Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of three-dimensional real Lie algebras. The Jacobi operators of these quantum algebras are explicitly calculated.
We analyse perturbative expansions of the invariants defined from unitary representations of the Quantum Lorentz Group in two different ways, namely using the Kontsevich Integral and weight systems, and the R-matrix in the Quantum Lorentz Group defined by Buffenoir and Roche. The two formulations are proved to be equivalent; and they both yield ℂ[[h]]h-valued knot invariants related with the Melvin-Morton expansion of the Coloured Jones Polynomial.
We study dimensional left-covariant differential calculi on the quantum group . In this way we obtain four classes of differential calculi which are algebraically much simpler as the bicovariant calculi. The algebra generated by the left-invariant vector fields has only quadratic-linear relations and posesses a Poincaré-Birkhoff-Witt basis. We use the concept of universal (higher order) differential calculus associated with a given left-covariant first order differential calculus. It turns out...
Generalizing the notion of matched pair of groups, we define and study matched pairs of locally compact groupoids endowed with Haar systems, in order to give new examples of measured quantum groupoids.
We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in...