The adjoint braid arrangement as a combinatorial Lie algebra via the Steinmann relations

TL;DR

This paper introduces a combinatorial Lie algebra structure on the adjoint braid arrangement using Steinmann relations, revealing new algebraic connections relevant to quantum field theory.

Contribution

It establishes a novel Lie algebra structure on the adjoint braid arrangement via forest derivatives and Steinmann relations, linking combinatorics and quantum field theory.

Findings

Forest derivatives factorize under Steinmann relations.

Functions satisfying Steinmann relations form a Lie coalgebra comodule.

The adjoint braid arrangement acquires a Lie algebra structure in vector species.

Abstract

We study a certain discrete differentiation of piecewise-constant functions on the adjoint of the braid hyperplane arrangement, defined by taking finite-differences across hyperplanes. In terms of Aguiar-Mahajan's Lie theory of hyperplane arrangements, we show that this structure is equivalent to the action of Lie elements on faces. We use layered binary trees to encode flags of adjoint arrangement faces, allowing for the representation of certain Lie elements by antisymmetrized layered binary forests. This is dual to the well-known use of (delayered) binary trees to represent Lie elements of the braid arrangement. The discrete derivative then induces an action of layered binary forests on piecewise-constant functions, which we call the forest derivative. Our main result states that forest derivatives of functions factorize as external products of functions precisely if one restricts to functions which satisfy the Steinmann relations, which are certain four-term linear relations appearing in the foundations of axiomatic quantum field theory. We also show that the forest derivative satisfies the Lie properties of antisymmetry the Jacobi identity. It follows from these Lie properties, and also crucially factorization, that functions which satisfy the Steinmann relations form a left comodule of the Lie cooperad, with the coaction given by the forest derivative. Dually, this endows the adjoint braid arrangement modulo the Steinmann relations with the structure of a Lie algebra internal to the category of vector species. This work is a first step towards describing new connections between Hopf theory in species and quantum field theory.