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Set theoretic Yang Baxter operators and their deformations

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126 pages
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


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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, . . . , sn1sisjsi=sjsisjif|ij|=12. sisj=sisjif|ij| ≥
Yang-Baxter representations
Each automorphism
ci
=
cAut(EE)acts onEn
as
Yang-Baxter representations
Each automorphism
ci
=
cAut(EE)acts onEn
as
The braid relation now becomes the Yang-Baxter equation:
=
(cid)(idc)(idc) = (idc)(cid)(idc)
Yang-Baxter representations
Each automorphismcAut(EE)acts onEn
ci=
as
The braid relation now becomes the Yang-Baxter equation:
=
(cid)(idc)(idc) = (idc)(cid)(idc)
Corollary (of Artin’s theorem)
Every Yang-Baxter operatorcinduces a braid group representation ρcn:BnAut(En)given bysi7→ci.
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