Gauss-Lusztig Decomposition for $G{L}_q^{+}(N,R)$ and Representation by q-Tori

Abstract

We found an explicit construction of a representation of the positive quantum group $G{L}_q^{+}(N,\mathbb{R})$ and its modular double $GL_{q\til[q]}^+(N,\R)$ by positive essentially self-adjoint operators. Generalizing Lusztig's parametrization, we found a Gauss type decomposition for the totally positive quantum group $G{L}_q^{+}(N,\mathbb{R})$ for $|q|=1$, parametrized by the standard decomposition of the longest element $w_0\in W=S_{N-1}$. 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 $uv=q^{2}vu$ by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an $L^2(GL_{q\til[q]}^+(N,\R))$ space in the von Neumann setting can also be defined.

Reorganizing the contents involving positivity. Renewed references

References in corpus (2)

• Positive Representations of Split Real Simply-laced Quantum Groups
• Representation of the Quantum Plane, its Quantum Double and Harmonic Analysis on $G{L}_q^{+}(2,R)$

Cited by in corpus (2)

• Positive Representations of Split Real Simply-laced Quantum Groups
• Positive representations, multiplier Hopf algebra, and continuous canonical basis