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We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with boundary into the symplectic group Sp(2n,R). These coordinates provide a noncommutative generalization of the parameterizations of the spaces of representations into SL(2,R) or PSL(2,R) given by Thurston, Penner, Kashaev and Fock-Goncharov. On the space of decorated symplectic representations the coordinates give a geometric realization of the noncommutative cluster-like structures introduced by Berenstein-Retakh. The locus of positive coordinates maps to the space of framed maximal representations. We use this to determine an explicit homeomorphism between the space of framed maximal representations and a quotient by the group O(n). This allows us to describe the homotopy type and, when n=2, to give an exact description of the singularities. Along the way, we establish a complete classification of pairs of nondegenerate quadratic forms.
We introduce the symplectic group $mathrm{Sp}_2(A,sigma)$ over a noncommutative algebra $A$ with an anti-involution $sigma$. We realize several classical Lie groups as $mathrm{Sp}_2$ over various noncommutative algebras, which provides new insights i
In arXiv:1802.02833 Guichard and Wienhard introduced the notion of $Theta$-positivity, a generalization of Lusztigs total positivity to real Lie groups that are not necessarily split. Based on this notion, we introduce in this paper $Theta$-positive
We consider, in a superspace, new operator dependent noncommutative (NC) geometries of the nonlinear quantum Hall limit related to classes of f-deformed Landau operators in the spherical harmonic well. Different NC coordinate algebras are determined
We show that the mapping class group acts properly on the space of maximal representations of the fundamental group of a closed Riemann surface into G when G = Sp(2n,R), SU(n,n), SO*(2n) or Spin(2,n).
We consider electrons in uniform external magnetic and electric fields which move on a plane whose coordinates are noncommuting. Spectrum and eigenfunctions of the related Hamiltonian are obtained. We derive the electric current whose expectation val