×

zbMATH — the first resource for mathematics

Symmetric Lie superalgebras and deformed quantum Calogero-Moser problems. (English) Zbl 1430.17034
Summary: The representation theory of symmetric Lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum Calogero-Moser systems. In the special case of symmetric pair \(\mathfrak{g} = \mathfrak{gl}(n,2m), \mathfrak{k} = \mathfrak{osp}(n,2m)\) we establish a natural bijection between projective covers of spherically typical irreducible \(\mathfrak{g}\)-modules and the finite dimensional generalised eigenspaces of the algebra of Calogero-Moser integrals \(\mathfrak{D}_{n,m}\) acting on the corresponding Laurent quasi-invariants \(\mathfrak{A}_{n,m}\).

MSC:
17B20 Simple, semisimple, reductive (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
17B80 Applications of Lie algebras and superalgebras to integrable systems
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alldridge, A.; Schmittner, S., Spherical representations of Lie supergroups, J. Funct. Anal., 268, 6, 1403-1453, (2015) · Zbl 1360.22018
[2] Alldridge, A.; Hilgert, J.; Zirnbauer, M. R., Chevalley’s restriction theorem for reductive symmetric superpairs, J. Algebra, 323, 4, 1159-1185, (2010) · Zbl 1238.17005
[3] Berezin, F. A.; Pokhil, G. P.; Finkelberg, V. M., Schrödinger equation for a system of one-dimensional particles with point interaction, Vestnik MGU, 1, 21-28, (1964)
[4] Bourbaki, N., Lie groups and Lie algebras, (2005), Springer, Chapters 7-9
[5] Brundan, J., Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra \(\mathfrak{gl}(m | n)\), J. Amer. Math. Soc., 16, 1, 185-231, (2003) · Zbl 1050.17004
[6] Brundan, J., Tilting modules for Lie superalgebras, Comm. Algebra, 32, 6, 2251-2268, (2004) · Zbl 1077.17006
[7] Brundan, J.; Stroppel, C., Highest weight categories arising from Khovanov’s diagram algebra IV: the general linear supergroup, J. Eur. Math. Soc. (JEMS), 14, 2, 373-419, (2012) · Zbl 1243.17004
[8] Chalykh, O. A.; Feigin, M. V.; Veselov, A. P., New integrable generalizations of Calogero-Moser quantum problem, J. Math. Phys., 39, 2, 695-703, (1998) · Zbl 0906.34061
[9] Dixmier, J., Enveloping algebras, Graduate Studies in Math., vol. 11, (1977), Amer. Math. Society
[10] Etingof, P., Calogero-Moser systems and representation theory, Zurich Lectures in Advanced Mathematics, (2000), EMS
[11] Goodman, R.; Wallach, N. R., Representations and invariants of the classical groups, Encyclopedia of Mathematics and Its Applications, vol. 68, (1998), Cambridge University Press Cambridge, 685 pp · Zbl 0901.22001
[12] Helgason, S., Differential geometry and symmetric spaces, (1962), Academic Press, Inc. · Zbl 0122.39901
[13] Helgason, S., Groups and geometric analysis, (1984), Academic Press, Inc.
[14] Jack, H., A class of symmetric polynomials with a parameter, Proc. Roy. Soc. Edinburgh Sect. A, 69, 1-18, (1970) · Zbl 0198.04606
[15] Kac, V. G., Lie superalgebras, Adv. Math., 26, 1, 8-96, (1977) · Zbl 0366.17012
[16] Kac, V. G., Characters of typical representations of classical Lie superalgebras, Comm. Algebra, 5, 8, 889-897, (1977) · Zbl 0359.17010
[17] Kac, V., Representations of classical Lie superalgebras, (Differential Geometrical Methods in Mathematical Physics, II, Lecture Notes in Math., vol. 676, (1978), Springer Berlin), 597-626 · Zbl 0388.17002
[18] Macdonald, I., Symmetric functions and Hall polynomials, (1995), Oxford Univ. Press · Zbl 0824.05059
[19] Molev, A., Yangians and classical Lie algebras, (1993), Amer. Math. Soc. · Zbl 0876.17014
[20] Olshanetsky, M. A.; Perelomov, A. M., Quantum systems related to root systems and radial parts of Laplace operators, Funct. Anal. Appl., 12, 121-128, (1978) · Zbl 0407.43012
[21] Olshanetsky, M. A.; Perelomov, A. M., Quantum integrable systems related to Lie algebras, Phys. Rep., 94, 313-404, (1983)
[22] Penkov, I.; Serganova, V., Representations of classical Lie superalgebras of type I, Indag. Math. (N.S.), 3, 419-466, (1992) · Zbl 0849.17030
[23] Serganova, V. V., Classification of simple real Lie superalgebras and symmetric superspaces, Funct. Anal. Appl., 17, 3, 200-207, (1983) · Zbl 0545.17001
[24] Serganova, V., Automorphisms of complex simple Lie superalgebras and affine Kac-Moody algebras, (1988), Leningrad State University, PhD thesis
[25] Sergeev, A., The invariant polynomials on simple Lie superalgebras, Represent. Theory, 3, 250-280, (1999) · Zbl 0999.17016
[26] Sergeev, A. N., The Calogero operator and Lie superalgebras, Theoret. Math. Phys., 131, 3, 747-764, (2002) · Zbl 1039.81028
[27] Sergeev, A. N.; Veselov, A. P., Deformed quantum Calogero-Moser problems and Lie superalgebras, Comm. Math. Phys., 245, 2, 249-278, (2004) · Zbl 1062.81097
[28] Sergeev, A. N.; Veselov, A. P., Generalised discriminants, deformed Calogero-Moser-Sutherland operators and super-Jack polynomials, Adv. Math., 192, 2, 341-375, (2005) · Zbl 1069.05076
[29] Sergeev, A. N.; Veselov, A. P., \(B C_\infty\) Calogero-Moser operator and super Jacobi polynomials, Adv. Math., 222, 1687-1726, (2009) · Zbl 1220.33014
[30] Sergeev, A. N.; Veselov, A. P., Jack-Laurent symmetric functions, Proc. Lond. Math. Soc., 111, 1, 63-92, (2015) · Zbl 1316.05124
[31] Sergeev, A. N.; Veselov, A. P., Dunkl operators at infinity and Calogero-Moser systems, Int. Math. Res. Not. IMRN, (2015) · Zbl 1329.81433
[32] Sergeev, A. N.; Veselov, A. P., Jack-Laurent symmetric functions for special values of parameters, Glasg. Math. J., 58, 3, 599-616, (2016) · Zbl 1343.05158
[33] Ujino, H.; Hikami, K.; Wadati, M., Integrability of the quantum Calogero-Moser model, J. Phys. Soc. Jpn., 61, 3425, (1992)
[34] (van Diejen, J. F.; Vinet, L., Calogero-Moser-Sutherland Models, Montreal, QC, 1997, CRM Ser. Math. Phys., (2000), Springer New York), 23-35
[35] Zirnbauer, M. R., Riemannian symmetric superspaces and their origin in random-matrix theory, J. Math. Phys., 37, 10, 4986-5018, (1996) · Zbl 0871.58005
[36] Zou, Y. M., Categories of finite-dimensional weight modules over type I classical Lie superalgebras, J. Algebra, 180, 2, 459-482, (1996) · Zbl 0843.17001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.