GAP (YangBaxter) - Chapter 1: Preliminaries

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1 Preliminaries
1.1 Definition and examples
1.1-1 IsSkewbrace
1.1-2 Skewbrace
1.1-3 SmallSkewbrace
1.1-4 TrivialBrace
1.1-5 TrivialSkewbrace
1.1-6 SmallBrace
1.1-7 IdSkewbrace
1.1-8 IdBrace
1.1-9 IsomorphismSkewbraces
1.1-10 DirectProductSkewbraces
1.1-11 IsTwoSided
1.1-12 IsClassical
1.1-13 IsTrivialSkewbrace
1.1-14 Skewbrace2YB
1.1-15 Brace2YB
1.1-16 SkewbraceSubset2YB

### 1 Preliminaries

In this section we define skew braces and list some of their main properties [GV17].

#### 1.1 Definition and examples

A skew brace is a triple (A,+,\circ), where (A,+) and (A,\circ) are two (not necessarily abelian) groups such that the compatibility a\circ (b+c)=a\circ b-a+a\circ c holds for all a,b,c\in A. Ones proves that the map \lambda\colon (A,\circ)\to\mathrm{Aut}(A,+), a\mapsto\lambda_a(b), \lambda_a(b)=-a+a\circ b, is a group homomorphism. Notation: For a,b\in A, we write a*b=\lambda_a(b)-b.

##### 1.1-1 IsSkewbrace

`‣ IsSkewbrace`( arg ) | ( filter )

Returns: `true` or `false`

##### 1.1-2 Skewbrace

`‣ Skewbrace`( list ) | ( operation )

Returns: a skew brace

The argument list is a list of pairs of elements in a group. By Proposition 5.11 of [GV17], skew braces over an abelian group A are equivalent to pairs (G,\pi), where G is a group and \pi\colon G\to A is a bijective 1-cocycle, a finite skew brace can be constructed from the set \{(a_j,g_j):1\leq j\leq n\}, where G=\{g_1,\dots,g_n\} and A=\{a_1,\dots,a_n\} are permutation groups. This function is used to construct skew braces.

    gap> Skewbrace([[(),()]]);
    <brace of size 1>
    gap> Skewbrace([[(),()],[(1,2),(1,2)]]);
    <brace of size 2>

##### 1.1-3 SmallSkewbrace

`‣ SmallSkewbrace`( n, k ) | ( operation )

Returns: a skew brace

The function returns the k-th skew brace from the database of skew braces of order n.

    gap> SmallSkewbrace(8,3);
    <brace of size 8>

##### 1.1-4 TrivialBrace

`‣ TrivialBrace`( abelian_group ) | ( operation )

Returns: a brace

This function returns the trivial brace over the abelian group abelian_group. Here abelian_group should be an abelian group!

    gap> TrivialBrace(CyclicGroup(IsPermGroup, 5));
    <brace of size 5>

##### 1.1-5 TrivialSkewbrace

`‣ TrivialSkewbrace`( group ) | ( operation )

Returns: a skew brace

This function returns the trivial skew brace over group.

    gap> TrivialSkewbrace(DihedralGroup(10));
    <skew brace of size 10>

##### 1.1-6 SmallBrace

`‣ SmallBrace`( n, k ) | ( operation )

Returns: a brace of abelian type

The function returns the k-th brace (of abelian type) from the database of braces of order n.

    gap> SmallBrace(8,3);
    <brace of size 8>

##### 1.1-7 IdSkewbrace

`‣ IdSkewbrace`( obj ) | ( attribute )

Returns: a list

The function returns [ n, k ] if the skew brace obj is isomorphic to SmallSkewbrace(n,k).

    gap> IdSkewbrace(SmallSkewbrace(8,5));
    [ 8, 5 ]

##### 1.1-8 IdBrace

`‣ IdBrace`( obj ) | ( attribute )

Returns: a list

The function returns [ n, k ] if the brace of abelian type obj is isomorphic to SmallBrace(n,k).

    gap> IdBrace(SmallBrace(8,5));
    [ 8, 5 ]

##### 1.1-9 IsomorphismSkewbraces

`‣ IsomorphismSkewbraces`( obj1, obj2 ) | ( function )

Returns: an isomorphism of skew braces if obj1 and obj2 are isomorphic and fail otherwise.

If A and B are skew braces, a skew brace homomorphism is a map f\colon A\to B such that

f(a+b)=f(a)+f(b)\quad f(a\circ b)=f(a)\circ f(b)

hold for all a,b\in A. A skew brace isomorphism is a bijective skew brace homomorphism. IsomorphismSkewbraces first computes all injective homomorphisms from (A,+) to (B,+) and then tries to find one f such that f(a\circ b)=f(a)\circ f(b) for all a,b\in A.

##### 1.1-10 DirectProductSkewbraces

`‣ DirectProductSkewbraces`( obj1, obj2 ) | ( operation )

Returns: the direct product of obj1 and obj2

    gap> br1 := SmallBrace(8,18);;
    gap> br2 := SmallBrace(12,2);;
    gap> br := DirectProductSkewbraces(br1,br2);;
    gap> IsLeftNilpotent(br);
    false
    gap> IsRightNilpotent(br);
    false
    gap> IsSolvable(br);
    true

##### 1.1-11 IsTwoSided

`‣ IsTwoSided`( obj ) | ( property )

Returns: true if the skew brace is two sided, false otherwise

A skew brace A is said to be two-sided if (a+b)\circ c=a\circ c-c+b\circ c holds for all a,b,c\in A.

    gap> IsTwoSided(SmallSkewbrace(8,2));
    false
    gap> IsTwoSided(SmallSkewbrace(8,4));
    true

##### 1.1-12 IsClassical

`‣ IsClassical`( obj ) | ( property )

Returns: true if the skew brace is of abelian type, false otherwise

Let \mathcal{X} be a property of groups. A skew brace A is said to be of \mathcal{X}-type if its additive group belongs to \mathcal{X}. In particular, skew braces of abelian type are those skew braces with abelian additive group. Such skew braces were introduced by Rump in [Rum07].

##### 1.1-13 IsTrivialSkewbrace

`‣ IsTrivialSkewbrace`( obj ) | ( property )

Returns: true if the skew brace is trivial, false otherwise

The function returns true if the skew brace A is trivial, i.e., a\circ b=a+b for all a,b\in A.

##### 1.1-14 Skewbrace2YB

`‣ Skewbrace2YB`( obj ) | ( attribute )

Returns: the set-theoretic solution associated with the skew brace obj

If A is a skew brace, the map r_A\colon A\times A\to A\times A

r_A(a,b)=(\lambda_a(b),\lambda_a(b)'\circ a\circ b)

is a non-degenerate set-theoretic solution of the Yang--Baxter equation. Furthermore, r_A is involutive if and only if A is of abelian type (i.e., the additive group of A is abelian).

    gap> Skewbrace2YB(TrivialBrace(CyclicGroup(6)));
    <A set-theoretical solution of size 6>

##### 1.1-15 Brace2YB

`‣ Brace2YB`( arg ) | ( attribute )

##### 1.1-16 SkewbraceSubset2YB

`‣ SkewbraceSubset2YB`( obj ) | ( operation )

Returns: the set-theoretic solution associated with a given subset of a skew brace

    gap> br := TrivialSkewbrace(SymmetricGroup(3));;
    gap> AsList(br);
    [ <()>, <(2,3)>, <(1,2)>, <(1,2,3)>, <(1,3,2)>, <(1,3)> ]
    gap> SkewbraceSubset2YB(br, last{[4,5]});
    <A set-theoretical solution of size 2>

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