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Yang-Baxter and Braid Dynamics

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Yang-Baxter and Braid Dynamics Yang-Baxter and braid dynamics Alexander Veselov, Loughborough University GADUDIS, Glasgow, April 1, 2009 Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I Links with Painleve-VI and Frobenius manifolds I Markov dynamics I Diophantine analysis and Markov equation I Yang-Baxter versus braid relations I Braid group and modular group Plan I Dynamics and groups Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I Links with Painleve-VI and Frobenius manifolds I Markov dynamics I Diophantine analysis and Markov equation I Yang-Baxter versus braid relations Plan I Dynamics and groups I Braid group and modular group Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I Links with Painleve-VI and Frobenius manifolds I Markov dynamics I Diophantine analysis and Markov equation Plan I Dynamics and groups I Braid group and modular group I Yang-Baxter versus braid relations Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I Links with Painleve-VI and Frobenius manifolds I Markov dynamics Plan I Dynamics and groups I Braid group and modular group I Yang-Baxter versus braid relations I Diophantine analysis and Markov equation Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I Links with Painleve-VI and Frobenius manifolds Plan I Dynamics and groups I Braid group and modular group I Yang-Baxter versus braid relations I Diophantine analysis and Markov equation I Markov dynamics Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. Plan I Dynamics and groups I Braid group and modular group I Yang-Baxter versus braid relations I Diophantine analysis and Markov equation I Markov dynamics I Links with Painleve-VI and Frobenius manifolds Plan I Dynamics and groups I Braid group and modular group I Yang-Baxter versus braid relations I Diophantine analysis and Markov equation I Markov dynamics I Links with Painleve-VI and Frobenius manifolds Thanks to Leonid Chekhov, Boris Dubrovin, Andy Hone, Oleg Lisovyy for illuminating discussions and to Vsevolod Adler for very useful comments and help with computer simulations. I would like to discuss today the actions of the braid group B3, which has exponential growth. Examples: Julia, Fatou, Ritt ; AV: commuting maps Okamoto; Noumi, Yamada: affine Weyl group actions INTEGRABILITY ∼ POLYNOMIAL GROWTH = ALMOST NILPOTENT GROUPS (Gromov’s theorem) Groups and dynamics Dynamical Erlangen programme: GROUPS ——– DYNAMICS Usual discrete dynamical system can be considered as nonlinear representation of Z. I would like to discuss today the actions of the braid group B3, which has exponential growth. Examples: Julia, Fatou, Ritt ; AV: commuting maps Okamoto; Noumi, Yamada: affine Weyl group actions Groups and dynamics Dynamical Erlangen programme: GROUPS ——– DYNAMICS Usual discrete dynamical system can be considered as nonlinear representation of Z. INTEGRABILITY ∼ POLYNOMIAL GROWTH = ALMOST NILPOTENT GROUPS (Gromov’s theorem) I would like to discuss today the actions of the braid group B3, which has exponential growth. Groups and dynamics Dynamical Erlangen programme: GROUPS ——– DYNAMICS Usual discrete dynamical system can be considered as nonlinear representation of Z. INTEGRABILITY ∼ POLYNOMIAL GROWTH = ALMOST NILPOTENT GROUPS (Gromov’s theorem) Examples: Julia, Fatou, Ritt ; AV: commuting maps Okamoto; Noumi, Yamada: affine Weyl group actions Groups and dynamics Dynamical Erlangen programme: GROUPS ——– DYNAMICS Usual discrete dynamical system can be considered as nonlinear representation of Z. INTEGRABILITY ∼ POLYNOMIAL GROWTH = ALMOST NILPOTENT GROUPS (Gromov’s theorem) Examples: Julia, Fatou, Ritt ; AV: commuting maps Okamoto; Noumi, Yamada: affine Weyl group actions I would like to discuss today the actions of the braid group B3, which has exponential growth. If we add the relations 2 σi = 1 the braid group Bn is reduced to the symmetric group Sn. It is generated by n − 1 elements σ1, . , σn−1 with the following relations σi σi+1σi = σi+1σi σi+1 σi σj = σj σi , |i − j| > 1. Braid group Braid group Bn can be defined as the fundamental group π1(Mn) of the configuration space Mn of n different (unordered) points in C. If we add the relations 2 σi = 1 the braid group Bn is reduced to the symmetric group Sn. Braid group Braid group Bn can be defined as the fundamental group π1(Mn) of the configuration space Mn of n different (unordered) points in C. It is generated by n − 1 elements σ1, . , σn−1 with the following relations σi σi+1σi = σi+1σi σi+1 σi σj = σj σi , |i − j| > 1. Braid group Braid group Bn can be defined as the fundamental group π1(Mn) of the configuration space Mn of n different (unordered) points in C. It is generated by n − 1 elements σ1, . , σn−1 with the following relations σi σi+1σi = σi+1σi σi+1 σi σj = σj σi , |i − j| > 1. If we add the relations 2 σi = 1 the braid group Bn is reduced to the symmetric group Sn. Braid group B3 bx bz bz bx b b x = z bz bx 3 b bx b = b b b z z x z x (bx bz) braidgroupdefiningrelations A natural homomorphism ∼ PSL(2, Z) → SL(2, Z2) = S3 maps it to the symmetric group. The quotient B3/Z is isomorphic to the modular group PSL(2, Z) = SL(2, Z)/{±I }, which is itself the free product Z2 ∗ Z3. Braid group B3 and the modular group Braid group B3 is generated by σ1 and σ2 with the only relation σ1σ2σ1 = σ2σ1σ2. It has the centre Z generated by 3 (σ1σ2) = (σ1σ2σ1)(σ2σ1σ2). A natural homomorphism ∼ PSL(2, Z) → SL(2, Z2) = S3 maps it to the symmetric group. Braid group B3 and the modular group Braid group B3 is generated by σ1 and σ2 with the only relation σ1σ2σ1 = σ2σ1σ2. It has the centre Z generated by 3 (σ1σ2) = (σ1σ2σ1)(σ2σ1σ2). The quotient B3/Z is isomorphic to the modular group PSL(2, Z) = SL(2, Z)/{±I }, which is itself the free product Z2 ∗ Z3. Braid group B3 and the modular group Braid group B3 is generated by σ1 and σ2 with the only relation σ1σ2σ1 = σ2σ1σ2. It has the centre Z generated by 3 (σ1σ2) = (σ1σ2σ1)(σ2σ1σ2). The quotient B3/Z is isomorphic to the modular group PSL(2, Z) = SL(2, Z)/{±I }, which is itself the free product Z2 ∗ Z3. A natural homomorphism ∼ PSL(2, Z) → SL(2, Z2) = S3 maps it to the symmetric group. The reversible Yang-Baxter maps satisfy the relation R21R = Id, corresponding to 2 Si = Id. The twisted maps Si = Pii+1Rii+1 satisfy the braid relations Si Si+1Si = Si+1Si Si+1, S12S23S12 = S23S12S23. The map R is called Yang-Baxter map if it satisfies the Yang-Baxter relation R12R13R23 = R23R13R12. Yang-Baxter versus braid relations Let X be any set and R be a map: R : X × X → X × X . n n n Let Rij : X → X , X = X × X × ..... × X be the maps which acts as R on i-th and j-th factors and identically on the others. If P : X 2 → X 2 is the permutation: P(x, y) = (y, x), then R21 = PRP. The reversible Yang-Baxter maps satisfy the relation R21R = Id, corresponding to 2 Si = Id. The twisted maps Si = Pii+1Rii+1 satisfy the braid relations Si Si+1Si = Si+1Si Si+1, S12S23S12 = S23S12S23. Yang-Baxter versus braid relations Let X be any set and R be a map: R : X × X → X × X . n n n Let Rij : X → X , X = X × X × ..... × X be the maps which acts as R on i-th and j-th factors and identically on the others. If P : X 2 → X 2 is the permutation: P(x, y) = (y, x), then R21 = PRP. The map R is called Yang-Baxter map if it satisfies the Yang-Baxter relation R12R13R23 = R23R13R12. The reversible Yang-Baxter maps satisfy the relation R21R = Id, corresponding to 2 Si = Id. Yang-Baxter versus braid relations Let X be any set and R be a map: R : X × X → X × X . n n n Let Rij : X → X , X = X × X × ..... × X be the maps which acts as R on i-th and j-th factors and identically on the others. If P : X 2 → X 2 is the permutation: P(x, y) = (y, x), then R21 = PRP. The map R is called Yang-Baxter map if it satisfies the Yang-Baxter relation R12R13R23 = R23R13R12. The twisted maps Si = Pii+1Rii+1 satisfy the braid relations Si Si+1Si = Si+1Si Si+1, S12S23S12 = S23S12S23. Yang-Baxter versus braid relations Let X be any set and R be a map: R : X × X → X × X . n n n Let Rij : X → X , X = X × X × ..... × X be the maps which acts as R on i-th and j-th factors and identically on the others. If P : X 2 → X 2 is the permutation: P(x, y) = (y, x), then R21 = PRP. The map R is called Yang-Baxter map if it satisfies the Yang-Baxter relation R12R13R23 = R23R13R12. The twisted maps Si = Pii+1Rii+1 satisfy the braid relations Si Si+1Si = Si+1Si Si+1, S12S23S12 = S23S12S23.
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