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# YangBaxter

## Combinatorial Solutions for the Yang-Baxter equation

0.7.0

23 February 2019

Leandro Vendramin
Email: lvendramin@dm.uba.ar
Homepage: http://mate.dm.uba.ar/~lvendram
Address:
Departamento de matem'atica, FCEN, UBA
Ciudad Universitaria, Pab. 1
Buenos Aires, Argentina

Alexander Konovalov
Email: alexander.konovalov@st-andrews.ac.uk
Homepage: https://alexk.host.cs.st-andrews.ac.uk/
Address:
School of Computer Science
University of St Andrews
Jack Cole Building, North Haugh,
St Andrews, Fife, KY16 9SX, Scotland

### Contents

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
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 LinearRepresentationOfStructureGroup
3 Ideals and left ideals
3.1 Left ideals
3.1-1 LeftIdeals
3.1-2 IsLeftIdeal
3.2 Ideals
3.2-1 IsIdeal
3.2-2 Ideals
3.2-3 AsIdeal
3.2-4 IdealGeneratedBy
3.2-5 IntersectionOfTwoIdeals
3.2-6 SumOfTwoIdeals
3.3 Sequences (left) ideals
3.3-1 LeftSeries
3.3-2 RightSeries
3.3-3 IsLeftNilpotent
3.3-4 IsSimpleSkewbrace
3.3-5 IsRightNilpotent
3.3-6 LeftNilpotentIdeals
3.3-7 RightNilpotentIdeals
3.3-8 SmoktunowiczSeries
3.3-9 Socle
3.4 Mutipermutation skew braces
3.4-1 SocleSeries
3.4-2 MultipermutationLevel
3.4-3 IsMultipermutation
3.4-4 Fix
3.4-5 KernelOfLambda
3.4-6 Quotient
3.5 Prime and semiprime ideals
3.5-1 IsPrimeBrace
3.5-2 IsPrimeIdeal
3.5-3 PrimeIdeals
3.5-4 IsSemiprime
3.5-5 IsSemiprimeIdeal
3.5-6 SemiprimeIdeals
3.5-7 BaerRadical
3.5-8 IsBaer
3.5-9 WedderburnRadical
3.5-10 SolvableSeries
3.5-11 IsMinimalIdeal
3.5-12 MinimalIdeals
References
Index
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