Weight functions and Drinfeld currents.(English)Zbl 1140.17010
The nested Bethe ansatz consists in the construction of certain rational functions with values in a representation of a quantum affine algebra or its rational or elliptic analogue. In [P. P. Kulish andN. Yu. Reshetikhin, J. Phys. A 16, L591–L596 (1983;Zbl 0545.35082)] by induction procedure it is proved that for the quantum affine algebra \(U_q(\widehat{gl}_N)\) these rational functions called the off-shell Bethe vectors have the form of the product of the monodromy matrices (a generating series for elements of the algebra) and a highest weight vector of a finite-dimensional representation of the corresponding \(U_q\). Such products defines a weight function for a quantum affine algebra.
The goal of the paper is to give a direct construction of weight functions, independent of an inductive procedure. The authors introduce the notion of a universal weight function as a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties. The action of the universal weight function on the highest weight vector defines a weight function with values in a representation of the quantum affine algebra.
The quantum affine algebras, as well as affine Kac-Moody Lie algebras, admit two different realizations. One is based on the description of the quantum affine algebra by means of Chevalley generators, satisfying \(q\)-analogues of the defining relations for Kac-Moody Lie algebras. In the other, generators are the components of the Drinfeld currents, and the relations are the deformations of the loop algebra presentation of the affine Lie algebra. The quantum affine algebra is equipped with two coproducts: standard and “Drinfeld”. It is proved in [S. Khoroshkin andV. Tolstoy, The Cartan-Weyl basis and the universal \({\mathcal R}\)-matrix for quantum Kac-Moody algebras and superalgebras. In: Quantum Symmetries, River Edge, NJ: World Sci. Publ. 1993, pp. 336–351 (1993), J. Geom. Phys. 11, 445–452 (1993;Zbl 0784.17022),J. Beck, Commun. Math. Phys. 165, No .3, 555–568 (1994;Zbl 0807.17013),I. Damiani, Ann. Sci. Éc. Norm. Supér. (4) 31, No. 4, 493–523 (1998;Zbl 0911.17005),J. Ding, S. Pakulyak andS. Khoroshkin, Theor. Math. Phys. 124, No. 2, 1007–1037 (2000); translation from Teor. Mat. Fiz. 124, No. 2, 179–214 (2000;Zbl 1112.17300)] that these coproducts are related by a twist. In correspondence with these structures the weight functions have different construction. Each realization determines a decomposition of the algebra as the product of two opposite Borel subalgebras (so there are four subalgebras). So, each Borel subalgebra decomposes as the product of intersections with two Borel subalgebras of the other type, and determines two projection operators which map it to these intersections. By means of these projections the twist is defined.
In the paper under review the authors give a new topological proof of these results. The main result of the paper is Theorem 3, that says that collection of images of products of Drinfeld currents by the projection defines a universal weight function. This allows to compute the weight function in some particular cases using techniques of complex analysis and conformal algebras. A conjecture on the general form of the universal weight function is given and some results from this conjecture for finite dimensional modules are proposed.
The goal of the paper is to give a direct construction of weight functions, independent of an inductive procedure. The authors introduce the notion of a universal weight function as a family of functions with values in a quotient of its Borel subalgebra, satisfying certain coalgebraic properties. The action of the universal weight function on the highest weight vector defines a weight function with values in a representation of the quantum affine algebra.
The quantum affine algebras, as well as affine Kac-Moody Lie algebras, admit two different realizations. One is based on the description of the quantum affine algebra by means of Chevalley generators, satisfying \(q\)-analogues of the defining relations for Kac-Moody Lie algebras. In the other, generators are the components of the Drinfeld currents, and the relations are the deformations of the loop algebra presentation of the affine Lie algebra. The quantum affine algebra is equipped with two coproducts: standard and “Drinfeld”. It is proved in [S. Khoroshkin andV. Tolstoy, The Cartan-Weyl basis and the universal \({\mathcal R}\)-matrix for quantum Kac-Moody algebras and superalgebras. In: Quantum Symmetries, River Edge, NJ: World Sci. Publ. 1993, pp. 336–351 (1993), J. Geom. Phys. 11, 445–452 (1993;Zbl 0784.17022),J. Beck, Commun. Math. Phys. 165, No .3, 555–568 (1994;Zbl 0807.17013),I. Damiani, Ann. Sci. Éc. Norm. Supér. (4) 31, No. 4, 493–523 (1998;Zbl 0911.17005),J. Ding, S. Pakulyak andS. Khoroshkin, Theor. Math. Phys. 124, No. 2, 1007–1037 (2000); translation from Teor. Mat. Fiz. 124, No. 2, 179–214 (2000;Zbl 1112.17300)] that these coproducts are related by a twist. In correspondence with these structures the weight functions have different construction. Each realization determines a decomposition of the algebra as the product of two opposite Borel subalgebras (so there are four subalgebras). So, each Borel subalgebra decomposes as the product of intersections with two Borel subalgebras of the other type, and determines two projection operators which map it to these intersections. By means of these projections the twist is defined.
In the paper under review the authors give a new topological proof of these results. The main result of the paper is Theorem 3, that says that collection of images of products of Drinfeld currents by the projection defines a universal weight function. This allows to compute the weight function in some particular cases using techniques of complex analysis and conformal algebras. A conjecture on the general form of the universal weight function is given and some results from this conjecture for finite dimensional modules are proposed.
Reviewer: Valentina Golubeva (Moskva)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
Keywords:
universal weight functions;solutions of the \(q\)KZ equations;Bethe vectors;Yangian;representation of quantum affine algebraCite
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