Recent developments in quantum information led to a generalised notion of reference frames transformations, relevant when reference frames are associated to quantum systems. In this talk, I discuss whether such quantum reference frame transformations could realise a notion of deformed symmetries formalised as quantum group transformations. In particular, I show the correspondence between quantum reference frame transformations and transformations generated by a quantum deformation of the Galilei group with commutative time, taken at the first order in the quantum deformation parameter. This correspondence is found once the group noncommutative transformation parameters are represented on the phase space of a quantum particle, and upon setting the quantum deformation parameter to be proportional to the inverse of the mass of the particle serving as the quantum reference frame.[-]

This talk introduces a class of Hopf algebras, called T -Poincaré, which represent, arguably, the simplest small scale/high energy quantum group deformations of the Poincaré group. Starting from some reasonable assumptions on the structure of the commutators, I am able to show that these models arise from a class of classical r-matrices on the Poincaré group. These have been known since the work of Zakrzewski and Tolstoy, and allow me to identify 16 multiparametric models. Each T -Poincaré model admits a canonical 4-dimensional quantum homogeneous spacetime, T -Minkowski, which is left invariant by the coaction of the group. A key result is the systematic unification provided by this framework, which incorporates well-established non-commutative spacetimes like Moyal, lightlike κ-Minkowski, and ρ-Minkowski as specific instances. I will then outline all the mathematical structures that are necessary in order to study field theory on these spaces: differential and integral calculus, noncommutative Fourier theory, and braided tensor products. I will then discuss how to describe (classical) Standard Model fields within this framework, and the challenges associated with quantum field theory. Particular focus is placed on the Poincar´e covariance of these models, with the goal of finding a mathematically consistent model of physics at the Planck scale that preserves the principle of Special Relativity while possessing a noncommutativity length scale.[-]

The classification of topological phases in each Altland-Zirnbauer symmetry class is related to one of 2 complex or 8 real $\mathrm{K}$-theory by Kitaev. A more general framework, in which we deal with systems with an arbitrary symmetry of quantum mechanics specified by Wigner's theorem, is introduced by Freed and Moore by using a generalization of twisted $\mathrm{K}$-theory. In this talk, we introduce the definition of twisted $\mathrm{K}$-theory in the sense of Freed-Moore for $C^*$-algebras, which gives a framework for the study of topological phases of non-periodic systems with a symmetry of quantum mechanics. Moreover, we introduce uses of basic tools in $\mathrm{K}$-theory of operator algebras such as inductions and the Green-Julg isomorphism for the study of topological phases.[-]

Let $\mathfrak{h}$ be a finite dimensional real Leibniz algebra. Exactly as the linear dual space of a Lie algebra is a Poisson manifold with respect to the Kostant-Kirillov-Souriau (KKS) bracket, $\mathfrak{h}^*$ can be viewed as a generalized Poisson manifold. The corresponding bracket is roughly speaking the evaluation of the KKS bracket at $0$ in one variable. This (perhaps strange looking) bracket comes up naturally when quantizing $\mathfrak{h}^*$ in an analoguous way as one quantizes the dual of a Lie algebra. Namely, the product $X \vartriangleleft Y = exp(ad_X)(Y)$ can be lifted to cotangent level and gives than a symplectic micromorphism which can be quantized by Fourier integral operators. This is joint work with Benoit Dherin (2013). More recently, we developed with Charles Alexandre, Martin Bordemann and Salim Rivire a purely algebraic framework which gives the same star-product.[-]