Symmetry Lie Algebras of Varieties with Applications to Algebraic Statistics

The motivation for this paper is to detect when an irreducible projective variety [Formula: see text] is not toric. This is achieved by analyzing a Lie group and a Lie algebra associated with [Formula: see text]. If the dimension of [Formula: see text] is strictly less than the dimension of the aforementioned objects, then [Formula: see text] is not a toric variety. We provide an algorithm to compute the Lie algebra of an irreducible variety and use it to present examples of nontoric statistical models in algebraic statistics.