ABOUT THE DYNAMICAL YANG BAXTER EQUATION S

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Niveau: Supérieur, Doctorat, Bac+8
ABOUT THE DYNAMICAL YANG-BAXTER EQUATION(S) AN INVITATION TO DYNAMICAL QUANTUM GROUPS DAMIEN CALAQUE Abstra t. These are the notes of a short talk given on some aspe ts of the dynami al Yang-Baxter equation during the meeting of the GDR Tresses in Clermont-Ferrand (September 3-6, 2006). It is largely inspired from the le ture notes of ICM talks by Felder [2? and Etingof [1?. I thank the organizors for giving me the o asion to give this talk, and for the ex ellent atmosphere during the onferen e. 1. The (quantum) dynami al Yang-Baxter equation Let h be a nite dimensional abelian Lie algebra, V a semi-simple h-module and ~ ? C?. For any (meromorphi ) fun tion R(?, z) : h? ? C ? Endh(V ? V ), the quantum dynami al Yang-Baxter equation (QDYBE) with step ~ reads: R1,2(?? ~h(3), z1 ? z2)R1,3(?, z1 ? z3)R2,3(?? ~h(1), z2 ? z3) = R2,3(?, z2 ? z3)R1,3(? ? ~h(2), z1 ? z3)R1,2(?, z1 ? z2) .

  • ve tor

  • lie algebra

  • quantum group

  • cdybe

  • standard theta-fun

  • fun tion

  • dynami al

  • graded lie


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h V h
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1,2 (3) 1,3 2,3 (1)
R (λ−~h ,z −z )R (λ,z −z )R (λ−~h ,z −z )1 2 1 3 2 3
2,3 1,3 (2) 1,2=R (λ,z −z )R (λ−~h ,z −z )R (λ,z −z ).2 3 1 3 1 2
1,2 (3)R (λ−~h ,z)
1,2 (3)R (λ−~h ,z)(v ⊗v ⊗v ) :=R(λ−~μ,z)(v ⊗v )⊗v ,1 2 3 1 2 3
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R(λ,z) = E ⊗E + E ⊗E E ⊗E ,ii ii ii jj ij ji
θ(λ −λ )θ(z−~) θ(z−~)θ(λ −λ )i j j ii=1 1≤i=j≤n
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~ r(λ,z)
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arXiv:hep-th/9407154
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el
P
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On
able
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al
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Y
h
ang-Baxter

e
Enriquez,
quation
Etingof,
,
quations
Pro
gr

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of

the
to
ICM,
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,
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arXiv:math.QA/0207008
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.


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elder,
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eld
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ory
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