# Multiplicatively closed Markov models must form Lie algebras

@article{Sumner2017MultiplicativelyCM,
title={Multiplicatively closed Markov models must form Lie algebras},
author={Jeremy G. Sumner},
journal={Anziam Journal},
year={2017},
volume={59},
pages={240-246}
}
• J. Sumner
• Published 4 April 2017
• Mathematics
• Anziam Journal
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.
11 Citations

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## References

SHOWING 1-10 OF 19 REFERENCES

### Lie Algebra Solution of Population Models Based on Time-Inhomogeneous Markov Chains

The Lie algebraic method is presented, and applied to three biologically well motivated examples, and the result is a solution form that is often highly computationally advantageous.

### On the ideals of equivariant tree models

• Mathematics, Computer Science
• 2007
Equivariant tree models in algebraic statistics are introduced, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group-based modelssuch as the Jukes–Cantor and Kimura models and yield generators of the full ideal.

### On a Lie-theoretic approach to generalized doubly stochastic matrices and applications

In this article, we study generalized doubly stochastic matrices using the theory of Lie groups and Lie algebras. Applications to the inverse eigenvalue problem for symmetric doubly stochastic

### On a Lie-theoretic approach to generalized doubly stochastic matrices and applications

In this article, we study generalized doubly stochastic matrices using the theory of Lie groups and Lie algebras. Applications to the inverse eigenvalue problem for symmetric doubly stochastic

### Markov‐type Lie groups in GL(n,R)

The general linear group GL(n,R) is decomposed into a Markov‐type Lie group and an abelian scale group. The Markov‐type Lie group basis is shown to generate all singly stochastic matrices which are

### Lie Markov models.

• Mathematics
Journal of theoretical biology
• 2012

### SEMIGROUPS IN LIE GROUPS, SEMIALGEBRAS IN LIE ALGEBRAS

• Mathematics
• 1985
Consider a subsemigroup of a Lie group containing the identity and being ruled by one-parameter semigroups near the identity. We associate with it the set W of its tangent vectors at the identity and

### Lie Markov models with purine/pyrimidine symmetry

• Mathematics
Journal of mathematical biology
• 2015
This paper derives the complete hierarchy of Lie Markov models that respect the grouping of nucleotides into purines and pyrimidines—that is, models with purine/pyrimidine symmetry.

### A New Hierarchy of Phylogenetic Models Consistent with Heterogeneous Substitution Rates

• Economics
Systematic biology
• 2015
Compared against the benchmark of the ever-popular GTR model, it is found that as a whole the Lie Markov models perform well, with the best performing models having 8–10 parameters and the ability to recognize the distinction between purines and pyrimidines.

### Is the general time-reversible model bad for molecular phylogenetics?

• Biology
Systematic biology
• 2012
Examples are given that demonstrate why the general time-reversible GTR model may pose a problem for phylogenetic analysis and add GTR to the growing list of factors that are known to cause model misspecification in phylogenetics.