QM Research Seminar: Higher Yang—Baxter equations via Steenrod squares

Speaker: Guillaume Laplante-Anfossi (University of Melbourne)

Abstract:

On the one hand, simplex equations are higher dimensional generalizations of the Yang—Baxter (aka triangle) equation. The tetrahedron equation was introduced by Zamolodchikov in 1980, and the equations for higher dimensional simplices can be defined via higher Bruhat orders, a family of posets introduced by Manin and Schechtman in 1989. On the other hand, Steenrod squares are operations acting on the mod 2 cohomology of a topological space, introduced by Steenrod in 1947. They arise from correcting homotopically the lack of cocommutativity of the Alexander—Whitney diagonal at the cochain level. Together with Nicholas Williams, we found a surprising bijection between the terms of these two constructions: each side of a higher Yang—Baxter equation defines a Steenrod cup-i coproduct, and vice-versa. A conceptual explanation for, as well as consequences of this result remain unknown to us, and I hope to stimulate discussions about what could grow out of this new dictionary between pieces of mathematical physics and classical algebraic topology.