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Quantum group symmetryof integrable models
on the half line
Talk presented at the Workshop on Integrable Theories, Solitons and Duality
at IFT, Sao Paulo, 4 July 2002
Gustav W DeliusAlan GeorgeNiall MacKay
hep-th/0112023and unpublished
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ContentsI. Boundary quantum groups
Describing an algebraic structure that leads to the solutions of the reflection equation
II. Non-local charges in affine Toda field theory on the half-lineUsing boundary perturbation theory to find non-local charges unbroken by the boundary conditions
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Boundary quantum groups Drinfeld and Jimbo explained how to
obtain solutions of the Yang-Baxter equation as intertwiners of representations of quantum groups
We will introduce boundary quantum groups to similarly obtain solutions of the reflection equation as intertwiners
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Yang-Baxter equation
=
where =
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Reflection equation
=
where
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Drinfeld-Jimbo quantum groups
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Boundary quantum groups
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Properties required of boundary quantum group
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Finding boundary quantum groupsTo find boundary quantum groups and their
representations we determine the symmetry algebras of physical models on the half line:
Rational: Principal chiral models This leads to twisted Yangians Y(g,h) as explained by Niall MacKay in his talk at this conference.
Trigonometric: Affine Toda field theorieswe will derive boundary quantum groups
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Sine-Gordon model
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Non-local charges
[Bernard and LeClair, Comm.Math.Phys. 142 (1991) 99]
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Quantum affine sl2
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Neumann boundary condition
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Free field two-point functions
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Perturbing operator
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Boundary quantum group
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General boundary condition
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Boundary perturbation theory
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Boundary quantum group
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Affine Toda theory action
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Non-local charges
[Bernard and LeClair, Comm.Math.Phys. 142 (1991) 99]
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Quantum affine algebra
[Felder and LeClair, Int.J.Mod.Phys. A7 (1992) 239]
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Boundary conditions
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Non-local charges
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Conventions
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Boundary quantum groups
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How do we establish non-local charges non-perturbatively? We calculated quantum group charges
using first order perturbation theory only.
We then used these to calculate the reflection matrices in the vector representation.
But how can we rule out that there are higher order corrections which will be needed to treat higher representations?
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Reconstructing symmetry generators from reflection matrix
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Reconstructing symmetry generators from reflection matrix
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The an(1) case
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Reflection equation algebras
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Summary We introduced boundary quantum groups as certain coideal
subalgebras of quantum affine algebras Reflection matrices arise as intertwiners of representations of
boundary quantum groups We used boundary perturbation theory to find the non-local
charges in affine Toda field on the half-line. They give us boundary quantum groups for trigonometric reflection matrices .
We determined the reflection matrices for the vector representation of an
(1), cn(1), dn
(1), and a2n-1(2).
We showed that our expressions for the boundary non-local charges are exact.
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Things to do next Establish the boundary quantum groups for bn
(1), a2n(2), dn+1
(2) and the exceptional cases (in progress with Alan George)
Find the boundary quantum groups corresponding to Dirichlet type boundary conditions
Apply the tensor product graph method to the construction of higher reflection matrices
Study the representation theory of boundary quantum groups (multiplets of boundary states)
Study the mathematical structure of boundary quantum groups (Universal K matrix? Classification?)