Finite quotients of groups of I-type or Quantum Yang-Baxter groups Finite quotients of groups of I-type or Fabienne Quantum Yang-Baxter groups Chouraqui General Introduction to the QYBE Fabienne Chouraqui Garside groups and the QYBE University of Haifa, Campus Oranim Coxeter-like finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
Finite quotients of groups of I-type or Quantum Yang-Baxter groups Finite quotients of groups of I-type or Quantum Yang-Baxter groups Fabienne Chouraqui Joint work with Eddy Godelle, Caen General Finite quotients of groups of I-type 2014 Introduction to the QYBE Garside groups and the QYBE Coxeter-like finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
Properties of a solution ( X , r ) Finite quotients of groups of I-type or Let X = { x 1 , ..., x n } and let r be defined in the following way: Quantum Yang-Baxter r ( i , j ) = ( σ i ( j ) , γ j ( i )), where σ i , γ i : X → X . groups Fabienne Proposition [P.Etingof, T.Schedler, A.Soloviev - 1999] Chouraqui General ( X , r ) is non-degenerate ⇔ σ i and γ i are bijective, Introduction 1 ≤ i ≤ n . to the QYBE ( X , r ) is involutive ⇔ r 2 = Id X × X . Garside groups and the QYBE ( X , r ) is braided ⇔ Coxeter-like ( Id × r )( r × Id )( Id × r ) = ( r × Id )( Id × r )( r × Id ) finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
Multipermutations solutions of level m ≥ 1 Finite quotients of groups of I-type or Quantum Yang-Baxter groups A retract relation ≡ on X is defined by: Fabienne Chouraqui General Introduction to the QYBE Garside groups and the QYBE Coxeter-like finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
Multipermutations solutions of level m ≥ 1 Finite quotients of groups of I-type or Quantum Yang-Baxter groups A retract relation ≡ on X is defined by: Fabienne Chouraqui x i ≡ x j if and only if σ i = σ j . General Introduction to the QYBE Garside groups and the QYBE Coxeter-like finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
Multipermutations solutions of level m ≥ 1 Finite quotients of groups of I-type or Quantum Yang-Baxter groups A retract relation ≡ on X is defined by: Fabienne Chouraqui x i ≡ x j if and only if σ i = σ j . General ( X , r ) is a multipermutation solution of level m or retractable if: Introduction to the QYBE Garside groups and the QYBE Coxeter-like finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
Multipermutations solutions of level m ≥ 1 Finite quotients of groups of I-type or Quantum Yang-Baxter groups A retract relation ≡ on X is defined by: Fabienne Chouraqui x i ≡ x j if and only if σ i = σ j . General ( X , r ) is a multipermutation solution of level m or retractable if: Introduction to the QYBE There exits m ≥ 1 such that Ret m ( G ) is a cyclic group and m is Garside groups and the smallest such integer, where Ret k +1 ( G ) = Ret 1 (Ret k ( G )). the QYBE Coxeter-like finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
The QYBE group: the structure group of ( X , r ) Finite quotients of groups of I-type or Assumption: ( X , r ) is a non-degenerate, involutive and braided Quantum solution. Yang-Baxter groups Fabienne Chouraqui General Introduction to the QYBE Garside groups and the QYBE Coxeter-like finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
The QYBE group: the structure group of ( X , r ) Finite quotients of groups of I-type or Assumption: ( X , r ) is a non-degenerate, involutive and braided Quantum solution. Yang-Baxter groups The structure group G of ( X , r ) [Etingof, Schedler, Soloviev] Fabienne Chouraqui The generators: X = { x 1 , x 2 , .., x n } . General Introduction to the QYBE Garside groups and the QYBE Coxeter-like finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
The QYBE group: the structure group of ( X , r ) Finite quotients of groups of I-type or Assumption: ( X , r ) is a non-degenerate, involutive and braided Quantum solution. Yang-Baxter groups The structure group G of ( X , r ) [Etingof, Schedler, Soloviev] Fabienne Chouraqui The generators: X = { x 1 , x 2 , .., x n } . General Introduction The defining relations: x i x j = x k x l whenever to the QYBE r ( i , j ) = ( k , l ) Garside groups and the QYBE Coxeter-like finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
The QYBE group: the structure group of ( X , r ) Finite quotients of groups of I-type or Assumption: ( X , r ) is a non-degenerate, involutive and braided Quantum solution. Yang-Baxter groups The structure group G of ( X , r ) [Etingof, Schedler, Soloviev] Fabienne Chouraqui The generators: X = { x 1 , x 2 , .., x n } . General Introduction The defining relations: x i x j = x k x l whenever to the QYBE r ( i , j ) = ( k , l ) Garside groups and the QYBE Coxeter-like There are exactly n ( n − 1) defining relations. finite 2 quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
The example Finite Let X = { x 1 , x 2 , x 3 , x 4 } . quotients of groups of I-type or The functions that define r Quantum Yang-Baxter σ 1 = γ 1 = σ 3 = γ 3 = (1 , 2 , 3 , 4) groups σ 2 = γ 2 = σ 4 = γ 4 = (1 , 4 , 3 , 2) Fabienne Chouraqui ( X , r ) is a non-degenerate, involutive and braided solution. General ( X , r ) is a multipermutation of level 2. Introduction to the QYBE Garside groups and the QYBE Coxeter-like finite quotients Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
The example Finite Let X = { x 1 , x 2 , x 3 , x 4 } . quotients of groups of I-type or The functions that define r Quantum Yang-Baxter σ 1 = γ 1 = σ 3 = γ 3 = (1 , 2 , 3 , 4) groups σ 2 = γ 2 = σ 4 = γ 4 = (1 , 4 , 3 , 2) Fabienne Chouraqui ( X , r ) is a non-degenerate, involutive and braided solution. General ( X , r ) is a multipermutation of level 2. Introduction to the QYBE Garside groups and The defining relations in G and in M the QYBE x 2 1 = x 2 x 2 3 = x 2 Coxeter-like 2 4 finite x 1 x 2 = x 3 x 4 x 1 x 3 = x 4 x 2 quotients x 2 x 4 = x 3 x 1 x 2 x 1 = x 4 x 3 Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
The example Finite Let X = { x 1 , x 2 , x 3 , x 4 } . quotients of groups of I-type or The functions that define r Quantum Yang-Baxter σ 1 = γ 1 = σ 3 = γ 3 = (1 , 2 , 3 , 4) groups σ 2 = γ 2 = σ 4 = γ 4 = (1 , 4 , 3 , 2) Fabienne Chouraqui ( X , r ) is a non-degenerate, involutive and braided solution. General ( X , r ) is a multipermutation of level 2. Introduction to the QYBE Garside groups and The defining relations in G and in M the QYBE x 2 1 = x 2 x 2 3 = x 2 ( x 1 x 4 = x 1 x 4 x 2 x 3 = x 2 x 3 ) Coxeter-like 2 4 finite x 1 x 2 = x 3 x 4 x 1 x 3 = x 4 x 2 quotients x 2 x 4 = x 3 x 1 x 2 x 1 = x 4 x 3 ( x 3 x 2 = x 3 x 2 x 4 x 1 = x 4 x 1 ) Orderability of groups Remarks and questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
The example Finite Let X = { x 1 , x 2 , x 3 , x 4 } . quotients of groups of I-type or The functions that define r Quantum Yang-Baxter σ 1 = γ 1 = σ 3 = γ 3 = (1 , 2 , 3 , 4) groups σ 2 = γ 2 = σ 4 = γ 4 = (1 , 4 , 3 , 2) Fabienne Chouraqui ( X , r ) is a non-degenerate, involutive and braided solution. General ( X , r ) is a multipermutation of level 2. Introduction to the QYBE Garside groups and The defining relations in G and in M the QYBE x 2 1 = x 2 x 2 3 = x 2 ( x 1 x 4 = x 1 x 4 x 2 x 3 = x 2 x 3 ) Coxeter-like 2 4 finite x 1 x 2 = x 3 x 4 x 1 x 3 = x 4 x 2 quotients x 2 x 4 = x 3 x 1 x 2 x 1 = x 4 x 3 ( x 3 x 2 = x 3 x 2 x 4 x 1 = x 4 x 1 ) Orderability of groups There are n ( n − 1) relations (and n trivial relations) Remarks and 2 questions to conclude Fabienne Chouraqui Finite quotients of groups of I-type or Quantum Yang-Baxter groups
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