Set theoretic Yang Baxter operators and their deformations

Set theoretic Yang Baxter operators and their deformations

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Michael Eisermann

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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

Publié par

Nombre de lectures

22

Langue

English

Washington (District de Columbia) Baxter Knot theory

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:

si

=

Artin’s braid group (1925)

Braids form a group:

Standard generators:

Obvious relations:

si

=

=

,

=

Artin’s braid group (1925)

Braids form a group:

Standard generators:

Obvious relations:

si=

=

,

=

Theorem (Artin 1925) The groupBnof braids onnstrands is presented by Bn=s1, . . . , sn−1sisjsi=sjsisjif|i−j|=12. sisj=sisjif|i−j| ≥

Yang-Baxter representations

Each automorphism

ci

=

c∈Aut(E⊗E)acts onE⊗n

as

Yang-Baxter representations

Each automorphism

ci

=

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

ci=

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 ρcn:Bn→Aut(E⊗n)given bysi7→ci.

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