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Separation of Variables for

Published online by Cambridge University Press:  20 November 2018

Samuel A. Lopes
Samuel A. Lopes*
Affiliation:
Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal e-mail: slopes@fc.up.pt
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Abstract

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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.

Keywords

17B3716W3517B1016D60
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 4 , 01 December 2005 , pp. 587 - 600
Copyright
Copyright © Canadian Mathematical Society 2005

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