Abstract
Recently, A. A. Kirillov introduced an important notion of classical and quantum family algebras. Here the criterion of commutativity is given. The quantum eigenvalues of \(\mathfrak{s}\mathfrak{l}_2 \left( \mathbb{C} \right)\) are computed.
Similar content being viewed by others
References
Dixmier, J.: Enveloping Algebras, Grad. Stud. Math. 11, Amer. Math. Soc., Providence, RI, 1996.
Gould, M. D.: Characteristic identities for semisimple Lie algebras, J. Austral. Math. Soc. B 26(3) (1985), 257–283.
Green, H. S.: Characteristic identities for generators of GL(n), O(n) and SP(n), J. Math. Phys. 12 (1971), 2106–2113.
Hesselink, W. H.: Characters of the Nullcone, Math. Ann. 252(3) (1980), 179–182.
Gupta, R. K.: Characters and the q-analogue of weight multiplicities, J. London Math. Soc. (2) 36(1) (1987), 68–76.
Gupta, R. K.: Generalized exponents via Hall–Littlewood symmetric functions, Bull. Amer. Math. Soc. (N.S.) 16(2) (1987), 287–291.
Kato, S.: Spherical functions and a q-analogue of Kostant's weight multiplicity formula, Invent. Math. 66(3) (1982), 461–468.
Kirillov, A. A.: Family algebras, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 7–20 (electronic).
Kirillov, A. A.: Introduction to family algebras, Moscow Math. J. 1(1) (2001), 49–63.
Kostant, B.: On the tensor product of a finite and an infinite dimensional representation, J. Funct. Anal. 20(4) (1975), 257–285.
Kostant, B.: Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.
Kostant, B.: A formula for the multiplicity of a weight, Trans. Amer. Math. Soc. 93 (1959), 53–73.
Lusztig, G.: Singularities, character formulas, and a q-analog of weight multiplicities, In: Analyse et Topologie sur les Espaces Singuliers (II-III), Asterisque, 1983, 101–102, pp. 208–227.
Molev, A.: Sklyanin determinant, Laplace operators, and characteristic identities for classical Lie algebras, J. Math. Phys. 36(1) (1995), 923–943.
Nazarov, M.: Capelli elements in the classical universal enveloping algebras, In: Combinatorial Methods in Representation Theory (Kyoto, 1998), Adv. Stud. Pure Math. 28, Kinokuniya, Tokyo, 2000, pp. 261–285.
Perelomov, A. M. and Popov, V. S.: Casimir operators for semi-simple Lie groups, Izv. Akad. Nauk SSSR, Ser. Mat. 32 (1968), 1368–1390.
Rozhkovskaya, N.: Family algebras of representations with simple spectrum, ESI preprint No. 1045, http://www.esi.ac.at/preprints/ESI-Preprints.html.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rozhkovskaya, N. Commutativity of Quantum Family Algebras. Letters in Mathematical Physics 63, 87–103 (2003). https://doi.org/10.1023/A:1023037100013
Issue Date:
DOI: https://doi.org/10.1023/A:1023037100013