Separation of Variables for

Lopes, Samuel A.

Canadian mathematical bulletin, Tome 48 (2005) no. 4, pp. 587-600

Résumé

Let ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$ be the positive part of the quantized enveloping algebra ${{U}_{q}}\left( \mathfrak{s}{{\Iota }_{n+1}} \right)$ . Using results of Alev–Dumas and Caldero related to the center of ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$ , we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra $U\left( \mathfrak{g} \right)$ of a complex semisimple Lie algebra g, and also of an analogous result of Joseph–Letzter for the quantum algebra ${{\overset{\scriptscriptstyle\smile}{U}}_{q}}\left( \mathfrak{g} \right)$ . Of greater importance to its representation theory is the fact that ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$ is free over a larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$ to ${{U}_{q}}{{\left( \mathfrak{s}{{\Iota }_{n+1}} \right)}^{+}}$ provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules.

Détail

DOI : 10.4153/CMB-2005-054-8

Mots-clés : 17B37, 16W35, 17B10, 16D60

Bibliographie

[1] [1] Alev, J. and Dumas, F., Sur le corps des fractions de certaines algèbres quantiques. J. Algebra 170(1994), 229–265. Google Scholar

[2] [2] Benkart, G. and Gorelik, M., The separation and annihilation theorems for down-up algebras, preprint. Google Scholar

[3] [3] Benkart, G. and Roby, T., Down-up algebras. J. Algebra 209(1998), 305–344. Google Scholar

[4] [4] Caldero, P., Générateurs du centre de Ŭ(sl(N + 1)). Bull. Sci. Math. 118(1994), 177–208. Google Scholar

[5] [5] Caldero, P., Sur le centre de U(n+). Beiträge Algebra Geom. 35(1994), 13–24, Google Scholar

[6] [6] Caldero, P., Étude des q-commutations dans l’algèbre U(n+). J. Algebra 178(1995), 444–457. Google Scholar

[7] [7] Caldero, P., On harmonic elements for semi-simple Lie algebras. Adv. Math. 166(2002), 73–99. Google Scholar

[8] [8] De Concini, C. and Kac, V. G., Representations of quantum groups at roots of 1. In: Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Progr. Math. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 471–506. Google Scholar

[9] [9] Dixmier, J., Sur les représentations unitaires des groupes de Lie nilpotents. IV. Canad. J. Math. 11(1959), 321–344. Google Scholar

[10] [10] Drinfel’d, V. G., Hopf algebras and the quantum Yang-Baxter equation. Dokl. Akad. Nauk SSSR 283(1985), 1060–1064. Google Scholar

[11] [11] Futorny, V. and Ovsienko, S., Kostant's theorem for special filtered algebras. Bull. London Math. Soc. 37(2005), 187–199. Google Scholar

[12] [12] Jantzen, J. C., Lectures on Quantum Groups. Graduate Studies in Mathematics 6, American Mathematical Society, Providence, RI, 1996. Google Scholar

[13] [13] Jimbo, M., A q-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10(1985), 63–69. Google Scholar

[14] [14] Joseph, A. and Letzter, G., Separation of variables for quantized enveloping algebras. Amer. J. Math. 116(1994), 127–177. Google Scholar

[15] [15] Kostant, B., Lie group representations on polynomial rings. Amer. J. Math. 85(1963), 327–404. Google Scholar

[16] [16] Lusztig, G., Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra. J. Amer.Math. Soc. 3(1990), 257–296. Google Scholar

[17] [17] Ringel, C. M., PBW-bases of quantum groups. J. Reine Angew.Math. 470(1996), 51–88. Google Scholar

[18] [18] Takeuchi, M., The q-bracket product and quantum enveloping algebras of classical types. J. Math. Soc. Japan 42(1990), 605–629. Google Scholar

[19] [19] Yamane, H., A Poincaré-Birkhoff-Witt theorem for quantized universal enveloping algebras of type AN. Publ. Res. Inst.Math. Sci. 25(1989), 503–520. Google Scholar