Dihedral solutions of the set theoretical Yang-Baxter equation
Alex W. Nowak, Anna Zamojska-Dzienio
TL;DR
The paper develops a unified framework for set-theoretic Yang–Baxter equation solutions by introducing braided dihedral sets (BDS) and braided triality sets (BTS), focusing on Latin variants LBDS and LBTS. It characterizes LBDS via uniquely $2$-divisible Bruck loops and LBTS via commutative Moufang loops of exponent 3, establishing correspondences between isomorphism classes and involutions in automorphism groups. Detailed classifications are provided for LBDS and LBTS at prime, prime-square, and small composite orders, with extensive computational enumeration (GAP) for orders $27$ and $81$. The results illuminate the structure of dihedral and triality symmetries in set-theoretic YBE solutions and reveal how loop-theoretic properties govern the corresponding algebraic objects. Overall, the work offers a concrete, computable taxonomy of LBDS/LBTS linked to loop theory, with potential implications for representations of dihedral and symmetric groups in YBE-related constructions.
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.