Invariants of the half-liberated orthogonal group

TL;DR

This paper studies the algebraic properties and representation theory of the half-liberated orthogonal quantum group $O_n^*$, revealing its structure and growth characteristics.

Contribution

It classifies irreducible representations of $O_n^*$ using a twisting relation and Lie algebra methods, and analyzes its dual's polynomial growth.

Findings

Classified irreducible representations of $O_n^*$.

Established polynomial growth of the dual quantum group.

Connected $O_n^*$ to Lie algebra techniques.

Abstract

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 for $O_n^*$, 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 $O_n^*$ has polynomial growth.