ABOUT THE DYNAMICAL YANG BAXTER EQUATION S

Description

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

Sujets

• Clermont-Ferrand
• Baxter
• Lie algebra
• Quantum group

Informations

• Publié par: mijec
• Nombre de lectures: 41
• Langue: English

Extrait

h V h
×~∈C
∗R(λ,z) : h ×C→ End (V ⊗V)h
~
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
v ,v ∈V v ∈V[μ] V[μ] μ1 2 3
(1)
A Vn−1
gl h Eijn
∗(E ) =δ δ λ = (λ ,...,λ )∈ hij kl ik jl 1 n
∗λ =Ei ii
nX X θ(λ −λ +~)θ(z) θ(z−λ +λ )θ(~)i j j i
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
θ(z) :=θ(z|τ) ∂ θ(0) = 1z
θ(z) sin(z) z
z→∞ z
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1g
∗C h λ∈ h
M λ vλ λ
∗vλ
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v vΦ < Φ v∈V |v| =λ−μ>=vλ λ
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2,1−1R(λ) := J (λ) J (λ) zVV V,V
1
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J C M
1⊗J (λ) J (λ)V ,V V ,V ⊗V2 3 1 2 3
−⊗(−⊗(−⊗•)) −⊗((−⊗−)⊗•) (−⊗(−⊗−))⊗•
(3)J (λ−h )V ,V1 2
J (λ)V ⊗V ,V1 2 3
(−⊗−)⊗(−⊗•) ((−⊗−)⊗−)⊗•
2 ∗R(λ,z) = −~r(λ,z)+O(~ )∈Mer(h ×C,End (V⊗V))V⊗V h
~ r(λ,z)
1,2 2,3 1,2 1,3 1,3 2,3[r (λ,z −z ),r (λ,z −z )]+[r (λ,z −z ),r (λ,z −z )]+[r (λ,z −z ),r (λ,z −z )]1 2 2 3 1 2 1 3 1 3 2 3
2,3 1,3 1,2X ∂r ∂r ∂r(1) (2) (3)+ h (λ,z −z )−h (λ,z −z )+h (λ,z −z ) = 02 3 1 3 1 2ν ν νν ν ν∂λ ∂λ ∂λ
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1.3.X X ∂Fi,j (i)∂ F(z ,...,z ) = r (λ,z −z )·F − h · (i = 1,...,n),z 1 n i ji ν ν∂λ
νj|j=i
n ⊗nF(z ,...,z ) :C →V1 n
x y 1 ≤ i ≤ n 1 ti i ij
1≤i =j ≤n 2
[x ,x ] = [y