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A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL

Year 2021, , 1 - 15, 17.07.2021
https://doi.org/10.24330/ieja.969570

Abstract

Let $q$ be a scalar that is not a root of unity. We show that any
nonzero polynomial in the Casimir element of the Fairlie-Odesskii
algebra $U_q'(\mathfrak{so}_3)$ cannot be expressed in terms of
only Lie algebra operations performed on the generators
$I_1,I_2,I_3$ in the usual presentation of
$U_q'(\mathfrak{so}_3)$. Hence, the vector space sum of the center
of $U_q'(\mathfrak{so}_3)$ and the Lie subalgebra of
$U_q'(\mathfrak{so}_3)$ generated by $I_1,I_2,I_3$ is direct.

References

  • G. Bergman,The diamond lemma for ring theory, Adv. in Math., 29(2) (1978), 178-218.
  • R. Cantuba, A Lie algebra related to the universal Askey-Wilson algebra, Matimyas Matematika, 38(2) (2015), 51-75.
  • R. Cantuba, Lie polynomials in $q$-deformed Heisenberg algebras, J. Algebra, 522 (2019), 101-123.
  • R. Cantuba, Compactness property of Lie polynomials in the creation and annihilation operators of the $q$-oscillator, Lett. Math. Phys., 110(10) (2020), 2639-2657.
  • R. Cantuba and M. Merciales, An extension of a $q$-deformed Heisenberg algebra and its Lie polynomials, Expo. Math., (2020), https://doi.org/10.1016/j.exmath.2019.12.001.
  • R. Cantuba and S. Silvestrov, Torsion-type $q$-deformed Heisenberg algebra and its Lie polynomials, In: S. Silvestrov, A. Malyarenko, M. Rancic (eds), Algebraic Structures and Applications, SPAS 2017, Springer Proceedings in Mathematics & Statistics, vol 317, Springer, Cham., (2020), 575-592.
  • R. Cantuba and S. Silvestrov, Lie polynomial characterization problems, In: S. Silvestrov, A. Malyarenko, M. Rancic (eds), Algebraic Structures and Applications, SPAS 2017, Springer Proceedings in Mathematics & Statistics, 317, Springer, Cham., (2020), 593-601.
  • L. Chekhov and V. Fock, Observables in 3D gravity and geodesic algebras, Czechoslovak J. Phys., 50(11) (2000), 1201-1208.
  • D. Fairlie, Quantum deformations of $SU(2)$, J. Phys. A., 23(5) (1990), L183-L187.
  • M. Havlicek, U. Klimyk and S. Posta, Representations of the cyclically symmetric q-deformed algebra $so_{q}(3)$, J. Math. Phys., 40(4) (1999), 2135-2161.
  • M. Havlicek and S. Posta, On the classification of irreducible finite-dimensional representations of U'q (so3), J. Math. Phys., 42(1) (2001), 472-500.
  • M. Havlicek and S. Posta, Center of quantum algebra U'q (so3), J. Math. Phys., 52(4) (2011), 943521 (15 pp).
  • L. Hellstrom and S. Silvestrov, Commuting elements in $q$-deformed Heisenberg algebras, World Scient Publ Co., River Edge, NJ, 2000.
  • L. Hellstrom and S. Silvestrov, Two-sided ideals in $q$-deformed Heisenberg algebras, Expo. Math., 23(2) (2005), 99-125.
  • N. Iorgov, On the center of the $q$-deformed algebra U'q (so3) related to quantum gravity at $q$ a root of $1$, Proceedings of Institute of Mathematics of NAS of Ukraine, 43(2) (2002), 449-455.
  • T. Ito, P. Terwilliger and C. Weng, The quantum algebra Uq (sl2) and its equitable presentation, J. Algebra, 298(1) (2006), 284-301.
  • A. Odesskii, An analog of the Sklyanin algebra, Funct. Anal. Appl., 20 (1986), 152-154.
  • P. Terwilliger, The universal Askey-Wilson algebra, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 069 (24 pp).
  • A. Zhedanov, "Hidden symmetry" of the Askey-Wilson polynomials, Theoret. and Math. Phys., 89(2) (1991), 1146-1157.
Year 2021, , 1 - 15, 17.07.2021
https://doi.org/10.24330/ieja.969570

Abstract

References

  • G. Bergman,The diamond lemma for ring theory, Adv. in Math., 29(2) (1978), 178-218.
  • R. Cantuba, A Lie algebra related to the universal Askey-Wilson algebra, Matimyas Matematika, 38(2) (2015), 51-75.
  • R. Cantuba, Lie polynomials in $q$-deformed Heisenberg algebras, J. Algebra, 522 (2019), 101-123.
  • R. Cantuba, Compactness property of Lie polynomials in the creation and annihilation operators of the $q$-oscillator, Lett. Math. Phys., 110(10) (2020), 2639-2657.
  • R. Cantuba and M. Merciales, An extension of a $q$-deformed Heisenberg algebra and its Lie polynomials, Expo. Math., (2020), https://doi.org/10.1016/j.exmath.2019.12.001.
  • R. Cantuba and S. Silvestrov, Torsion-type $q$-deformed Heisenberg algebra and its Lie polynomials, In: S. Silvestrov, A. Malyarenko, M. Rancic (eds), Algebraic Structures and Applications, SPAS 2017, Springer Proceedings in Mathematics & Statistics, vol 317, Springer, Cham., (2020), 575-592.
  • R. Cantuba and S. Silvestrov, Lie polynomial characterization problems, In: S. Silvestrov, A. Malyarenko, M. Rancic (eds), Algebraic Structures and Applications, SPAS 2017, Springer Proceedings in Mathematics & Statistics, 317, Springer, Cham., (2020), 593-601.
  • L. Chekhov and V. Fock, Observables in 3D gravity and geodesic algebras, Czechoslovak J. Phys., 50(11) (2000), 1201-1208.
  • D. Fairlie, Quantum deformations of $SU(2)$, J. Phys. A., 23(5) (1990), L183-L187.
  • M. Havlicek, U. Klimyk and S. Posta, Representations of the cyclically symmetric q-deformed algebra $so_{q}(3)$, J. Math. Phys., 40(4) (1999), 2135-2161.
  • M. Havlicek and S. Posta, On the classification of irreducible finite-dimensional representations of U'q (so3), J. Math. Phys., 42(1) (2001), 472-500.
  • M. Havlicek and S. Posta, Center of quantum algebra U'q (so3), J. Math. Phys., 52(4) (2011), 943521 (15 pp).
  • L. Hellstrom and S. Silvestrov, Commuting elements in $q$-deformed Heisenberg algebras, World Scient Publ Co., River Edge, NJ, 2000.
  • L. Hellstrom and S. Silvestrov, Two-sided ideals in $q$-deformed Heisenberg algebras, Expo. Math., 23(2) (2005), 99-125.
  • N. Iorgov, On the center of the $q$-deformed algebra U'q (so3) related to quantum gravity at $q$ a root of $1$, Proceedings of Institute of Mathematics of NAS of Ukraine, 43(2) (2002), 449-455.
  • T. Ito, P. Terwilliger and C. Weng, The quantum algebra Uq (sl2) and its equitable presentation, J. Algebra, 298(1) (2006), 284-301.
  • A. Odesskii, An analog of the Sklyanin algebra, Funct. Anal. Appl., 20 (1986), 152-154.
  • P. Terwilliger, The universal Askey-Wilson algebra, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 069 (24 pp).
  • A. Zhedanov, "Hidden symmetry" of the Askey-Wilson polynomials, Theoret. and Math. Phys., 89(2) (1991), 1146-1157.
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Rafael Reno S. Cantuba This is me

Publication Date July 17, 2021
Published in Issue Year 2021

Cite

APA Cantuba, R. R. S. (2021). A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL. International Electronic Journal of Algebra, 30(30), 1-15. https://doi.org/10.24330/ieja.969570
AMA Cantuba RRS. A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL. IEJA. July 2021;30(30):1-15. doi:10.24330/ieja.969570
Chicago Cantuba, Rafael Reno S. “A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL”. International Electronic Journal of Algebra 30, no. 30 (July 2021): 1-15. https://doi.org/10.24330/ieja.969570.
EndNote Cantuba RRS (July 1, 2021) A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL. International Electronic Journal of Algebra 30 30 1–15.
IEEE R. R. S. Cantuba, “A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL”, IEJA, vol. 30, no. 30, pp. 1–15, 2021, doi: 10.24330/ieja.969570.
ISNAD Cantuba, Rafael Reno S. “A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL”. International Electronic Journal of Algebra 30/30 (July 2021), 1-15. https://doi.org/10.24330/ieja.969570.
JAMA Cantuba RRS. A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL. IEJA. 2021;30:1–15.
MLA Cantuba, Rafael Reno S. “A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL”. International Electronic Journal of Algebra, vol. 30, no. 30, 2021, pp. 1-15, doi:10.24330/ieja.969570.
Vancouver Cantuba RRS. A CASIMIR ELEMENT INEXPRESSIBLE AS A LIE POLYNOMIAL. IEJA. 2021;30(30):1-15.