Abstract
Let g be a simple finite-dimensional complex Lie algebra and letG be the corresponding simply-connected algebraic group. A theorem of Kostant states that the universal enveloping algebra of g is a free module over its center. A theorem of Richardson states that the algebra of regular functions ofG is a free module over the subalgebra of regular class functions. Joseph and Letzter extended Kostant's theorem to the case of the quantized enveloping algebra of g. Using the theory of crystal bases as the main tool, we prove a quantum analogue of Richardson's theorem. From it, we recover Joseph and Letzter's result by a kind of “quantum duality principle”.
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References
[BS] P. Baumann, F. Schmitt,Classification of bicovariant differential calculi on quantum groups (a representation-theoretic approach), Commun. Math. Phys.194 (1998), 71–86.
[Ca1] P. Caldero,Eléments ad-finis de certains groupes quantiques, C. R. Acad. Sci. Paris Sér. I Math.316 (1993), 327–329.
[Ca2] P. CalderoOn harmonic elements for semi-simple Lie algebras, preprint Université Lyon I, 1999.
[DKP] C. De Concini, V. G. Kac, and C. Procesi,Quantum coadjoint action, J. Amer. Math. Soc.5 (1992), 151–189.
[Dr1] V. G. Drinfeld,Quantum groups, in Proceeding of the International Congress of Mathematicians (Berkeley, CA, 1986, A. M. Gleason, ed.), 798–820; American Mathematical Society, Providence, RI, 1987.
[Dr2] В. Г. Дринфельд, О почми кокоммумамивных алгебрах Хопфа Алгебра и Анализ1 (1989), 30–46. English translation: V. G. Drinfeld,On almost cocommutative Hopf algebras Leningrad Math. J.1 (1990), 321–342.
[Jo] A. Joseph,Quantum Groups and Their Primitive Ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 29, Springer-Verlag, Heidelberg, 1995.
[JL] A. Joseph and G. Letzter,Separation of variables for quantized enveloping algebras, Amer. J. Math.116 (1994), 127–177.
[Li] P. LittelmannA Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math.116 (1994), 329–346.
[KK] S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki,Affine crystals and vertex models, in Infinite analysis (Kyoto, 1991, A. Tsuchiya, T. Eguchi, and M. Jimbo, eds.), Part A, 449–484; Adv. Ser. Math. Phys., vol. 16, World Scientific Publishing, River Edge, NJ, 1992.
[Ka1] M. Kashiwara,On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J.63 (1991), 465–516.
[Ka2] M. Kashiwara,Crystal bases of modified quantized enveloping algebra, Duke Math. J.73 (1994), 383–413.
[Ka3] M. Kashiwara,On crystal bases, in Representations of groups Banff, AB, 1994, B. N. Allison and G. H. Cliff, eds.), 155–197; CMS Conference Proceedings, vol. 16, American Mathematical Society, Providence, RI, 1995.
[Ko] B. Kostant,Lie group representations on polynomial rings, Amer. J. Math.85 (1963), 327–404.
[Ma] S. Majid,Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group, Commun. Math. Phys.156 (1993), 607–638.
[RS] N. Yu. Reshetikhin and M. A. Semenov-Tian-Shansky,Quantum R-matrices and factorization problems, J. Geom. Phys.5 (1988), 533–550.
[Ri] R. W. Richardson,The conjugating representation of a semisimple group, Invent. Math.54, (1979), 229–245.
[Ro] M. Rosso,Représentations des groupes quantiques, Séminaire Bourbaki no. 744, Astérisque201–202–203 (1991), 443–483.
[Sw] M. E. Sweedler,Hopf Algebras, Mathematics Lecture Note Series, Benjamin, New York, 1969.
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Baumann, P. Another proof of Joseph and Letzter's separation of variables theorem for quantum groups. Transformation Groups 5, 3–20 (2000). https://doi.org/10.1007/BF01237175
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DOI: https://doi.org/10.1007/BF01237175