Quantum anchor: \(U_q (sl(2))\) case
This is an overview article on a noncommutative analog of a hyperboloid together with the action of the Lie algebra \(\mathfrak s\mathfrak l(2, \mathbb C)\) on its complexified tangent bundle. The basic idea is that the fundamental vector fields for the latter action generate the space of vector fields on the hyperboloid as a module over functions. Starting from the quantum group \(U_{q}(\mathfrak s\mathfrak l(2))\) (for generic \(q\)), the noncommutative hyperboloid is defined as a quotient of the tensor algebra of the fundamental representation of \(U_{q}(\mathfrak s\mathfrak l(2))\). Next, vector fields on the quantum hyperboloid are defined as a quotient of functions with values in the \(q\)-analog of the three-dimensional representation of \(\mathfrak s\mathfrak l(2, \mathbb C)\). This turns out to be a flat deformation of its classical counterpart. This left module admits a ``quantum anchor'', i.e. a right module structure such that three specific elements realize an action of the braided Lie algebra \(\mathfrak s\mathfrak l(2)_{q}\).
scientific article; zbMATH DE number 1502064