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Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds

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Abstract

We suggest a method for describing some types of degenerate orbits of orthogonal and unitary groups in the corresponding Lie algebras as level surfaces of a special collection of polynomial functions. This method allows one to describe orbits of the types SO(2n)/SO(2k)×SO(2)n−k, SO(2n+1)/SO(2k+1)×SO(2)n−k, and (S)U(n)/(S)(U(2k)×U(2)n−k) in so(2n), so(2n+1), and (s)u(n), respectively. In addition, we show that the orbits of minimal dimensions of the groups under consideration can be described in the corresponding algebras as intersections of quadries. In particular, this approach is used for describing the orbit CP n−1⊂u(n).

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References

  1. D. P. Zhelobenko, Compact Lie Groups and Their Representations [in Russian], Nauka, Moscow, (1970).

    MATH  Google Scholar 

  2. N. E. Hurt, Geometric Quantization in Action: Applications of Harmonic Analysis in Quantum Statistical Mechanics and Quantum Field Theory. Reidel, Dordrecht-Boston-London (1983).

    MATH  Google Scholar 

  3. A. A. Kirillov, “Noncommutative harmonic analysis,” in: VINITI Series [in Russian], Vol. 22, VINITI, Moscow (1988), pp. 5–162.

    Google Scholar 

  4. A. Borel, “Kählerian coset spaces of semisimple Lie groups,” Proc. Natl., Acad. Sci. USA, 40, 1147 (1954).

    Article  MATH  MathSciNet  Google Scholar 

  5. F. R. Gantmakher, Matrix Theory [in Russian], Nauka, Moscow, (1967).

    Google Scholar 

  6. E. Montroll, “Lectures on the Ising model,” in: Stability and Phase Transitions [Russian translation], Mir, Moscow (1973), pp. 92–163.

    Google Scholar 

  7. J. Wolf, “Representation associated to minimal coadjoint orbits,” Lect. Notes Math., 676, 329 (1978).

    Article  Google Scholar 

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Authors
  1. O. M. Boyars'kyi
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  2. T. V. Skrypnik
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Additional information

Institute for Theoretical Physics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7. pp. 895–905, July, 1997.

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Boyars'kyi, O.M., Skrypnik, T.V. Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds. Ukr Math J 49, 1003–1015 (1997). https://doi.org/10.1007/BF02528745

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  • DOI: https://doi.org/10.1007/BF02528745

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