Most of my recent research has revolved around algebraic structures arising in the description of the dynamics of open quantum systems. This theme can be traced back at least to the work of Lindblad, who showed that derivations occur naturally in the master equation of Markovian open quantum systems. The propagator of this Lindblad master equation is called a quantum Markov semigroup. The interplay between quantum Markov semigroups, derivations and bimodules becomes particularly rich if one moves beyond the realm of of matrix algebras to general von Neumann algebras. Far from being restricted to applications in mathematical physics, quantum Markov semigroups in this general setting touch upon a variety of purely mathematical fields, from von Neumann algebra theory and free probability to unitary representations and group cocycles. Another algebraic structure that has begun to interest me recently is the connection between Lie algebra representations and quantum Markov semigroups. One fascinating aspect of this circle of ideas is that they yield not only qualitative structural insights, but have also been instrumental in the quantitative analysis of the decay to equilibrium of open quantum systems in recent years.