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Skew braces and the Yang-Baxter equation
L. Guarnieri, L. Vendramin
TL;DR
The paper asks how brace methods can handle non-involutive non-degenerate set-theoretical Yang–Baxter solutions and generalizes braces to the non-commutative setting. It develops skew-brace theory, classification algorithms, and a small-size database, with canonical solutions and explicit computational enumeration as principal outcomes.
Problem
The paper addresses the need for an algebraic structure and effective construction methods for non-involutive solutions and finite Yang–Baxter solutions.
Method
The paper introduces skew braces, relates them to bijective 1-cocycles, and uses prior classification results to algorithmically enumerate and construct small classical and skew braces.
Results
The paper provides a canonical Yang–Baxter solution for each skew left brace, constructs associated solutions, and determines the number of skew left braces of sizes n ≤30.
Takeaways & Limitations
Skew braces supply a common framework for involutive and non-involutive braidings, while the database offers examples for studying Yang–Baxter problems.
Takeaways & Limitations
The paper leaves open computational problems and conjectures, including values of b(32), b(64), b(81), and b(96) and a conjecture about quaternion braces.
Abstract
from arXiv · showhide
Braces were introduced by Rump to study non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. We generalize Rump's braces to the non-commutative setting and use this new structure to study not necessarily involutive non-degenerate set-theoretical solutions of the Yang-Baxter equation. Based on results of Bachiller and Catino and Rizzo, we develop an algorithm to enumerate and construct classical and non-classical braces of small size up to isomorphism. This algorithm is used to produce a database of braces of small size. The paper contains several open problems, questions and conjectures.
Introduction
The paper extends brace methods from involutive to non-involutive Yang–Baxter solutions by introducing skew braces, while addressing construction and classification problems for finite solutions.
- The Yang–Baxter equation originated in physics and connects to quantum groups, knot theory, tensor categories, and integrable systems.
- Braces provide an algebraic language for many problems concerning non-degenerate involutive Yang–Baxter solutions.
- A finite solvable group need not be an involutive Yang–Baxter group, as shown by Bachiller’s construction.
- Constructing all finite non-degenerate involutive solutions requires methods beyond brute force and depends on classifying finite left braces.
- Skew braces extend braces to the non-commutative setting and provide an algebraic framework for both involutive and non-involutive braidings.
- The paper develops small-size brace classification and construction algorithms, producing a database intended to support Yang–Baxter research.
1. Skew left braces
Skew left braces extend Rump’s braces from abelian to non-commutative multiplicative groups, providing an algebraic framework connected to Yang–Baxter solutions and bijective 1-cocycles.
- A skew left brace is a group with a second group operation satisfying the brace compatibility condition.
- Rump’s left braces are the special case in which the original group operation is abelian.
- Brace homomorphisms preserve both group operations, with kernels defined using the identity of the original group.
- Unique factorizations and group actions provide constructions of skew left braces, including the transported structure from A+ × A−.
- The associated map λ_a(b) = a^-1(a ◦ b) characterizes skew left braces through multiplicativity and the relation λ_a◦b = λ_aλ_b.
- Skew left brace structures over a group are equivalent to groups equipped with bijective 1-cocycles into that group.
2. Ideals and quotients
Ideals in skew left braces are compatible with both group structures and support quotient constructions, while the socle forms a central ideal of the original group.
- An ideal is a normal subgroup of the ◦-group that is stable under every λ_a.
- The kernel of every skew brace homomorphism is an ideal because λ-actions preserve its elements.
- Every ideal is also normal in the original group, and both the ideal and quotient A/I inherit skew brace structures.
- The socle is defined by elements on which the two operations agree in one interaction and which satisfy a commutation condition under the ◦-operation.
- The socle is an ideal contained in the center of the original group.
3. Braces and the Yang–Baxter equation
The paper connects skew left braces to non-degenerate Yang–Baxter solutions in the non-commutative setting, generalizing Rump’s construction. It also gives canonical, universal, and restricted-subset constructions of such solutions.
- Canonical solutions: Theorem 3.1 assigns every skew left brace A a canonical non-degenerate Yang–Baxter solution r_A.The associated solution is involutive exactly when the multiplicative group of A is abelian.
- Canonical solutions: The canonical solution r_A is involutive if and only if ab = ba for all a, b ∈A.
- Restricted solutions: A subset X of A yields a non-degenerate solution by restriction when bλ_a(x)b−1 remains in X for every x ∈X and a, b ∈A.
- Related constructions: The guitar map T_n is invertible and intertwines the braid-group actions induced by the canonical solution r and a related solution s.Its coordinates are built by successive λ-actions.
- Universal construction: For each non-degenerate solution (X, r), its structure group G(X, r) has a unique skew left brace structure whose associated solution extends r compatibly.The construction also satisfies a universal property for maps from X into any skew left brace.
- Universal construction: For finite non-degenerate solutions, G(X, r)/Soc(G(X, r)) is a finite skew left brace.
4. Constructing skew braces
The paper characterizes skew left braces through regular subgroups of the holomorph of a group. This yields a bijective classification framework in which isomorphic brace structures correspond to conjugate regular subgroups.
- Holomorph framework: The holomorph of a group A is Hol(A) = Aut(A)⋉A, with its natural action on A given by (f, x) · a = xf(a).This action is transitive, and the stabilizer of any element is isomorphic to Aut(A).
- Holomorph framework: A subgroup H of Hol(A) is regular when each a ∈A has a unique element (f, x) ∈H satisfying xf(a) = 1.
- Holomorph framework: For a regular subgroup H of Hol(A), the projection π2|H : H →A is bijective.This bijection allows the group operation of H to be transported onto A.
- Brace–subgroup correspondence: Every skew left brace A determines a regular subgroup {(λ_a, a) : a ∈A} of Hol(A, ·), and every regular subgroup determines a skew left brace.The transported brace operation has multiplicative group isomorphic to the regular subgroup.
- Brace–subgroup correspondence: Skew left brace structures over A correspond bijectively to regular subgroups of Hol(A), while isomorphic structures correspond to conjugate subgroups under Aut(A).
5. Computational results
The paper presents an algorithm based on regular subgroups of holomorphs to enumerate skew left braces, and applies it to compute brace counts and runtimes for small sizes.
- Enumeration algorithm: Algorithm 5.1 enumerates skew left brace structures over a finite group by computing regular subgroups of Hol(A) up to Aut(A)-conjugacy.For each representative subgroup, the algorithm constructs the corresponding brace multiplication through a bijection with the base group.
- Enumeration algorithm: The third construction step is unnecessary when the goal is only to enumerate all skew left brace structures over A.
- Implementation: Algorithm 5.1 was implemented in both GAP and Magma and run on an Intel Core i5-4440 system with 16 GB of RAM under Linux.
- Skew left braces: The number of non-isomorphic skew left braces was determined for every size n ≤30, with the calculation taking about twenty minutes.Table 5.1 reports selected values of c(n), the number of non-isomorphic skew left braces of size n.
- Left braces: The number of left braces up to isomorphism was determined for sizes n ≤120, while constructing them required considerably more CPU time.The paper reports selected counts and separate runtime data for enumeration and construction.
6. Further questions
The paper formulates computational problems and conjectures about brace counts, including explicit conjectures for prime-related sizes and quaternion braces. Several conjectures are verified over finite ranges, while broader classification questions remain open.
- Left braces: The authors ask for b(32), b(64), b(81), and b(96), where b(n) counts non-isomorphic left braces of size n.
- Left braces: Conjectures 6.2 and 6.3 give prime-dependent formulas, with stated values determined by congruence classes of the parameters.The supplied passages list values including 11 and 9 in one case, and 14, 4, and 11 in another.
- Left braces: Conjecture 6.4 states that b(p^2q) = 4 when p < q are primes and q ≠1 mod p.The conjecture was subsequently proved by Agata Smoktunowicz.
- Quaternionic braces: For m > 2, Conjecture 6.6 assigns q(4m), the number of quaternion-brace isomorphism classes of size 4m, values based on m modulo 8 or its parity.The stated values are 2 for odd m, 7 for m ≡0 mod 8, 9 for m ≡4 mod 8, and 6 for m ≡2 or 6 mod 8.
- Quaternionic braces: Conjecture 6.6 was checked for all m ≤512, and the additive groups observed for sizes m ∈{2, …, 512} belong to five listed families.These include Z4m, Z2m × Z2, Zm × Z2 × Z2, Zm × Z4, and Zm/2 × Z2 × Z2 × Z2.
- Quaternionic braces: The paper asks which finite abelian groups occur as additive groups of quaternion braces and how many such braces occur for a prescribed additive group.It also states Conjecture 6.9 that seven isomorphism classes occur for quaternion braces of size 2^k when k > 4, verified for k = 5, 6, 7, 8, 9.