Classification of braces of order p⁴

Abstract

The Yang-Baxter equation holds a fundamental place in both physics and mathematics, connecting to various fields such as knot theory, quantum integrable systems, quantum algebras, braid theory, and Hopf algebras, to name a few. In 2007 W. Rump introduced the brace, a generalisation of a nilpotent ring, as an algebraic tool for the study of solutions of the set-theoretic Yang-Baxter equation. More specifically, W. Rump showed that there is a one-to-one correspondence between braces and non-degenerate, involutive, set-theoretic solutions to the Yang-Baxter equation, thereby offering a pathway for finding novel solutions through the classification of braces. Since then, braces have been studied in connection to other algebraic structures, such as Hopf-Galois extensions, trusses, braided groups - and, of specific relevance to this thesis, pre-Lie rings and pre-Lie algebras. Pre-Lie rings and pre-Lie algebras are types of algebras over a ring whose product is left-symmetric. The connection between braces and pre-Lie algebras was first described in 2014 by W. Rump, in the case of R-braces and pre-Lie algebras over R. Later, in 2022, A. Smoktunowicz showed that right nilpotent braces of prime power cardinality correspond to pre-Lie rings of the same cardinality, for sufficiently large primes. In this case, right nilpotent braces can be explicitly obtained from the corresponding pre-Lie rings through the construction of the, so-called, group of flows. Hence, the classification of right nilpotent braces of prime power order can be achieved through the description of the corresponding pre-Lie rings. This thesis achieves the classification of braces of cardinality p⁴ for sufficiently large primes p⁴. The classification is separated into two parts, based on the right nilpotency of the braces under consideration. For braces of cardinality p⁴ that are right nilpotent, their correspondence to pre-Lie rings is leveraged. Working with pre-Lie rings is advantageous due to their comparative simplicity, primarily stemming from pre-Lie rings’ distributivity, which does not hold in braces. It is not known whether braces that are not right nilpotent correspond to pre-Lie rings, in which case we describe the braces directly, through a consideration of their defining generators and relations. We first determine the right nilpotency of braces of cardinality p⁴ for primes p such that p≥5. Then we describe all braces of cardinality p⁴ which are not right nilpotent for primes p such that p≥5. We also show that the constructed braces are prime, and contain a non-zero strongly nilpotent ideal. Using the constructed braces we produce examples of finite-dimensional pre-Lie algebras which are left nilpotent but not right nilpotent. Finally, we classify nilpotent pre-Lie rings of cardinality p⁴, and thereby braces of the same cardinality, for primes p greater than 5⁵. This thesis is based on three papers, each included in its entirety. The included content largely retains its original presentation, albeit with some changes to the sequence of section inclusions, and with occasional changes in wording, or minor corrections. The mathematical content of the referenced papers is unchanged.