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MULTIPLICATIVELY CLOSED MARKOV MODELS MUST FORM LIE ALGEBRAS

Part of: Markov processes Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  23 October 2017

JEREMY G. SUMNER Open the ORCID record for JEREMY G. SUMNER [Opens in a new window]
JEREMY G. SUMNER*
Affiliation:
University of Tasmania, Hobart 7000, Australia email jsumner@utas.edu.au
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Abstract

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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.

Keywords

Lie algebrascontinuous-time Markov chainssemigroupsphylogenetics

MSC classification

Primary: 17B45: Lie algebras of linear algebraic groups
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 2 , October 2017 , pp. 240 - 246
Copyright
© 2017 Australian Mathematical Society 

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