Skew braces and the Yang–Baxter equation

Guarnieri, L.; Vendramin, L.

doi:10.1090/mcom/3161

Cited by 273 publications

(345 citation statements)

References 41 publications

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“…If

A

is a skew left brace, the multiplicative group acts on the additive group by automorphisms. The map

λ : (A, \circ) \to Aut (A, +)

a \mapsto λ_{a}

, where

λ_{a} false(b false) = - a + a \circ b

, is a group homomorphism, see [, Corollary 1.10]. Remark Let

A

be a skew left brace.…”

Section: Preliminariesmentioning

confidence: 99%

“…The socle of a skew left brace

A

is defined as

Soc (A) = {x \in A : x \circ a = x + a and x + a = a + x, for all a \in A} .

Clearly

Soc (A) = ker (λ) \cap Z (A, +)

. In [, Lemma 2.5] it is proved that

Soc (A)

is an ideal of

A

. Lemma Let

A

be a skew left brace and

a \in Soc (A)

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Braces provide a powerful algebraic framework to work with set‐theoretic solutions. In [, Theorem 3.1] it is proved that for a skew left brace

A

, the map

r_{A} : A \times A \to A \times A, r_{A} false(a, b false) = false(- a + a \circ b, {false(- a + a \circ b false)}^{'} \circ a \circ b false)

is a non‐degenerate set‐theoretic solution of the Yang–Baxter equation. In [, Theorem 4.5] it is proved that if

(X, r)

be a non‐degenerate solution of the Yang–Baxter equation, then there exists a unique skew left brace structure over the group

G = G (X, r) = ⟨ X : x \circ y = u \circ v whenever r false(x, y false) = false(u, v false) ⟩

such that

where

ι : X \to G (X, r)

is the canonical map.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Skew left braces of nilpotent type

Cedó

Smoktunowicz

Vendramin

2018

Proc. London Math. Soc.

105

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“…If

A

is a skew left brace, the multiplicative group acts on the additive group by automorphisms. The map

λ : (A, \circ) \to Aut (A, +)

a \mapsto λ_{a}

, where

λ_{a} false(b false) = - a + a \circ b

, is a group homomorphism, see [, Corollary 1.10]. Remark Let

A

be a skew left brace.…”

Section: Preliminariesmentioning

confidence: 99%

“…The socle of a skew left brace

A

is defined as

Soc (A) = {x \in A : x \circ a = x + a and x + a = a + x, for all a \in A} .

Clearly

Soc (A) = ker (λ) \cap Z (A, +)

. In [, Lemma 2.5] it is proved that

Soc (A)

is an ideal of

A

. Lemma Let

A

be a skew left brace and

a \in Soc (A)

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Braces provide a powerful algebraic framework to work with set‐theoretic solutions. In [, Theorem 3.1] it is proved that for a skew left brace

A

, the map

r_{A} : A \times A \to A \times A, r_{A} false(a, b false) = false(- a + a \circ b, {false(- a + a \circ b false)}^{'} \circ a \circ b false)

is a non‐degenerate set‐theoretic solution of the Yang–Baxter equation. In [, Theorem 4.5] it is proved that if

(X, r)

be a non‐degenerate solution of the Yang–Baxter equation, then there exists a unique skew left brace structure over the group

G = G (X, r) = ⟨ X : x \circ y = u \circ v whenever r false(x, y false) = false(u, v false) ⟩

such that

where

ι : X \to G (X, r)

is the canonical map.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Skew left braces of nilpotent type

Cedó

Smoktunowicz

Vendramin

2018

Proc. London Math. Soc.

105

View full text Add to dashboard Cite

show abstract

“…The purpose of the this work is to introduce Hopf braces, a new algebraic structure related to the Yang-Baxter equation, which include Rump's braces and their non-commutative generalizations [13] as particular cases. Our generalization is based on Hopf algebras.…”

Section: (C ⊗ Id)(id ⊗ C)(c ⊗ Id) = (C ⊗ Id)(id ⊗ C)(c ⊗ Id)mentioning

confidence: 99%

“…Example 1.4. Recall from [13] that a skew left brace is a group A with an additional group structure given by (a,…”

Section: Hopf Bracesmentioning

confidence: 99%