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Lopes, Samuel A.
2006.
Primitive Ideals of.
Communications in Algebra,
Vol. 34,
Issue. 12,
p.
4523.
Article contents
Separation of Variables for 
Published online by Cambridge University Press: 20 November 2018
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.
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