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.
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