TY - JOUR

T1 - The adjoint braid arrangement as a combinatorial Lie algebra via the Steinmann relations

AU - Liu, Zhengwei

AU - Norledge, William

AU - Ocneanu, Adrian

N1 - Funding Information:
The authors are grateful to the Templeton Religion Trust, which supported this research with grant TRT 0159 for the Mathematical Picture Language Project at Harvard University . This made possible the visiting appointment of Adrian Ocneanu and the postdoctoral fellowship of William Norledge for the academic year 2017-2018. Adrian Ocneanu wants to thank Penn State for unwavering support during decades of work, partly presented for the first time during his visiting appointment. We also thank Nick Early for discussions related to permutohedral cones and the Steinmann relations [12] , [13] . After a literature search, Early discovered that the relations, which were conjectured by Ocneanu to characterize the span of characteristic functions of permutohedral cones, were known in axiomatic quantum field theory as the Steinmann relations. Zhengwei Liu would like to thank Arthur Jaffe for many helpful suggestions.
Funding Information:
Supported by the Templeton Religion Trust grant TRT 0159 for the Mathematical Picture Language Project at Harvard University . The Harvard course of Adrian Ocneanu described work done by him at Penn State University, 1990-2017, partly supported by NSF grants DMS-9970677 , DMS-0200809 , DMS-0701589 .
Publisher Copyright:
© 2023 Elsevier B.V.

PY - 2023/8

Y1 - 2023/8

N2 - 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.

AB - 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.

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U2 - 10.1016/j.jpaa.2023.107349

DO - 10.1016/j.jpaa.2023.107349

M3 - Article

AN - SCOPUS:85148673535

SN - 0022-4049

VL - 227

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 8

M1 - 107349

ER -