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Ukrainian Mathematical Journal

, Volume 62, Issue 8, pp 1213–1233 | Cite as

Regular orthoscalar representations of the extended Dynkin graph \( {\tilde{E}_8} \) AND ∗-algebra associatedwith it

  • S. A. Kruhlyak
  • I. V. Livins’kyi
Article

We obtain a classification of regular orthoscalar representations of the extended Dynkin graph \( {\tilde{E}_8} \) with special character. Using this classification, we describe triples of self-adjoint operators A, B, and C such that their spectra are contained in the sets {0, 1, 2, 3, 4, 5}, {0, 2, 4}, and {0, 3}, respectively, and the equality A + B + C = 6I is true.

Keywords

Hilbert Space Unitary Transformation Zero Element Imaginary Root Indecomposable Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    V. Ostrovskyi and Yu. Samoilenko, Introduction to the Theory of Representations of Finitely Presented-Algebras. I. Representations by Bounded Operators, Harwood Academic Publishers (1999).Google Scholar
  2. 2.
    S. Albeverio, V. Ostrovskyi, and Yu. Samoilenko, “On functions on graphs and representations of a certain class of ∗-algebras,” J. Algebra, 308, 567–582 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    S. A. Kruglyak, V. I. Rabanovich, and Yu. S. Samoilenko, “On sums of projectors,” Funkts. Anal. Prilozhen., 36, Issue 3, 20–35 (2002).MathSciNetGoogle Scholar
  4. 4.
    S. A. Kruglyak and A. V. Roiter, “Locally scalar representations of graphs in the category of Hilbert spaces,” Funkts. Anal. Prilozhen., 39, Issue 2, 13–30 (2005).MathSciNetGoogle Scholar
  5. 5.
    I. V. Redchuk and A. V. Roiter, “Singular locally scalar representations of quivers in Hilbert spaces and separating functions,” Ukr. Mat. Zh., 56, No. 6, 796–809 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    S. A. Kruglyak, L. A. Nazarova, and A. V. Roiter, “Orthoscalar representations of quivers in the category of Hilbert spaces,” Zap. Nauch. Sem. POMI, 338, 180–199 (2006).zbMATHGoogle Scholar
  7. 7.
    A. V. Roiter, S. A. Kruglyak, and L. A. Nazarova, “Orthoscalar representations of quivers corresponding to extended Dynkin graphs in the category of Hilbert spaces,” Funkts. Anal. Prilozhen., 44, Issue 1, 57–73 (2010).MathSciNetGoogle Scholar
  8. 8.
    A. S. Mellit, “On the case where the sum of three partial maps is equal to zero,” Ukr. Mat. Zh., 55, No. 9, 1277–1283 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    V. L. Ostrovs’kyi, “Representation of an algebra associated with Dynkin graph \( {\tilde{E}_7} \),” Ukr. Mat. Zh., 56, No. 9, 1193–1202 (2004).Google Scholar
  10. 10.
    A. Mellit, Certain Examples of Deformed Preprojective Algebras and Geometry of Their ∗-Representations, arXiv:math/0502055v1 [math.RT].Google Scholar
  11. 11.
    V. L. Ostrovs’kyi and Yu. S. Samoilenko, “On spectral theorems for families of linearly connected self-adjoint operators with given spectra associated with extended Dynkin graphs,” Ukr. Mat. Zh., 58, No 11, 1556–1570 (2006).Google Scholar
  12. 12.
    V. G. Kac, “Infinite root systems, representations of graphs and invariant theory. II,” J. Algebra, 78, 141–162 (1982).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    W. Crawley-Boevey, Lectures on Representations of Quivers, http://www.maths.leeds.uk/pintwe/dintwe/quivlecs.pdf.
  14. 14.
    L. A. Nazarova and A. V. Roiter, “Representations of posets,” Zap. Nauch. Sem. LOMI, 28, 5–31 (1972).MathSciNetGoogle Scholar
  15. 15.
    S. A. Kruglyak, S. V. Popovych, and Yu. S. Samoilenko, “Representations of ∗-algebras associated with Dynkin graphs and Horn’s problem,” Uchen. Zap. Tavr. Nats. Univ., 16(55), No. 2, 133–139 (2003).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • S. A. Kruhlyak
    • 1
  • I. V. Livins’kyi
    • 2
  1. 1.Institute of Mathematics, Ukrainian National Academy of SciencesKyivUkraine
  2. 2.Shevchenko Kyiv National UniversityKyivUkraine

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