Dihedral solutions of the set theoretical Yang-Baxter equation

Abstract

We introduce the notion of a \emph{braided dihedral set} (BDS) to describe set-theoretical solutions of the Yang-Baxter equation (YBE) that furnish representations of the infinite dihedral group on the Cartesian square of the underlying set. BDS which lead to representations of the symmetric group on three objects are called \emph{braided triality sets} (BTS). Basic examples of BDS come from symmetric spaces. We show that Latin BDS (LBDS) can be described entirely in terms of involutions of uniquely 2-divisible Bruck loops. We show that isomorphism classes of LBDS are in one-to-one correspondence with conjugacy classes of involutions of uniquely 2-divisible Bruck loops. We describe all LBDS of prime, prime-square and 3 times prime-order, up to isomorphism. Using \texttt{GAP}, we enumerate isomorphism classes of LBDS of orders 27 and 81. Latin BTS, or LBTS, are shown to be in one-to-one correspondence with involutions of commutative Moufang loops of exponent 3 (CML3), and, as with LBDS, isomorphisms classes of LBTS coincide with conjugacy classes of CML3-involutions. We classify all LBTS of order at most 81.