GAP (YangBaxter) - Chapter 2: Algebraic Properties of Braces

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2 Algebraic Properties of Braces
2.1 Braces and Radical Rings
2.1-1 AdditiveGroupOfRing
2.1-2 IsJacobsonRadical
2.2 Braces and Yang-Baxter Equation
2.2-1 Evaluate
2.2-2 LyubashenkoYB
2.2-3 DehornoyClass
2.2-4 DehornoyRepresentationOfStructureGroup
2.2-5 IdYB
2.2-6 LinearRepresentationOfStructureGroup

### 2 Algebraic Properties of Braces

#### 2.1 Braces and Radical Rings

##### 2.1-1 AdditiveGroupOfRing

`‣ AdditiveGroupOfRing`( ring ) | ( attribute )

Returns: a group

This function returns a permutation representation of the additive group of the given ring.

    gap> rg := SmallRing(8,10);;
    gap> StructureDescription(AdditiveGroupOfRing(rg));
    "C4 x C2"

##### 2.1-2 IsJacobsonRadical

`‣ IsJacobsonRadical`( ring ) | ( attribute )

Returns: true if the ring is radical and false otherwise.

This function checks whether a ring is Jacobson radical.

    gap> rg := SmallRing(8,11);;
    gap> IsJacobsonRadical(rg);
    true
    gap> rg := SmallRing(8,20);;
    gap> IsJacobsonRadical(rg);
    false

#### 2.2 Braces and Yang-Baxter Equation

##### 2.2-1 Evaluate

`‣ Evaluate`( obj, pair ) | ( operation )

Returns: a pair of two integers

Given the pair (x,y) this function returns r(x,y).

    gap> cs := SmallCycleSet(4,13);;
    gap> yb := CycleSet2YB(cs);;
    gap> Permutations(yb);
    [ [ (3,4), (1,3,2,4), (1,4,2,3), (1,2) ],
      [ (2,4), (1,4,3,2), (1,2,3,4), (1,3) ] ]
    gap> Evaluate(yb, [1,2]);
    [ 2, 4 ]
    gap> Evaluate(yb, [1,3]);
    [ 4, 2 ]

##### 2.2-2 LyubashenkoYB

`‣ LyubashenkoYB`( size, f, g ) | ( operation )

Returns: a permutation solution to the YBE

Finite Lyubashenko (or permutation) solutions are defined as follows: Let X=\{1,\dots,n\} and f,g\colon X\to X be bijective functions such that fg=gf. Then (X,r), where r(x,y)=(f(y),g(x)), is a set-theoretic solution to the YBE.

    gap> yb := LyubashenkoYB(4, (1,2),(3,4));
    <A set-theoretical solution of size 4>
    gap> Permutations(last);
    [ [ (1,2), (1,2), (1,2), (1,2) ], [ (3,4), (3,4), (3,4), (3,4) ] ]

##### 2.2-3 DehornoyClass

`‣ DehornoyClass`( obj ) | ( attribute )

Returns: The class of an involutive solution

    gap> cs := SmallCycleSet(4,13);;
    gap> yb := CycleSet2YB(cs);;
    gap> DehornoyClass(yb);
    2
    gap> cs := SmallCycleSet(4,19);;
    gap> yb := CycleSet2YB(cs);;
    gap> DehornoyClass(yb);
    4

##### 2.2-4 DehornoyRepresentationOfStructureGroup

`‣ DehornoyRepresentationOfStructureGroup`( obj, variable ) | ( operation )

Returns: A faithful linear representaiton of the structure group of obj

    gap> cs := SmallCycleSet(4,13);;
    gap> yb := CycleSet2YB(cs);;
    gap> Permutations(yb);
    [ [ (3,4), (1,3,2,4), (1,4,2,3), (1,2) ],
      [ (2,4), (1,4,3,2), (1,2,3,4), (1,3) ] ]
    gap> field := FunctionField(Rationals, 1);;
    gap> q := IndeterminatesOfFunctionField(field)[1];;
    gap> G := DehornoyRepresentationOfStructureGroup(yb, q);;
    gap> x1 := G.1;;
    gap> x2 := G.2;;
    gap> x3 := G.3;;
    gap> x4 := G.4;;
    gap> x1*x2=x2*x4;
    true
    gap> x1*x3=x4*x2;
    true
    gap> x1*x4=x3*x3;
    true
    gap> x2*x1=x3*x4;
    true
    gap> x2*x2=x4*x1;
    true
    gap> x3*x1=x4*x3;
    true

##### 2.2-5 IdYB

`‣ IdYB`( obj ) | ( attribute )

Returns: the identification number of obj

    gap> cs := SmallCycleSet(5,10);;
    gap> IdCycleSet(cs);
    [ 5, 10 ]
    gap> cs := SmallCycleSet(4,3);;
    gap> yb := CycleSet2YB(cs);;
    gap> IdYB(yb);
    [ 4, 3 ]

##### 2.2-6 LinearRepresentationOfStructureGroup

`‣ LinearRepresentationOfStructureGroup`( obj ) | ( attribute )

Returns: the permutation brace of the involutive solution of obj a linear representation of the structure group of a finite involutive solution

    gap> yb := SmallIYB(5,86);;
    gap> IdBrace(IYBBrace(yb));
    [ 6, 2 ]

    gap> yb := SmallIYB(5,86);;
    gap> gr := LinearRepresentationOfStructureGroup(yb);;
    gap> gens := GeneratorsOfGroup(gr);;
    gap> Display(gens[1]);
    [ [  0,  1,  0,  0,  0,  1 ],
      [  1,  0,  0,  0,  0,  0 ],
      [  0,  0,  0,  0,  1,  0 ],
      [  0,  0,  1,  0,  0,  0 ],
      [  0,  0,  0,  1,  0,  0 ],
      [  0,  0,  0,  0,  0,  1 ] ]

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