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MULTIPLICATIVELY CLOSED MARKOV MODELS MUST FORM LIE ALGEBRAS

Part of: Lie algebras and Lie superalgebras Markov processes

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 

References

Campbell, J. E., “On a law of combination of operators”, Proc. Lond. Math. Soc. (3) 29 (1898) 14–32; doi:10.1112/plms/s1-29.1.14.Google Scholar
Draisma, J. and Kuttler, J., “On the ideals of equivariant tree models,”, Math. Ann. 344 (2009) 619–644; doi:10.1007/s00208-008-0320-6.Google Scholar
Felsenstein, J., Inferring phylogenies (Sinauer Associates, Sunderland, 2004);https://global.oup.com/ushe/product/inferring-phylogenies-9780878931774?.Google Scholar
Fernández-Sánchez, J., Sumner, J. G., Jarvis, P. D. and Woodhams, M. D., “Lie Markov models with purine/pyrimidine symmetry”, J. Math. Biol. 70 (2015) 855–891; doi:10.1007/s00285-014-0773-z.Google Scholar
Hasegawa, M., Kishino, H. and Yano, T., “Dating of human-ape splitting by a molecular clock of mitochondrial DNA”, J. Mol. Evol. 22 (1985) 160–174; doi:10.1007/BF02101694.CrossRefGoogle ScholarPubMed
Hilgert, J. and Hofmann, K. H., “Semigroups in Lie groups, semialgebras in Lie algebras”, Trans. Amer. Math. Soc. 288 (1985) 481–504; doi:10.1090/S0002-9947-1985-0776389-7.Google Scholar
House, T., “Lie algebra solution of population models based on time-inhomogeneous Markov chains”, J. Appl. Probab. 49 (2012) 472–481; doi:10.1239/jap/1339878799.Google Scholar
Johnson, J. E., “Markov-type Lie groups in $\text{GL}(n,R)$ ”, J. Math. Phys. 26 (1985) 252–257; doi:10.1063/1.526654.CrossRefGoogle Scholar
Mourad, B., “On a Lie-theoretic approach to generalized doubly stochastic matrices and applications”, Linear Multilinear Algebra 52 (2004) 99–113; doi:10.1080/0308108031000140687.Google Scholar
Semple, C. and Steel, M., Phylogenetics (Oxford University Press, Oxford, 2003); https://global.oup.com/academic/product/phylogenetics-9780198509424?cc=au&lang=en&.Google Scholar
Steel, M., Phylogeny: Discrete and random processes in evolution (SIAM, Philadelphia, PA, 2016); doi:10.1137/1.9781611974485.Google Scholar
Sumner, J. G., Fernández-Sánchez, J. and Jarvis, P. D., “Lie Markov models”, J. Theoret. Biol. 298 (2012) 16–31; doi:10.1016/j.jtbi.2011.12.017.CrossRefGoogle ScholarPubMed
Sumner, J. G., Jarvis, P. D., Fernández-Sánchez, J., Kaine, B. T., Woodhams, M. D. and Holland, B. R., “Is the general time-reversible model bad for molecular phylogenetics?”, Syst. Biol. 61 (2012) 1069–1074; doi:10.1093/sysbio/sys042.Google Scholar
Tavaré, S., “Some probabilistic and statistical problems in the analysis of DNA sequences”, Lect. Math. Life Sci. (American Society) 17 (1986) 57–86; https://www.scopus.com/record/display.uri?eid=2-s2.0-60649089791&origin=inward&txGid=47ea59cb4b0bb9183bac8afba8e05dc1.Google Scholar
Woodhams, M. D., Fernández-Sánchez, J. and Sumner, J. G., “A new hierarchy of phylogenetic models consistent with heterogeneous substitution rates”, Syst. Biol. 64 (2015) 638–650;doi:10.1093/sysbio/syv021.Google Scholar