Lie geometry of 2×2 Markov matrices.

In recent work discussing model choice for continuous-time Markov chains, we have argued that it is important that the Markov matrices that define the model are closed under matrix multiplication [6, 7]. The primary requirement is then that the associated set of rate matrices form a Lie algebra. For the generic case, this connection to Lie theory seems to have first been made by [3], with applications for specific models given in [1] and [2]. Here we take a different perspective: given a model that forms a Lie algebra, we apply existing Lie theory to gain additional insight into the geometry of the associated Markov matrices. In this short note, we present the simplest case possible of 2×2 Markov matrices. The main result is a novel decomposition of 2×2 Markov matrices that parameterises the general Markov model as a perturbation away from the binary-symmetric model. This alternative parameterisation provides a useful tool for visualising the binary-symmetric model as a submodel of the general Markov model. keywords: Lie algebras, algebra, symmetry, Markov chains, phylogenetics 1 Results Consider the set of real 2× 2 Markov matrices