On the quantum $\mathfrak{osp}(1|2\ell)$ Toda chain
Teoretičeskaâ i matematičeskaâ fizika, Tome 208 (2021) no. 2, pp. 180-195 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The orthosymplectic Lie superalgebra $\mathfrak{osp}(1|\,2\ell)$ is the closest superanalogue of standard Lie algebras. We demonstrate that the corresponding $\mathfrak{osp}(1|\,2\ell)$ Toda chain is an instance of the $BC_\ell$ Toda chain. The underlying reason for this relation is discussed.
Keywords: orthosymplectic Lie superalgebra, $BC$-type root system, quantum Toda chain.
@article{TMF_2021_208_2_a1,
     author = {A. A. Gerasimov and D. R. Lebedev and S. V. Oblezin},
     title = {On the~quantum $\mathfrak{osp}(1|2\ell)$ {Toda} chain},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {180--195},
     year = {2021},
     volume = {208},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2021_208_2_a1/}
}
TY  - JOUR
AU  - A. A. Gerasimov
AU  - D. R. Lebedev
AU  - S. V. Oblezin
TI  - On the quantum $\mathfrak{osp}(1|2\ell)$ Toda chain
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2021
SP  - 180
EP  - 195
VL  - 208
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2021_208_2_a1/
LA  - ru
ID  - TMF_2021_208_2_a1
ER  - 
%0 Journal Article
%A A. A. Gerasimov
%A D. R. Lebedev
%A S. V. Oblezin
%T On the quantum $\mathfrak{osp}(1|2\ell)$ Toda chain
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2021
%P 180-195
%V 208
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2021_208_2_a1/
%G ru
%F TMF_2021_208_2_a1
A. A. Gerasimov; D. R. Lebedev; S. V. Oblezin. On the quantum $\mathfrak{osp}(1|2\ell)$ Toda chain. Teoretičeskaâ i matematičeskaâ fizika, Tome 208 (2021) no. 2, pp. 180-195. http://geodesic.mathdoc.fr/item/TMF_2021_208_2_a1/

[1] B. Kostant, “Quantization and representation theory”, Representation Theory of Lie Groups, London Mathematical Society Lecture Note Series, 34, eds. M. F. Atiyah, R. Bott, S. Helgason, D. Kazhdan, B. Kostant, G. Lustztig, Cambridge Univ. Press, New York, 1980, 287–316 | DOI

[2] R. Goodman, N. R. Wallach, “Classical and quantum mechanical systems of Toda-lattice type. III. Joint eigenfunctions of the quantized systems”, Commun. Math. Phys., 105:3 (1986), 473–509 | DOI | MR

[3] M. A. Semenov-Tyan-Shanskii, “Kvantovanie nezamknutykh tsepochek Tody”: M. A. Olshanetskii, A. M. Perelomov, A. G. Reiman, M. A. Semenov-Tyan-Shanskii, Integriruemye sistemy. II, Dinamicheskie sistemy – 7, Itogi nauki i tekhn. Ser. Sovrem. probl. matem. Fundam. napravleniya, 16, ed. R. V. Gamkrelidze, VINITI, M., 1987, 194–226 | MR | MR

[4] M. Hashizume, “Whittaker functions on semisimple Lie groups”, Hiroshima Math. J., 12:2 (1982), 259–293 | DOI | MR

[5] A. A. Gerasimov, D. R. Lebedev, S. V. Oblezin, “Novye integralnye predstavleniya funktsii Uittekera dlya klassicheskikh grupp Li”, UMN, 67:1(403) (2012), 3–96, arXiv: 0705.2886 | DOI | DOI | MR | Zbl

[6] A. Gerasimov, D. Lebedev, S. Oblezin, “Quantum Toda chains intertwined”, Algebra i analiz, 22:3 (2010), 107–141, arXiv: 0907.0299 | DOI | MR | Zbl

[7] S. Khelgason, Differentsialnaya geometriya, gruppy Li i simmetricheskie prostranstva, KhKh vek. Matematika i mekhanika, ed. A. L. Onischik, Faktorial Press, M., 2005 | MR

[8] O. Loos, Cimmetricheskie prostranstva, Nauka, M., 1985 | MR

[9] H. P. B. Jacquet, “Fonctions de Whittaker associees aux groupes de Chevalley”, Bull. Soc. Math. France, 95 (1967), 243–309 | DOI | MR | Zbl

[10] A. G. Reiman, M. A. Semenov-Tyan-Shanskii, Integriruemye sistemy (teoretiko-gruppovoi podkhod), RKhD, M., Izhevsk, 2003 | MR

[11] E. K. Sklyanin, “Boundary conditions for integrable quantum systems”, J. Phys. A: Math. Gen., 21:10 (1988), 2375–2389 | DOI | MR

[12] V. G. Kac, “Lie superalgebras”, Adv. Math., 26:1 (1977), 8–96 | DOI | MR

[13] V. G. Kac, “A sketch of Lie superalgebra theory”, Commun. Math. Phys., 53:1 (1977), 31–64 | DOI | MR

[14] B. Kostant, “On Whittaker vectors and representation theory”, Invent. Math., 48:2 (1978), 101–184 | DOI | MR

[15] P. Etingof, “Whittaker functions on quantum groups and $q$-deformed Toda operators”, Differential Topology, Infinite-dimensional Lie Algebras, and Applications: D. B. Fuchs' 60th Anniversary Collection, American Mathematical Society Translations. Ser. 2, 194, eds. A. Astashkevich, S. Tabachnikov, AMS, Providence, RI, 1999, 9–25 | DOI | MR | Zbl

[16] P. Deligne, J. W. Morgan, “Notes on supersymmetry (following Joseph Berenstein)”, Quantum Fields and Strings: A Course for Mathematicians, v. 1, eds. P. Deligne, P. Etingof, D. S. Freed, L. Jeffrey, D. Kazhdan J. W. Morgan D. R. Morrison E. Witten, AMS, Providence, RI, 1999, 41–97 | MR