Geometry of the tangent space on the quantum hyperboloid (Q2719721)

This paper is devoted to the braided geometry of a braided hyperboloid. The main goal of the paper is to define a tangent space on the braided hyperboloid and to introduce the analogues of a metric and a connection. Let \(U_q (\text{sl}(2))\) be the quantum group and \({\mathcal A}^c_{h,q}\) a two-parameter noncommutative braided algebra which is the result of a double quantification of a Poisson bracket on the hyperboloid embedded as an orbit in \(\text{sl}(2)\). The author considers the tangent vector space \(T(H_q)\) of the quantum hyperboloid \(H_q\) as a factor \({\mathcal A}^c_{0,q}\)-module by analogy with the classical expression \(\{aU+bV+cW; a,b,c \in {\mathcal A}^c_{0,1}\}\) modulo \(\{f(2uW+vV+2wU); f \in {\mathcal A} ^c_{0,1} \}\), where \(U,V,W\) are the infinitesimal hyperbolic rotations corresponding to the generators \(u,v,w\) of \(\text{sl}(2)\). To obtain \(T(H_q)\) as a vector space of braided vector fields acting on \({\mathcal A}^c_{0,q}\) as operators, the author introduces a braided analogue of the adjoint action of \(\text{sl}(2)\). Finally, the author defines a \(U_q(\text{sl} (2))\)-covariant braided metric on \(T(H_q)\) and generalizes the notion of a linear connection \(\nabla\) without torsion partially defined on \(T(H_q)\).