Abstract
Let H be a bialgebra and H LH be the category of Long H-dimodules defined, for a commutative and co-commutative H, by F. W. Long and studied in connection with the Brauer group of a so-called H-dimodule algebra. For a commutative and co-commutative H, H LH =H YDH (the category of Yetter–Drinfel'd modules), but for an arbitrary H, the categories H LH and H YDH are basically different. Keeping in mind that the category H YDH is deeply involved in solving the quantum Yang–Baxter equation, we study the category H LH of H-dimodules in connection with what we have called the D-equation: R12 R23 = R23 R12, where R ∈ Endk(M ⊗ M) for a vector space M over a field k. The main result is a FRT-type theorem: if M is finite-dimensional, then any solution R of the D-equation has the form R = R(M, ⋅, ρ), where (M, ⋅, ρ) is a Long D(R)-dimodule over a bialgebra D(R) and R(M, ⋅, ρ) is the special map R(M, ⋅, ρ)(m ⊗ n) : = ∑ n〈1〉 ⋅ m ⊗ n〈0〉. In the last section, if C is a coalgebra and I is a coideal of C, we introduce the notion of D-map on C, that is a k-bilinear map σ : C ⊗ C / I → k satisfying a condition which ensures on the one hand that, for any right C-comodule, the special map Rσ is a solution of the D-equation and, on the other, that, in the finite case, any solution of the D-equation has this form.
Similar content being viewed by others
References
Abe, E.: Hopf Algebras, Cambridge University Press, Cambridge, 1977.
Baaj, S. and Skandalis, G.: Unitaries multiplicatifs et dualité pour les produits croises de C *-algèbres, Ann. Sci. Ecole Norm. Sup. 26 (1993), 425-488.
Caenepeel, C., Militaru, G. and Zhu, S.: Crossed modules and Doi-Hopf modules, Israel J. Math. 100 (1997), 221-247.
Caenepeel, S., Militaru, G. and Zhu, S.: Doi-Hopf modules, Yetter-Drinfel'd modules and Frobenius type properties, Trans. Amer. Math. Soc. 349 (1997), 4311-4342.
Caenepeel, S., Van Oystaeyen, F. and Zhou, B.: Making the category of Doi-Hopf modules into a braided monoidal category, Algebras Represent. Theory 1 (1998), 75-96.
Doi, Y.: Unifying Hopf modules, J. Algebra 153 (1992), 373-385.
Faddeev, L. D., Reshetikhin, N. Y. and Takhtajan, L. A.: Quantization of Lie groups and Lie algebras, LOMI preprint E-14-87.
Kassel, C.: Quantum Groups, Springer-Verlag, Berlin, 1995.
Lambe, L. A. and Radford, D.: Algebraic aspects of the quantum Yang-Baxter equation, J. Algebra 54 (1992), 228-288.
Long, F. W.: The Brauer group of dimodule algebras, J. Algebra 31 (1974), 559-601.
Militaru, G.: The Hopf modules category and the pentagonal equations, Comm. Algebra 26 (1998), 3071-3097.
Montgomery, S.: Hopf Algebras and their Actions on Rings, Amer. Math. Soc., Providence, 1993.
Radford, D.: Solutions to the quantum Yang-Baxter equation and the Drinfel'd double, J. Algebra 161 (1993), 20-32.
Radford, D.: Solutions to the quantum Yang-Baxter equation arising from pointed bialgebras, Trans. Amer. Math. Soc. 343 (1994), 455-477.
Takesaki, M.: Duality and von Neumann algebras, In: Lecture Notes in Math. 247, Springer-Verlag, New York, 1972, pp. 665-779.
Yetter, D. N.: Quantum groups and representations of monoidal categories, Math. Proc. Cambridge Philos. Soc. 108 (1990), 261-290.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Militaru, G. The Long Dimodules Category and Nonlinear Equations. Algebras and Representation Theory 2, 177–200 (1999). https://doi.org/10.1023/A:1009905324871
Issue Date:
DOI: https://doi.org/10.1023/A:1009905324871