Universal Weight Function

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Universal Weight Function
Plan
- Nested (off-shell) Bethe vectors
- Borel subalgebras in the quantum affine algebras
- Projections and an Universal weight function
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Algebraic Bethe Ansatz (
Part I Weight functions and the Hierarchical
Bethe ansatz
case)
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The Hierarchical Bethe Ansatz
(short review)
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Weight function as specific element of monodromy
matrix ( case)
A.Varchenko, V.Tarasov. Jackson integrals for the
solutions to Knizhnik-Zamolodchikov equation,
Algebra and Analysis 2 (1995) no.2, 275-313
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Weight function
Let be -module. A vector
is called a weight singular vector with respect
to the action of if for
and If
we call
a vector-valued weight function of the weight
associated with the vector
Let be the set of indices of
the simple roots for the Lie
algebra A -multiset is a collection of
indexes together with a map
, where and
We associate a formal variable to the index
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We call the element
the
Universal Weight Function
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Part II Universal weight function and
Drinfelds current
Different realizations of the QAA
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Orthogonal decompositions of Hopf algebras
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Projections and the weight function
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Coproduct property of the weight
function
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Ding-Frenkel isomorphism
J. Ding, I. Frenkel. Isomorphism of two
realizations of quantum
affine algebra Comm.Math. Phys.
156 (1993) 277-300
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Composed currents and projections
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Calculation of the projections
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The element for a collection of times
satisfies