Sur les semi-invariants d'une sous-algèbre parabolique d'une algèbre enveloppante quantifiée
Identifieur interne : 001178 ( Main/Merge ); précédent : 001177; suivant : 001179Sur les semi-invariants d'une sous-algèbre parabolique d'une algèbre enveloppante quantifiée
Auteurs : F. Fauquant-Millet [France] ; A. Joseph [France, Israël]Source :
- Transformation Groups [ 1083-4362 ] ; 2001-06-01.
Abstract
Abstract: Let Ŭ q (g π) be the simply connected quantized enveloping algebra of the complex semisimple finite dimensional Lie algebrag π. Letp be a parabolic subalgebra ofg π and $$\mathcal{P}: = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{U} _q \left( {g_\pi } \right)$$ the Hopf subalgebra of Ŭ q (g π) associated top. LetY $$\left( \mathcal{P} \right)$$ be the space generated by the semi-invariants of $$\mathcal{P}$$ under adjoint action. We describe a basis forY $$\left( \mathcal{P} \right)$$ , show that it is a polynomial algebra and describe its rank, which we compute explicitly in a number of cases. Except for the Borel and some special cases a corresponding result is not known for the semi-centre of the enveloping algebra ofp. However we show here that the latter has the same Gelfand-Kirillov dimension asY $$\left( \mathcal{P} \right)$$ . We also describe the multiplication rules of the above basos elements ofY $$\left( \mathcal{P} \right)$$ . These are analogous to “fusion rules” in tensor product decomposition and their derivation obtains from an analysis of theR-matrix.
Url:
DOI: 10.1007/BF01597132
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<front><div type="abstract" xml:lang="en">Abstract: Let Ŭ q (g π) be the simply connected quantized enveloping algebra of the complex semisimple finite dimensional Lie algebrag π. Letp be a parabolic subalgebra ofg π and $$\mathcal{P}: = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{U} _q \left( {g_\pi } \right)$$ the Hopf subalgebra of Ŭ q (g π) associated top. LetY $$\left( \mathcal{P} \right)$$ be the space generated by the semi-invariants of $$\mathcal{P}$$ under adjoint action. We describe a basis forY $$\left( \mathcal{P} \right)$$ , show that it is a polynomial algebra and describe its rank, which we compute explicitly in a number of cases. Except for the Borel and some special cases a corresponding result is not known for the semi-centre of the enveloping algebra ofp. However we show here that the latter has the same Gelfand-Kirillov dimension asY $$\left( \mathcal{P} \right)$$ . We also describe the multiplication rules of the above basos elements ofY $$\left( \mathcal{P} \right)$$ . These are analogous to “fusion rules” in tensor product decomposition and their derivation obtains from an analysis of theR-matrix.</div>
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