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Multiplicatively closed Markov models must form Lie algebras

Sumner, JG ORCID: 0000-0001-9820-0235 2017 , 'Multiplicatively closed Markov models must form Lie algebras' , ANZIAM Journal, vol. 59, no. 2 , pp. 240-246 , doi: 10.1017/S1446181117000359.

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Abstract

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.

Item Type: Article
Authors/Creators:Sumner, JG
Keywords: multiplicative closure, Markov models, semigroups, phylogenetics, lie algebras, continuous-time Markov chains
Journal or Publication Title: ANZIAM Journal
Publisher: Australian Mathematics Publ Assoc Inc
ISSN: 1446-1811
DOI / ID Number: 10.1017/S1446181117000359
Copyright Information:

Copyright 2017 Australian Mathematical Society

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