Multiplicatively closed Markov models must form Lie algebras

Jeremy G. Sumner


We 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 Baker–Campbell–Haursdorff formula.



Lie algebras, continuous-time Markov chains, semigroups, phylogenetics.


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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.