Multiplicatively closed Markov models must form Lie algebras
Keywords:Lie algebras, continuous-time Markov chains, semigroups, phylogenetics.
AbstractWe prove that the probability substitution matrices obtained from a continuous-time Markov chain form a multiplicatively closed set if and only if the rate matrices associated with the chain form a linear space spanning a Lie algebra. The key original contribution we make is to overcome an obstruction, due to the presence of inequalities that are unavoidable in the probabilistic application, which prevents free manipulation of terms in the Bakerâ€“Campbellâ€“Haursdorff formula. doi:10.1017/S1446181117000359
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