Geodesic
Comultiplication in $ABCD$ algebra and scalar products of Bethe wave functions
Teoretičeskaâ i matematičeskaâ fizika, Tome 100 (1994) no. 1, pp. 113-118 
A. Mikhailov. Comultiplication in $ABCD$ algebra and scalar products of Bethe wave functions. Teoretičeskaâ i matematičeskaâ fizika, Tome 100 (1994) no. 1, pp. 113-118. http://geodesic.mathdoc.fr/item/TMF_1994_100_1_a10/
@article{TMF_1994_100_1_a10,
     author = {A. Mikhailov},
     title = {Comultiplication in $ABCD$ algebra and scalar products of {Bethe} wave functions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {113--118},
     year = {1994},
     volume = {100},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1994_100_1_a10/}
}
TY  - JOUR
AU  - A. Mikhailov
TI  - Comultiplication in $ABCD$ algebra and scalar products of Bethe wave functions
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1994
SP  - 113
EP  - 118
VL  - 100
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_1994_100_1_a10/
LA  - ru
ID  - TMF_1994_100_1_a10
ER  - 
%0 Journal Article
%A A. Mikhailov
%T Comultiplication in $ABCD$ algebra and scalar products of Bethe wave functions
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1994
%P 113-118
%V 100
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_1994_100_1_a10/
%G ru
%F TMF_1994_100_1_a10

Voir la notice de l'article provenant de la source Math-Net.Ru

The representation of scalar products of Bethe wave functions in terms of dual fields, proved by A. G. Izergin and V. E. Korepin in 1987, plays an important role in the theory of completely integrable models. The proof in [A. G. Izergin, Dokl. Akad. Nauk SSSR, 297, No. 2, 331 (1987)] and [V. E. Korepin, Commun. Math. Phys., 113, 177–190 (1978)] is based on the explicit expression for the senior coefficient, which was guessed in the Izergin paper and then proved to satisfy some recurrent relations, which determine it unambiguously. In this paper we present an alternative proof based on direct computation. It uses the operation of comultiplication in the $ABCD$-algebra.

[1] L. D. Faddeev, L. A. Takhtajan, Usp. Mat. Nauk, 34 (1979), 13–63 | MR

[2] V. E. Korepin, A. G. Izergin, N. M. Bogoliubov, Quantum Inverse Scattering Method, Correlation Functions and Algebraic Bethe Ansatz, Cambridge University Press, 1992 | MR

[3] N. Yu. Reshetikhin, L. A. Takhtajan, L. D. Faddeev, Algebra and Analysis, 1 (1989), 178–206 | MR

[4] V. E. Korepin, Commun. Math. Phys., 86 (1982), 391–418 | DOI | MR | Zbl

[5] V. E. Korepin, Commun. Math. Phys., 94 (1984), 93–113 | DOI | MR | Zbl

[6] A. G. Izergin, V. E. Korepin, Commun. Math. Phys., 94 (1984), 67–92 | DOI | MR | Zbl

[7] A. G. Izergin, Doklady AN SSSR, 297:2 (1987), 331 | MR | Zbl

[8] V. E. Korepin, Commun. Math. Phys., 113 (1978), 177–190 | DOI | MR

[9] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, “Transformation groups for soliton equations”, Proc. RIMS symp. Nonlinear integrable systems – classical theory and quantum theory

[10] G. Segal, G. Wilson, Publ. IHES, 61 (1985), 1 | DOI

[11] A. Yu. Morozov, Integrability and Matrix Models, preprint ITEP-M2/93

[12] M. Sato, M. Jimbo, T. Miwa, Y. Mori, Physica D, 1 (1980), 80–158 | DOI | MR | Zbl

[13] M. Jimbo, T. Miwa, K. Ueno, Physica D, 2 (1981), 306–352, 407–448 | DOI | MR | Zbl

[14] A. R. Its, A. G. Izergin, V. E. Korepin, N. A. Slavnov, Differential Equations for Quantum Correlation Functions, preprint CMA-R26-89, 1989 | MR

[15] A. R. Its, A. G. Izergin, V. E. Korepin, N. A. Slavnov, Quantum correlation function is the rfunction of classical differential equation, preprint ITP-SB-91-58, 1991 | MR