Free quantum analogue of Coxeter group $D_4$

We define the quantum group $D_4^+$ -- a free quantum version of the demihyperoctahedral group $D_4$ (the smallest representative of the Coxeter series $D$). In order to do so, we construct a free analogue of the property that a $4times4$ 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 $D_4^+$ is then defined by imposing this generalized determinant condition on the free hyperoctahedral group $H_4^+$. Moreover, we give a detailed combinatorial description of the representation category of $D_4^+$.