Symmetry Lie Algebras of Varieties with Applications to Algebraic Statistics

TL;DR

This paper introduces a method to determine whether a projective variety is toric by analyzing associated Lie algebras, providing an algorithm and examples relevant to algebraic statistics.

Contribution

It presents a novel algorithm for computing Lie algebras of varieties and applies it to identify non-toric models in algebraic statistics.

Findings

The algorithm effectively distinguishes non-toric varieties.

Examples demonstrate application to statistical models.

Lie algebra dimension comparison indicates non-toric nature.

Abstract

The motivation for this paper is to detect when an irreducible projective variety V is not toric. We do this by analyzing a Lie group and a Lie algebra associated to V. If the dimension of V is strictly less than the dimension of the above mentioned objects, then V is not a toric variety. We provide an algorithm to compute the Lie algebra of an irreducible variety and use it to provide examples of non-toric statistical models in algebraic statistics.