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