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arXiv : 0807.3256

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Journal of Lie Theory 19, 2 (2009) 339-370

Diamond representations of rank two semisimple Lie algebras

Boujemaa Agrebaoui 1, Didier Arnal 2, Olfa Khlifi 1

cmcu 06/S 1502 Collaboration(s)

(2009)

The present work is a part of a larger program to construct explicit combinatorial models for the (indecomposable) regular representation of the nilpotent factor $N$ in the Iwasawa decomposition of a semi-simple Lie algebra $\mathfrak g$, using the restrictions to $N$ of the simple finite dimensional modules of $\mathfrak g$. Such a description is given in \cite{[ABW]}, for the cas $\mathfrak g=\mathfrak{sl}(n)$. Here, we give the analog for the rank 2 semi simple Lie algebras (of type $A_1\times A_1$, $A_2$, $C_2$ and $G_2$). The algebra $\mathbb C[N]$ of polynomial functions on $N$ is a quotient, called reduced shape algebra of the shape algebra for $\mathfrak g$. Basis for the shape algebra are known, for instance the so called semi standard Young tableaux (see \cite{[ADLMPPrW]}). We select among the semi standard tableaux, the so called quasi standard ones which define a kind basis for the reduced shape algebra.

1 :	Unité de recherche UR 09-06, Faculté des Sciences de Sfax (FSS)
	University of Sfax, Tunisia
2 :	Institut de Mathématiques de Bourgogne (IMB)
	CNRS : UMR5584 – Université de Bourgogne

Domaine

:

Mathématiques/Algèbres quantiques

Mots Clés : Young tableaux – représentations des algèbres simples

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Contributeur : Didier Arnal <>

Soumis le : Lundi 21 Juillet 2008, 14:56:55

Dernière modification le : Lundi 19 Novembre 2012, 09:34:11