1. Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight function - Weight functions in theory Part I Weight functions and the Hierarchical Bethe ansatz Part II Universal weight function and Drinfeld’s currents

2. Algebraic Bethe Ansatz ( case) Letbe a -operator of some quantum integrable Due to the RTT relation is the generating series of quantum integrals of motion. the transfer matrix model associated with the problem of finding eigenfunctions for If we can f ind a vector in the form is reduced to the Bethe equations for parameters such that Part I Weight functions and the Hierarchical Bethe ansatz

3. The Hierarchical Bethe Ansatz P. Kulish, N. Reshetikhin. Diagonalization of invariant transfer matrices and quantum N -wave system (Lee model) J.Phys. A: Math. Gen. 16 (1983) L591-L596 (short review) The Hierarhical Bethe Ansatz starts from the decomposition into blocks whereis a scalar,-dimensional column and row, and monodromy and are is a matrix. Let be representations of with the highest and auxiliary spaces respectively. The monodromy is now The problem is to find eigenstates of the transfer matrix Letbe quantum weight

4. Let The column fundamental representation of where of the monodromy by the comultiplication be such that any vector satisfies may be considered as an operator valued We look for eigenvectors in the form These vectors are defined by the elements acting in the tensor product

5. Let In the fundamental be a rational -matrix, where is the representation the monodromy matrix acts by the-matrix permutation operator Examples LetFix Denote and Let Denote and

6. 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 Let It satisfies the -relation with Define an element Letbe nonnegative integers such that Rename the variables where is an -operator realization of Borel subalgebra of

7. Example: Define an element In the case of the rational R-matrix

8. 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 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

9. that is, is a formal power series over the variables For any -multiset we choose the formal series where with coefficients in the ring of polynomials : For any representation with singular weight vector converges to a meromorphic -valued weight function. If then and If are two weight singular vectors, then is a weight singular vector in the tensor product and for any -multiset the weight function satisfies the recurrent relation

10. We call the element the Universal Weight Function

11. Chevalley description: Standard Hopf structures: Current (Drinfeld’s «new») realization of Current Hopf structure: Different realizations of the QAA Part II Universal weight function and Drinfeld’s current

12. Here are the operators of the adjoint action: Relation between the two realizations Let be the longest root of the Lie algebra Let be The assignment establishes the isomorphism of the two realizations.

13. Different Borel Subalgebras in We call the STANDARD Borel subalgebras of We call and CURRENT Borel subalgebras of

14. of linear spaces; (i) (i) The algebra admits a decomposition that is, the multiplication map establishes an isomorphism Let be a bialgebra with unit 1 and counit We say that its subalgebras and determine an orthogonal decomposition of if Orthogonal decompositions of Hopf algebras curves. Israel J.Math 112 (1999) 61-108 B. Enriquez, V. Rubtsov. Quasi-Hopf algebras associated with and complex (ii) is a left coideal, is a right coideal:

15. The Universal a system of simple roots of a Lie algebra The Universal given by the projection weight function weight function for the quantum affine algebra is Let be an ordered -multiset, where is Projections and the weight function elements where Theorem. Theorem. Let be a left ideal of generated by the Let be a singular weight vector in a highest weight representation of then is a meromorphic -valued weight function

16. Coproduct property of the weight function representations with singular vectors They are eigen- vectors of the Cartan currents Let be the tensor product of highest weight

17. weight function as a projection representation of Let be a singular weight vector in a highest weight Then the weight function is equal to

18. Ding-Frenkel isomorphism affine algebra Comm.Math. Phys. 156 (1993) 277-300 J. Ding, I. Frenkel. Isomorphism of two realizations of quantum

19. Composed currents and projections Define the screening operators

20. Calculation of the projections First step Let us calculate using After -steps:

21. Taking into account we conclude that The element for a collection of times satisfies: