Set theoretic Yang Baxter operators and their deformations

Michael Eisermann

Description

Set-theoretic Yang-Baxter operators and their deformations Michael Eisermann Institut Fourier, Universite Grenoble www-fourier.ujf-grenoble.fr/˜eiserm January 5, 2009 AMS–MAA Joint Mathematics Meetings in Washington DC Special Session on Algebraic Structures in Knot Theory

• knot theory


• ujf grenoble


• obvious relations


• representations artin's braid


• yang


• www-fourier


• mathematics meetings


• artin's braid


• standard generators

Sujets

• Washington (District de Columbia)
• Baxter
• Knot theory

Informations

• Publié par: pefav
• Nombre de lectures: 22
• Langue: English

Extrait

Set-theoretic Yang-Baxter operators and their deformations

Michael Eisermann

Institut Fourier, Universit  ´e  Grenoble www-fourier.ujf-grenoble.fr/˜eiserm

January 5, 2009

AMS–MAA Joint Mathematics Meetings in Washington DC Special Session on Algebraic Structures in Knot Theory

Overview

1

2

3

4

Braid groups and Yang-Baxter representations

Yang-Baxter deformations

Yang-Baxter cohomology of racks

Conclusion and open questions

Overview

1

2

3

4

Braid groups and Yang-Baxter representations Artin’s braid group Yang-Baxter representations Set-theoretic operators

Yang-Baxter deformations

Yang-Baxter cohomology of racks

Conclusion and open questions

Artin’s braid group (1925)

Braids form a group:

Artin’s braid group (1925)

Braids form a group:

Standard generators:

s i

=

Artin’s braid group (1925)

Braids form a group:

Standard generators:

Obvious relations:

s i

=

=

,

=

Artin’s braid group (1925)

Braids form a group:

Standard generators:

Obvious relations:

s i =

=

,

=

Theorem (Artin 1925) The groupB n of braids onnstrands is presented by B n =  s 1 , . . . , s n−1  s i s j s i =s j s i s j if|i−j|=1 2  . s i s j =s i s j if|i−j| ≥

Yang-Baxter representations

Each automorphism

c i

=

c∈Aut(E⊗E)acts onE ⊗n

as

Yang-Baxter representations

Each automorphism

c i

=

c∈Aut(E⊗E)acts onE ⊗n

as

The braid relation now becomes the Yang-Baxter equation:

=

(c⊗id)(id⊗c)(id⊗c) = (id⊗c)(c⊗id)(id⊗c)

Yang-Baxter representations

Each automorphismc∈Aut(E⊗E)acts onE ⊗n

c i =

as

The braid relation now becomes the Yang-Baxter equation:

=

(c⊗id)(id⊗c)(id⊗c) = (id⊗c)(c⊗id)(id⊗c)

Corollary (of Artin’s theorem)

Every Yang-Baxter operatorcinduces a braid group representation ρ c n :B n →Aut(E ⊗n )given bys i 7→c i .