# Multiplicatively closed Markov models must form Lie algebras

## TL;DR

This paper establishes that continuous-time Markov models are multiplicatively closed if and only if their rate matrices form a Lie algebra, overcoming key obstacles in applying the Baker-Campbell-Hausdorff formula.

## Contribution

It proves a necessary and sufficient condition linking multiplicative closure of Markov models to Lie algebra structures of rate matrices, addressing a significant mathematical challenge.

## Key findings

- Markov models form a Lie algebra if multiplicatively closed
- Overcomes obstacles in applying Baker-Campbell-Hausdorff formula
- Provides a fundamental characterization of Markov model structure

## 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 to 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, that prevents free manipulation of terms in the Baker-Campbell-Haursdorff formula.