Recent work has discussed the importance of multiplicative closure for the Markov mod-
els used in phylogenetics. For continuous-time Markov chains, a sufficient condition for
multiplicative closure of a model class is ensured by demanding that the set of rate-matrices
belonging to the model class form a Lie algebra. It is the case that some well-known Markov
models do form Lie algebras and we refer to such models as “Lie Markov models”. However
it is also the case that some other well-known Markov models unequivocally do not form
Lie algebras (GTR being the most conspicuous example).
In this paper, we will discuss how to generate Lie Markov models by demanding that
the models have certain symmetries under nucleotide permutations. We show that the
Lie Markov models include, and hence provide a unifying concept for, “group-based” and
“equivariant” models. For each of two, three and four character states, the full list of
Lie Markov models with maximal symmetry is presented and shown to include interesting
examples that are neither group-based nor equivariant. We also argue that our scheme
is pleasing in the context of applied phylogenetics, as, for a given symmetry of nucleotide
substitution, it provides a natural hierarchy of models with increasing number of parameters.
