arXiv subjects

Elizabeth Xiao

Publications and source records attributed to Elizabeth Xiao.

3 recordsLinked to original sources

arXiv2026

A Hopf algebra on nonplanar binary forests

We equip the graded polynomial algebra generated by nonplanar rooted binary trees with a Hopf algebra structure by defining a coproduct which disallows cutting both children of any given vertex, refining Connes-Kreimer's notion of admissible cuts. We show that the terms in this coproduct have an additional combinatorial interpretation in terms of subsets of leaves, which facilitates the construction of Hopf algebra morphisms involving this Hopf algebra, and creates a connection with a Hopf algebra of Bruned used in the renormalization of stochastic processes. Finally, we show that this Hopf algebra is dual to the universal enveloping algebra of a Lie algebra arising from a pre-Lie operator on binary trees based on edge-insertion.

math.CO↗

arXiv2025

A Lie-algebraic perspective on Tree-Adjoining Grammars

We provide a novel mathematical implementation of tree-adjoining grammars using two combinatorial definitions of graphs. With this lens, we demonstrate that the adjoining operation defines a pre-Lie operation and subsequently forms a Lie algebra. We demonstrate the utility of this perspective by showing how one of our mathematical formulations of TAG captures properties of the TAG system without needing to posit them as additional components of the system, such as null-adjoining constraints and feature TAG.

cs.CL↗

arXiv2023

On the anisotropy theorem of Papadakis and Petrotou

We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen's g-conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic 2 follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field $\mathbb{Q}$. These results provide another self-contained proof of the g-conjecture for homology spheres in characteristic 2.

math.CO↗