Free quantum analogue of Coxeter group D4

We define the quantum group D4+ – a free quantum version of the demihyperoctahedral group D4 (the smallest representative of the Coxeter series D). In order to do so, we construct a free analogue of the property that a 4×4 matrix has determinant 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 D4+ is then defined by imposing this generalized determinant condition on the free hyperoctahedral group H4+. Moreover, we give a detailed combinatorial description of the representation category of D4+.