Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence

Steven Duplij

doi:10.3390/math9090972

Abstract

In this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.

Highlights

We begin by observing that the defining relations of the von Neumann regular semigroups (e.g., References [1,2,3]) and the Artin braid group [4,5] correspond to such properties of ternary matrices as idempotence and the orders of elements
The higher regular semigroups obtained in this way have appeared previously in semisupermanifold theory [6] and higher regular categories in Topological Quantum Field Theory [7]
The representations of the higher braid relations in vector spaces coincide with the higher braid equation and corresponding generalized R-matrix obtained in Reference [8], as do the ordinary braid group and the Yang-Baxter equation [9,10]

Summary

Introduction

We begin by observing that the defining relations of the von Neumann regular semigroups (e.g., References [1,2,3]) and the Artin braid group [4,5] correspond to such properties of ternary matrices (over the same set) as idempotence and the orders of elements (period). We generalize the correspondence introduced to the polyadic case and thereby obtain higher degree (in our definition) analogs of the former. The representations of the higher braid relations in vector spaces coincide with the higher braid equation and corresponding generalized R-matrix obtained in Reference [8], as do the ordinary braid group and the Yang-Baxter equation [9,10]. The proposed constructions use polyadic group methods and differ from the tetrahedron equation [11] and n-simplex equations [12] connected with the braid group representations [13,14], as well as from higher braid groups of Reference [15]. We define higher degree (in our sense) versions of the Coxeter group and the symmetric group and show that they are connected in the classical (i.e., non-higher) case only

Preliminaries

Ternary Matrix Group Corresponding to the Regular Semigroup

Polyadic Matrix Semigroup Corresponding to the Higher Regular Semigroup

Ternary Matrix Group Corresponding to the Braid Group

Generated k-Ary Matrix Group Corresponding the Higher Braid Group