Skip to main content
SpringerLink
Log in

Quantum dissipative systems. IV. Analogues of Lie algebras and groups

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

The condition of self-consistency for the quantum description of dissipative systems makes it necessary to exceed the limits of Lie algebras and groups, i.e., this requires the application of non-Lie algebras (in which the Jacobi identity does not hold) and analytic quasi-groups (which are nonassociative generalizations of groups). In the present paper, we show that the analogues of Lie algebras and groups for quantum dissipative systems are commutant-Lie algebras (anticommutative algebras whose communtants are Lie subalgebras) and analytic commutant-associative loops (whose commutants are associative subloops (groups)). It is proved that the tangent algebra of an analytic commutant-associative loop with invertibility (a Valya loop) is a commutant-Lie algebra (a Valya algebra). Some examples of commutant-Lie algebras are considered.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. S. Pontrjagin,Topological Groups, Gordon and Breach, New York (1966).

    Google Scholar 

  2. L. A. Skornyakov (ed.),General Algebra [in Russian], Vol. 1, Nauka, Moscow (1990).

    MATH  Google Scholar 

  3. V. E. Tarasov,Theor. Math. Phys.,110, 57 (1997).

    MATH  MathSciNet  Google Scholar 

  4. V. D. Belousov,Fundamentals in the Theory of Quasigroups and Loops [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  5. A. I. Mal’tsev,Mat. Sb.,36, 569 (1995);Selected Works. Vol. 1. Classical Algebra [in Russian], Nauka, Moscow (1976), pages 340–345.

    MathSciNet  Google Scholar 

  6. K. A. Zhelvakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov,Nearly Circular Rings [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  7. R. D. Shafer,An Introduction to Nonassociative Algebras, Academic Press, London-New York (1966).

    Google Scholar 

  8. V. E. Tarasov,Theor. Math. Phys.,100, 1100–1112 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  9. C. Godbillon,Géomètrie Différentielle et Méchanique Analytique, Hermann, Paris (1963).

    Google Scholar 

  10. M. V. Karasev and V. P. Maslov,Nonlinear Poisson Brackets. Geometry and Quantization, A. Math. Soc., Providence (1993).

    MATH  Google Scholar 

  11. R. G. Cooke,Infinite Matrices and Sequence Spaces, MacMillan, London (1950).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Research Institute for Nuclear Physics, Moscow State University, USSR

    V. E. Tarasov

Authors
  1. V. E. Tarasov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 214–227, February, 1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tarasov, V.E. Quantum dissipative systems. IV. Analogues of Lie algebras and groups. Theor Math Phys 110, 168–178 (1997). https://doi.org/10.1007/BF02630442

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02630442

Keywords

Search

Navigation