We describe some results concerning the phase space of 3-dimensional Einstein gravity when space is a torus and with negative cosmological constant. The approach uses the holonomy matrices of flat SL(2,R) connections on the torus to parametrise the geometry. After quantization, these matrices acquire noncommuting entries, in such a way that they satisfy q-commutation relations and exhibit inter...

The classical Baker-Campbell-Hausdorff formula gives a recursive way to compute the Hausdorff series H = ln(ee ) for noncommuting X, Y . Formally H lives in the graded completion of the free Lie algebra L generated by X, Y . We present a closed explicit formula for H = ln(ee ) in a linear basis of the graded completion of the free metabelian Lie algebra L̄ = L/[[L, L], [L, L]].

We prove in this paper that if r and s are two semilinear power series in commuting variables and s has bounded coefficients, then r−s is a rational series. This result can be thought of as extending a result of Eilenberg for series in noncommuting variables. However, our proof is combinatorial, avoiding the use of a deeply algebraic cross-section theorem as in the noncommutative case.

The algebraic reformulation of the BMV conjecture is equivalent to a family of dimensionfree tracial inequalities involving positive semidefinite matrices. Sufficient conditions for these to hold in the form of algebraic identities involving polynomials in noncommuting variables have been given by Markus Schweighofer and the second author. Later the existence of these certificates has been sett...

We introduce the concept of cloning for classes of observables and classify cloning machines for qubit systems according to the number of parameters needed to describe the class under investigation. A no-cloning theorem for observables is derived and the connections between cloning of observables and joint measurements of noncommuting observables are elucidated. Relationships with cloning of st...

The deformed Galilean generators are constructed on a space where both position and momentum coordinates are noncommutating operators. A dynamical model invariant under deformed phase space transformations is constructed. The Dirac brackets of this model reproduce the original non-commutative algebra. Also, the deformed generators are abstracted from this model in a consistent manner. Finally, ...

We construct an invariant of boundary links which is at least as strong as the Kontsevich integral, determines the S-equivalence class of a boundary link and which takes values in a space of trivalent graphs whose edges are decorated by rational functions in noncommuting variables. Our invariant is axiomatically characterized by a universal property closely related to the Homology Surgery view ...