms of a graded analogue of the Lie algebra \(sl(2,\mathbb{C})\). Ukr. Mat. Zhurn. 44(11), 1518–1524 (1992)
Rittenberg, V., Wyler, D.: Generalized superalgebras. Nuclear Phys. B 139, 189–202 (1978)
Scheunert, M.: Generalized Lie algebras. J. Math. Phys. 20, 712–720 (1979)
Scheunert, M.: The Theory of Lie Superalgebras, Lecture Notes in Math., vol. 716. Springer, Berlin Heidelberg New York (1979)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, XW., Silvestrov, S.D. & Van Oystaeyen, F. Representations and Cocycle Twists of Color Lie Algebras. Algebr Represent Theor 9, 633–650 (2006). https://doi.org/10.1007/s10468-006-9027-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-006-9027-0