Free quantum analogue of Coxeter group D4

Abstract

We define the quantum group D+ 4 – a free quantum version of the demihyperoctahedral group D4 (the smallest representa tive of the Coxeter series D). In order to do so, we construct a free analogue of the property that a 4 × 4 matrix has deter minant one. Such analogues of determinants are usually very hard to define for free quantum groups in general and our result only holds for the matrix size N = 4. The free D+ 4 is then defined by imposing this generalized determinant con dition on the free hyperoctahedral group H+ 4 . Moreover, we give a detailed combinatorial description of the representation category of D+4 .