2011/11/17 by Ivan Chi-Ho Ip, Ip, Ivan Chi-Ho
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1111.4025
openalex publication_date 2011/11/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We found an explicit construction of a representation of the positive quantum group GLq+(N,\R) and its modular double GLq\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 GLq+(N,\R) for |q|=1, parametrized by the standard decomposition of the longest element w0∈ 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=q2 vu by positive essentially self-adjoint operators. The modular double arises naturally from the transcendental relations, and an L2(GLq\til[q]+(N,\R)) space in the von Neumann setting can also be defined.