Skip to main content
Log in

Commutativity of Quantum Family Algebras

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Dixmier, J.: Enveloping Algebras, Grad. Stud. Math. 11, Amer. Math. Soc., Providence, RI, 1996.

    Google Scholar 

  2. Gould, M. D.: Characteristic identities for semisimple Lie algebras, J. Austral. Math. Soc. B 26(3) (1985), 257–283.

    Google Scholar 

  3. Green, H. S.: Characteristic identities for generators of GL(n), O(n) and SP(n), J. Math. Phys. 12 (1971), 2106–2113.

    Google Scholar 

  4. Hesselink, W. H.: Characters of the Nullcone, Math. Ann. 252(3) (1980), 179–182.

    Google Scholar 

  5. Gupta, R. K.: Characters and the q-analogue of weight multiplicities, J. London Math. Soc. (2) 36(1) (1987), 68–76.

    Google Scholar 

  6. Gupta, R. K.: Generalized exponents via Hall–Littlewood symmetric functions, Bull. Amer. Math. Soc. (N.S.) 16(2) (1987), 287–291.

    Google Scholar 

  7. Kato, S.: Spherical functions and a q-analogue of Kostant's weight multiplicity formula, Invent. Math. 66(3) (1982), 461–468.

    Google Scholar 

  8. Kirillov, A. A.: Family algebras, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 7–20 (electronic).

    Google Scholar 

  9. Kirillov, A. A.: Introduction to family algebras, Moscow Math. J. 1(1) (2001), 49–63.

    Google Scholar 

  10. Kostant, B.: On the tensor product of a finite and an infinite dimensional representation, J. Funct. Anal. 20(4) (1975), 257–285.

    Google Scholar 

  11. Kostant, B.: Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.

    Google Scholar 

  12. Kostant, B.: A formula for the multiplicity of a weight, Trans. Amer. Math. Soc. 93 (1959), 53–73.

    Google Scholar 

  13. 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.

    Google Scholar 

  14. Molev, A.: Sklyanin determinant, Laplace operators, and characteristic identities for classical Lie algebras, J. Math. Phys. 36(1) (1995), 923–943.

    Google Scholar 

  15. 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.

    Google Scholar 

  16. Perelomov, A. M. and Popov, V. S.: Casimir operators for semi-simple Lie groups, Izv. Akad. Nauk SSSR, Ser. Mat. 32 (1968), 1368–1390.

    Google Scholar 

  17. Rozhkovskaya, N.: Family algebras of representations with simple spectrum, ESI preprint No. 1045, http://www.esi.ac.at/preprints/ESI-Preprints.html.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints 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

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1023037100013

Navigation