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2 Algebraic Properties of Braces
 2.1 Braces and Radical Rings
 2.2 Braces and Yang-Baxter Equation

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 Table2YB
‣ Table2YB( table )( operation )

Returns: the solution given by the table

Given the table with \(r(x,y)\) in the position \((x,y)\) find the corresponding \(r\)

gap> l := Table(SmallIYB(4,13));;
gap> t := Table2YB(l);;
gap> IdCycleSet(YB2CycleSet(t));
[ 4, 13 ]

2.2-2 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 ]

2.2-3 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) ] ]

2.2-4 IsIndecomposable
‣ IsIndecomposable( X )( property )

Returns: true if the involutive solutions is indecomposable

2.2-5 Table
‣ Table( obj )( attribute )

Returns: a table with the image of the solution

The table shows the value of \(r(x,y)\) for each \((x,y)\)

gap> yb := SmallIYB(3,2);;
gap> Table(yb);
[ [ [ 1, 1 ], [ 2, 1 ], [ 3, 2 ] ], [ [ 1, 2 ], [ 2, 2 ], [ 3, 1 ] ], [ [ 2, 3 ], [ 1, 3 ], [ 3, 3 ] ] ]

2.2-6 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

2.2-7 DehornoyRepresentationOfStructureGroup
‣ DehornoyRepresentationOfStructureGroup( obj, variable )( operation )

Returns: A faithful linear representation 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

2.2-8 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 ]

2.2-9 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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