Gauss-Lusztig Decomposition for $GL_q^+(N,R)$ and Representation by q-Tori

Details Gauss-Lusztig Decomposition for $GL_q^+(N,R)$ and Representation by q-Tori Gauss-Lusztig Decomposition for $GL_q^+(N,R)$ and Representation by q-Tori

2011-11-17

arxiv.org/abs/1111.4025v2

Mathematics

Quantum Algebra

Added Remark 5.3, Theorem 5.7, replaced diagrams with tikz, renewed references and fixed minor typos

Scientific paper

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

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