Invariants of the half-liberated orthogonal group

Abstract

The half-liberated orthogonal group $O_n^{\ast }$ 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 $O_n^{\ast }$ and $U_n$, a non abelian discrete group playing the role of weight lattice for $O_n^{\ast }$, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that the discrete quantum group dual to $O_n^{\ast }$ has polynomial growth.

24 pages