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hal-00300612, version 1

## Diamond representations of rank two semisimple Lie algebras

Boujemaa Agrebaoui () 1, Didier Arnal () 2, Olfa Khlifi () 1

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

• 1:  Unité de recherche UR 09-06, Faculté des Sciences de Sfax (FSS)

• University of Sfax, Tunisia Route de Soukra, km 3,5 BP 1171, 3000Sfax, Tunisie Tunisia
• 2:  Institut de Mathématiques de Bourgogne (IMB)
• http://math.u-bourgogne.fr/IMB/
CNRS : UMR5584 – Université de Bourgogne 9, avenue Alain Savary - B.P. 47 870 - 21078 Dijon Cedex - France France

### Bibliographic reference

• Type of document: Articles in peer-reviewed journal
• Subject: Mathematics/Quantum Algebra
• Title: Diamond representations of rank two semisimple Lie algebras
• Abstract: 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.
• Fulltext language: English
• Production date: 2008-07-10
• Journal: Journal of Lie Theory
• Audience: not specified
• Publication date: 2009
• Submission date: 2009
• Volume: 19
• Issue: 2
• Page, identifiant, ...: 339-370
• Keyword(s): Young tableaux – représentations des algèbres simples
• Classification: 05E10; 05A15; 17B10
• Project(s), collaboration(s): cmcu 06/S 1502
• Contract, financing: partenariat Hubert Curien Utique

### Attached file list to this document:

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• hal-00300612, version 1
• oai:hal.archives-ouvertes.fr:hal-00300612
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• Submitted on: Monday, 21 July 2008 14:56:55
• Updated on: Monday, 19 November 2012 09:34:11