SKEW BRACES AND THE YANG–BAXTER EQUATION
L. GUARNIERI AND L. VENDRAMIN
Abstract. Braces were introduced by Rump to study non-degenerate in-
volutive 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 Yang–Baxter equation first appeared in theoretical physics and statistical
mechanics in the works of Yang [42] and Baxter [4, 5] and it has led to several
interesting applications in quantum groups and Hopf algebras, knot theory, ten-
sor categories and integrable systems, see for example [27], [30] and [39]. In [14],
Drinfeld posed the problem of studying this equation from the set-theoretical per-
spective.
Recall that a set-theoretical solution of the Yang–Baxter equation is a pair (X, r),
where X is a set and
r : X × X → X × X, r(x, y) = (σ
x
(y), τ
y
(x)), x, y ∈ X,
is a bijective map such that
(r × id)(id × r)(r × id) = (id × r)(r × id)(id × r).
Such a map r is usually called a braiding.
A solution (X, r) is said to be non-degenerate if the maps σ
x
and τ
x
are bijective
for each x ∈ X, and (X, r) is said to be involutive if r
2
= id
X×X
. The seminal works
of Etingof, Schedler and Soloviev [15], and Gateva-Ivanova and Van den Bergh [24],
discussed algebraic and geometrical interpretations and introduced several struc-
tures associated with the class of non-degenerate involutive solutions. Such solu-
tions have been intensively studied, see for example [17, 18, 19], [21, 22, 23], [25, 26],
[20], [32, 34], [6], [10], [11], [13], [28], and [40].
It was in studying involutive solutions that Rump introduced in [34] the brace
structure. In [12], Ced´o, Jespers and Okni´nski, defined a left brace as an abelian
group (A, +) with another group structure, defined via (a, b) 7→ ab, such that the
compatibility condition
a(b + c) + a = ab + ac
holds for all a, b, c ∈ A. This definition is equivalent to that of Rump.
This work is partially supported by CONICET, PICT-2014-1376, MATH-AmSud and ICTP.
1
2 L. GUARNIERI AND L. VENDRAMIN
Many of the problems related to involutive solutions can be restated in terms of
braces. Two prominent examples are the following:
• Is every finite solvable group an involutive Yang–Baxter group? Recall
that an involutive Yang–Baxter group is a group isomorphic to the group
generated by the set {σ
x
: x ∈ X}, where
r : X × X → X × X, r(x, y) = (σ
x
(y), τ
y
(x)),
is a non-degenerate involutive solution of the Yang–Baxter equation. Based
on a sketch of proof of Rump [36], Bachiller [2] found a solvable finite group
that is not an involutive Yang–Baxter group.
• Are there good methods to contruct all finite non-degenerate involutive
solutions to the Yang–Baxter equation? Brute force seems not to be good
enough. In [3], Bachiller, Ced´o and Jespers, give a method to construct all
finite solutions of a given size. For it to work, one needs the classification
of left braces.
Non-involutive solutions were studied by Soloviev [38] and Lu, Yan and Zhu [29].
Such solutions have applications in knot theory, since they produce powerful knot
and virtual knots invariants, see for example [31] and the references therein. The
following question naturally arises: Is there an algebraic structure similar to the
brace structure useful for studying non-involutive solutions? This paper introduces
the notion of skew brace and provides an affirmative answer to the above question.
Remarkably, this new structure provides the right algebraic framework to study
involutive and non-involutive braidings and allows us to restate the main results
of [29], [38] and [41].
As in the case of involutive solutions, the classification of finite skew braces is
one of the main steps needed for constructing finite solutions of the Yang–Baxter
equation. One of the main results of this paper is an explicit classification of classical
and skew braces of small size. An algorithm to construct all non-isomorphic classical
and skew braces of a given size is described. This heavily depends on results of
Bachiller [2] and Catino and Rizzo [9]. This algorithm was used to build a database
of classical and skew braces, a good source of examples that gives an explicit and
direct way to approach some of the problems related to the Yang–Baxter equation.
The database is available as a library for GAP [16] and Magma [8] immediately from
the authors on request.
The paper is organized as follows. In Section 1 we extend braces to the non-
commutative setting by defining skew braces, and state their main properties. We
prove in Proposition 1.11 that skew braces are equivalent to bijective 1-cocycles.
Section 2 is devoted to a study of quotients of skew braces. It is worth mentioning
that the proofs in Section 1 and 2 are basically the same as for classical braces.
In Section 3 the connection between skew braces and the Yang–Baxter equation is
explored. In Theorem 3.1 we generalize a result of Rump and produce a canonical
solution for each skew left brace. Some reconstruction theorems similar to those
of Etingof, Schedler and Soloviev [15], Lu, Yan and Zhu [29] and Soloviev [38]
are given at the end of this section. The method for constructing classical and
skew braces is given in Section 4. Section 5 discusses the algorithm that produces
and enumerates classical and skew left braces and some consequences. Problems,
questions and conjectures are discussed in Section 6.
SKEW BRACES AND THE YANG–BAXTER EQUATION 3
1. Skew left braces
Braces were introduced by Rump in [34] to study set-theoretical involutive solu-
tions of the Yang–Baxter equation. The following definition generalizes braces to
the non-commutative setting.
Definition 1.1. A skew left brace is a group A (written multiplicatively) with an
additional group structure given by (a, b) 7→ a ◦ b such that
(1.1) a ◦ (bc) = (a ◦ b)a
−1
(a ◦ c)
holds for all a, b, c ∈ A, where a
−1
denotes the inverse of a with respect to the group
structure given by (a, b) 7→ ab.
Of course Rump’s left braces are examples of skew braces. These are braces
where the group (A, ·) is abelian.
Definition 1.2. A homomorphism between two skew left braces A and B is a map
f : A → B such that f (ab) = f(a)f(b) and f(a ◦ b) = f(a) ◦ f(b) for all a, b ∈ A.
The kernel of f is
ker f = {a ∈ A : f(a) = 1},
where 1 denotes the identity of the group (A, ·) with multiplication a · b = ab for all
a, b ∈ A.
Example 1.3. Let (A, ·) be a group. Then A is a skew left brace with a ◦ b = ab
for all a, b ∈ A. Similarly, a ? b = ba defines a skew left brace structure over A.
These braces are isomorphic if and only if (A, ·) is abelian.
Example 1.4. Let A and B be groups and let α: A → Aut(B) be a group homo-
morphism. Then A × B has a skew left brace structure given by
(a, b)(a
0
, b
0
) = (aa
0
, bb
0
),
(a, b) ◦ (a
0
, b
0
) = (aa
0
, bα
a
(b
0
)),
where a, a
0
∈ A and b, b
0
∈ B.
Example 1.5. Let A and B be groups and let α: A → Aut(B) be a group homo-
morphism. Assume that A is abelian. Then A × B has a skew left brace structure
given by
(a, b)(a
0
, b
0
) = (aa
0
, bα
a
(b
0
)),
(a, b) ◦ (a
0
, b
0
) = (aa
0
, bb
0
),
where a, a
0
∈ A and b, b
0
∈ B.
Example 1.6. This example is motivated by the paper of Weinstein and Xu on the
Yang–Baxter equation, see [41]. Let A be a group and A
+
, A
−
be subgroups of A
such that A admits a unique factorization as A = A
+
A
−
. Thus each a ∈ A can be
written in a unique way as a = a
+
a
−
for some a
+
∈ A
+
and a
−
∈ A
−
. The map
A
+
× A
−
→ A, (a
+
, a
−
) 7→ a
+
(a
−
)
−1
,
is bijective. Using this map we transport the group structure of the direct product
A
+
× A
−
into the set A. For a = a
+
a
−
∈ A and b = b
+
b
−
∈ A let
a ◦ b = a
+
ba
−
.
Then (A, ◦) is a group. Furthermore, A is a skew left brace.
4 L. GUARNIERI AND L. VENDRAMIN
Lemma 1.7. Let A be a skew left brace. Then the following properties hold:
(1) 1 = 1
◦
, where 1
◦
denotes the unit of the group (A, ◦).
(2) a ◦ (b
−1
c) = a(a ◦ b)
−1
(a ◦ c) for all a, b, c ∈ A.
(3) a ◦ (bc
−1
) = (a ◦ b)(a ◦ c)
−1
a for all a, b, c ∈ A.
Proof. The first claim follows from (1.1) with c = 1
◦
. To prove the second claim
let d = bc. Then (1.1) becomes a ◦ d = (a ◦ b)a
−1
(a ◦ b
−1
d) and the claim follows.
The third claim is proved similarly.
Remark 1.8. Let A be a skew left brace. For each a ∈ A the map
λ
a
: A → A, b 7→ a
−1
(a ◦ b),
is bijective with inverse λ
−1
a
: A → A, b 7→ a ◦ (ab), where a is the inverse of a with
respect to ◦. It follows that
a ◦ b = aλ
a
(b), ab = a ◦ λ
−1
a
(b)
hold for all a, b ∈ A.
The following proposition extends results of Rump [34] and Gateva-Ivanova into
the non-commutative setting, see [17, Proposition 3.3].
Proposition 1.9. Let A be a set and assume that A has two operations such
that (A, ·) and (A, ◦) are groups. Assume that λ : A → S
A
, a 7→ λ
a
, is given by
λ
a
(b) = a
−1
(a ◦ b). The following are equivalent:
(1) A is a skew left brace.
(2) λ
a◦b
(c) = λ
a
λ
b
(c) for all a, b, c ∈ A.
(3) λ
a
(bc) = λ
a
(b)λ
a
(c) for all a, b, c ∈ A.
Proof. Let us first prove that (1) =⇒ (2). Let a, b, c ∈ A. Since A is a brace and
a ◦ b
−1
= a(a ◦ b)
−1
a by Lemma 1.7,
λ
a
λ
b
(c) = a
−1
(a ◦ λ
b
(c)) = a
−1
(a ◦ (b
−1
(b ◦ c)))
= a
−1
(a ◦ b
−1
)a
−1
(a ◦ b ◦ c) = (a ◦ b)
−1
(a ◦ b ◦ c) = λ
a◦b
(c).
Now we prove (2) =⇒ (3). Since ab = a ◦ λ
−1
a
(b) for all a, b ∈ A,
λ
a
(bc) = λ
a
(b ◦ λ
−1
b
(c)) = a
−1
(a ◦ b ◦ λ
−1
b
(c))
= a
−1
(a ◦ b)(a ◦ b)
−1
(a ◦ b ◦ λ
−1
b
(c))
= λ
a
(b)λ
a◦b
λ
−1
b
(c) = λ
a
(b)λ
a
λ
b
λ
−1
b
(c) = λ
a
(b)λ
a
(c).
Finally we prove that (3) =⇒ (1). Let a, b, c ∈ A. Then
a
−1
(a ◦ (bc)) = λ
a
(bc) = λ
a
(b)λ
a
(c) = a
−1
(a ◦ b)a
−1
(a ◦ c),
and hence a ◦ (bc) = (a ◦ b)a
−1
(a ◦ c).
Corollary 1.10. Let A be a skew left brace and
λ: (A, ◦) → Aut(A, ·), a 7→ λ
a
(b) = a
−1
(a ◦ b).
Then λ is a group homomorphism.
Proof. It follows immediately from Proposition 1.9.
SKEW BRACES AND THE YANG–BAXTER EQUATION 5
Let A and G be groups and assume that G×A → A, (g, a) 7→ g·a, is a left action
of G on A by automorphisms. A bijective 1-cocyle is a bijective map π : G → A
such that
(1.2) π(gh) = π(g)(g · π(h))
for all g, h ∈ G.
Proposition 1.11. Over any group (A, ·) the following data are equivalent:
(1) A group G and a bijective 1-cocycle π : G → A.
(2) A skew left brace structure over A.
Proof. Consider on A a second group structure given by
a ◦ b = π(π
−1
(a)π
−1
(b))
for all a, b ∈ A. Since π is a 1-cocycle and G acts on A by automorphisms,
a ◦ (bc) = π(π
−1
(a)π
−1
(bc)) = a(π
−1
(a) · (bc))
= a((π
−1
(a) · b)(π
−1
(a) · c)) = (a ◦ b)a
−1
(a ◦ c)
holds for all a, b, c ∈ A.
Conversely, assume that A is a skew left brace. Set G = A with the multiplication
(a, b) 7→ a◦b and π = id. By Corollary 1.10, a 7→ λ
a
, is a group homomorphism and
hence G acts on A by automorphisms. Then (1.2) holds and therefore π : G → A
is a bijective 1-cocycle.
Remark 1.12. The construction of Proposition 1.11 is categorical.
2. Ideals and quotients
Definition 2.1. Let A be a skew left brace. A normal subgroup I of (A, ◦) is said
to be an ideal of A if Ia = aI and λ
a
(I) ⊆ I for all a ∈ A.
Example 2.2. Let f : A → B be a skew brace homomorphism. Then ker f is an
ideal of A since
f(λ
a
(x)) = λ
f(a)
(f(x)) = 1
for all x ∈ ker f and a ∈ A.
Lemma 2.3. Let A be a skew left brace and I ⊆ A be an ideal. Then the following
properties hold:
(1) I is a normal subgroup of (A, ·).
(2) a ◦ I = aI for all a ∈ A.
(3) I and A/I are skew braces.
Proof. Let a, b ∈ I. Then a
−1
b = λ
a
(a ◦ b) ∈ I and hence I is a subgroup of (A, ·).
Remark 1.8 implies
aI = a ◦ I = I ◦ a = Ia
for all a ∈ A. Thus I is a normal subgroup of (A, ·) and hence it follows that I is
a skew left brace. Since the quotient groups A/I for both operations are the same,
A/I is a skew left brace.
Definition 2.4. Let A be a skew left brace. The socle of A is
Soc(A) = {a ∈ A : a ◦ b = ab, b(b ◦ a) = (b ◦ a)b for all b ∈ A}.
