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# Separation of Variables for $U_{q}(\mathfrak{sl}_{n+1})^{+}$

Published:2005-12-01
Printed: Dec 2005
• Samuel A. Lopes
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## Abstract

Let $U_{q}(\SL)^{+}$ be the positive part of the quantized enveloping algebra $U_{q}(\SL)$. Using results of Alev--Dumas and Caldero related to the center of $U_{q}(\SL)^{+}$, we show that this algebra is free over its center. This is reminiscent of Kostant's separation of variables for the enveloping algebra $U(\g)$ of a complex semisimple Lie algebra $\g$, and also of an analogous result of Joseph--Letzter for the quantum algebra $\Check{U}_{q}(\g)$. Of greater importance to its representation theory is the fact that $\U{+}$ is free over a larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$ to $\U{+}$ provides infinite-dimensional modules with good properties, including a grading that is inherited by submodules.
 MSC Classifications: 17B37 - Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 16W35 - Ring-theoretic aspects of quantum groups (See also 17B37, 20G42, 81R50)17B10 - Representations, algebraic theory (weights) 16D60 - Simple and semisimple modules, primitive rings and ideals

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