Invariants of the half-liberated orthogonal group

Banica, Teodor; Vergnioux, Roland

Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2137-2164.

Le groupe orthogonal semi-libéré $O_n^{*}$ est un groupe quantique intermédiaire entre le groupe orthogonal $O_n$ et sa version libre $O_n^{+}$. Nous discutons ici ses propriétés algébriques de base, et nous classifions ses représentations irréductibles. Cette classification est établie grâce à une mise en relation avec le groupe $U_n$ et des méthodes inspirées de la théorie des algèbres de Lie. Un groupe discret non abélien joue le rôle de réseau des poids. Nous utilisons ces résultats pour montrer que le groupe quantique discret dual est à croissance polynomiale.

The half-liberated orthogonal group $O_n^{*}$ appears as intermediate quantum group between the orthogonal group $O_n$, and its free version $O_n^{+}$. We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between $O_n^{*}$ and $U_n$, a non abelian discrete group playing the role of weight lattice, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that the dual discrete quantum group has polynomial growth.

Classification : 20G42, 16W30, 46L65
Keywords: Quantum group, maximal torus, root system
Mot clés : groupe quantique, tore maximal, système de racines

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