Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 055, 5 pages      math.RT/0605717      https://doi.org/10.3842/SIGMA.2006.055

On the Existence of Configurations of Subspaces in a Hilbert Space with Fixed Angles

Natasha D. Popova and Yurii S. Samoilenko
Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs'ka Str., Kyiv-4, 01601 Ukraine

Received December 01, 2005, in final form April 30, 2006; Published online May 29, 2006

Abstract
For a class of *-algebras, where *-algebra AΓ,τ is generated by projections associated with vertices of graph Γ and depends on a parameter τ (0 < τ ≤ 1), we study the sets ΣΓ of values of τ such that the algebras AΓ,τ have nontrivial *-representations, by using the theory of spectra of graphs. In other words, we study such values of τ that the corresponding configurations of subspaces in a Hilbert space exist.

Key words: representations of *-algebras; Temperley-Lieb algebras.

pdf (179 kb)   ps (141 kb)   tex (8 kb)

References

  1. Cvetkovic D.M., Doob M., Sachs H., Spectra of graphs. Theory and applications, Berlin, VEB Deutscher Verlag der Wissenschaften, 1980.
  2. Evans D.E., Kawahigashi Y., Quantum symmetries on operator algebras, Oxford University Press, 1998.
  3. Fan C.K., Green R.M., On the affine Temperley-Lieb algebras, J. London Math. Soc. (2), 1999, V.60, N 2, 366-380.
  4. Ostrovskyi V.L., Samolenko Yu.S., Introduction to the theory of representations of finitely presented *-algebras. I. Representations by bounded operators, Harwood Acad. Publ., 1999.
  5. Popova N., On the algebra of Temperley-Lieb type, in Proceedings of Fourth International Conference "Symmetry in Nonlinear Mathematical Physics" (July 9-15, 2001, Kyiv), Editors A.G. Nikitin, V.M. Boyko and R.O. Popovych, Proceedings of Institute of Mathematics, Kyiv, 2002, V.43, Part 2, 486-489.
  6. Temperley H.N.V., Lieb E.H., Relations between "percolations" and "colouring" problems and other graph theoretical problems associated with regular planar lattices: some exact results for the percolation problem, Proc. Roy. Soc. London Ser. A, 1971, V.322, N 1549, 251-280.
  7. Vlasenko M., On the growth of an algebra generated by a system of projections with fixed angles, Methods Funct. Anal. Topology, 2004, V.10, N 1, 98-104.
  8. Vlasenko M., Popova N., On configurations of subspaces of Hilbert space with fixed angles between them, Ukrain. Mat. Zh., 2004, V.56, N 5, 606-615 (English transl.: Ukrainian Math. J., 2004, V.56, N 5, 730-740).
  9. Wenzl H., On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada, 1987, V.9, N 1, 5-9.

Previous article   Next article   Contents of Volume 2 (2006)