Journal of Generalized Lie Theory and Applications

Matrix Bosonic realizations of a Lie colour algebra with three generators and five relations of Heisenberg Lie type

Gunnar Sigurdsson and Sergei D. Silvestrov

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Abstract

We describe realizations of a Lie colour algebra with three generators and five relations by matrices of power series in noncommuting indeterminates satisfying Heisenberg's canonical commutation relation of quantum mechanics. The obtained formulas are used to construct new operator representations of this Lie colour algebra using canonical representation of the Heisenberg commutation relation and creation and annihilation operators of the quantum mechanical harmonic oscillator.

Article information

Source
J. Gen. Lie Theory Appl., Volume 3, Number 4 (2009), 329-340.

Dates
First available in Project Euclid: 6 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.jglta/1281106600

Digital Object Identifier
doi:10.4303/jglta/S090406

Mathematical Reviews number (MathSciNet)
MR2602995

Zentralblatt MATH identifier
1235.17025

Subjects
Primary: 17B75: Color Lie (super)algebras 16G99: None of the above, but in this section 16S32: Rings of differential operators [See also 13N10, 32C38] 34K99: None of the above, but in this section 81S05: Canonical quantization, commutation relations and statistics

Keywords
Nonassociative Rings Nonassociative Algebras Color Lie Algebras Color Lie Superalgebras Associative Rings For The Commutative Case Associative Algebras For The Commutative Case Representation Theory Of Rings Rings Of Differential Operators Ordinary Differential Equations Functional-Differential Equations Differential-Difference Equations Quantum Theory General Quantum Mechanics General Problems Of Quantization Canonical Quantization Commutation Relations And Statistics

Citation

Sigurdsson, Gunnar; Silvestrov, Sergei D. Matrix Bosonic realizations of a Lie colour algebra with three generators and five relations of Heisenberg Lie type. J. Gen. Lie Theory Appl. 3 (2009), no. 4, 329--340. doi:10.4303/jglta/S090406. https://projecteuclid.org/euclid.jglta/1281106600


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References

  • V. K. Agrawala. Invariants of generalized Lie algebras. Hadronic J., 4 (1981), 444–496.
  • Y. Bahturin. Basic Structures of Modern Algebra. Math. Appl., 265. Kluwer Academic Publishers, Dordrecht, 1993.
  • Y. A. Bahturin, A. A. Mikhalev, V. M. Petrogradsky, and M. V. Zaicev. Infinite Dimensional Lie Superalgebras. Walter de Gruyter, Berlin, 1992.
  • R. Campoamor-Stursberg and M. Rausch de Traubenberg. Color Lie algebras and Lie algebras of order $F$. J. Gen. Lie Theory Appl., 3 (2009), 113–130.
  • H. S. Green and P. D. Jarvis. Casimir invariants, characteristic identities and Young diagrams for Colour algebras and superalgebras. J. Math. Phys., 24 (1983), 1681–1687.
  • L. Hellström. The diamond lemma for power series algebras. Doctoral Thesis, No. 23, Umeå University, 2002.
  • L. Hellström and S. D. Silvestrov. Commuting Elements in $q$-Deformed Heisenberg Algebras. World Scientific Publishing, New Jersey, 2000.
  • V. G. Kac. Lie Superalgebras. Adv. Math., 26 (1977), 8–96.
  • R. Kleeman. Commutation factors on generalized Lie algebras. J. Math. Phys., 26 (1985), 2405–2412.
  • A. K. Kwasniewski. Clifford- and Grassmann-like algebras – old and new. J. Math. Phys., 26 (1985), 2234–2238.
  • H. Ljungqvist and S. D. Silvestrov. Involutions in three-dimensional coloured Lie algebras. Research Reports 6, Department of Mathematics, Umeå University, 1996.
  • J. Lukierski, V. Rittenberg. Color-de Sitter and color-conformal superalgebras. Phys. Rev. D., 18 (1978), 385–389.
  • W. Marcinek. Colour extensions of Lie algebras and superalgebras. Preprint 746, University of Wroc\law, 1990.
  • W. Marcinek. Generalized Lie algebras and related topics, 1, 2. Acta Univ. Wratislaviensis (Matematyka, Fizyka, Astronomia), 55 (1991), 3–52.
  • M. V. Mosolova. Functions of non-commuting operators that generate a graded Lie algebra. Mat. Zametki, 29 (1981), 34–45.
  • V. L. Ostrovskii and S. D. Silvestrov. Representations of real forms of the graded analogue of a Lie algebra. Ukraïn. Mat. Zh., 44 (1992), 1518–1524 (Russian). Translation in Ukrainian Math. J., 44 (1992), 1395–1401.
  • L. Persson, S. D. Silvestrov, and P. Strunk. Central elements of the second order in three-dimensional generalised Lie algebras. Czech. J. Phys., 47 (1997), 99–106.
  • C. R. Putnam. Commutation Properties of Hilbert Space Operators and Related Topics. Springer-Verlag, New York, 1967.
  • V. Rittenberg and D. Wyler. Generalized Superalgebras. Nuclear Phys. B, 139 (1978), 189–202.
  • M. Scheunert. Generalized Lie algebras. J. Math. Phys., 20 (1979), 712–720.
  • M. Scheunert. Graded tensor calculus. J. Math. Phys., 24 (1983), 2658–2670.
  • F. Schwabl. Quantum Mechanics. 2nd ed. Springer-Verlag, Berlin, 1995.
  • G. Sigurdsson and S. D. Silvestrov. Canonical involutions in three-dimensional generalised Lie algebras. Czech. J. Phys., 50 (2000), 181–186.
  • G. Sigurdsson and S. D. Silvestrov. Bosonic Realizations of the Colour Heisenberg Lie Algebra. J. Nonlinear Math. Phys., 13 (2006), 110–128.
  • S. D. Silvestrov. Hilbert space representations of the graded analogue of the Lie algebra of the group of plane motions. Studia Math., 117 (1996), 195–203.
  • S. D. Silvestrov. Representations of commutation relations. A dynamical systems approach. Hadronic J. Suppl., 11 (1996), 1–116.
  • S. D. Silvestrov. On the classification of $3$-dimensional coloured Lie algebras. In “Quantum Groups and Quantum Spaces”. R. Budzyński, W. Pusz, and S. Zakrzewski, Eds. Banach Center Publ., 40. Polish Acad. Sci., Warsaw, 1997, 159–170.
  • H. Wielandt. Über die Unbeschränktheit der Operatoren der Quantenmechanik. Math. Ann., 121 (1949), 21.
  • A. Wintner. The unboundedness of quantum-mechanical matrices. Phys. Rev. (2), 71 (1947), 738–739. }