Gauss–Lusztig decomposition for positive quantum groups and representation by q-tori

We found an explicit construction of a representation of the positive quantum group GLq+(N,R) and its modular double GLqq˜+(N,R) by positive essentially self-adjoint operators. Generalizing Lusztig's parametrization, we found a Gauss type decomposition for the totally positive quantum group GLq+(N,R) parametrized by the standard decomposition of the longest element w0∈W=SN−1w0∈W=SN−1. Under this parametrization, we found explicitly the relations between the standard quantum variables, the relations between the quantum cluster variables, and realizing them using non-compact generators of the q  -tori uv=q2vuuv=q2vu by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an L2(GLqq˜+(N,R)) space in the von Neumann setting can also be defined.