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

SIGMA 9 (2013), 030, 16 pages      arXiv:1212.4462

Pentagon Relations in Direct Sums and Grassmann Algebras

Igor G. Korepanov and Nurlan M. Sadykov
Moscow State University of Instrument Engineering and Computer Sciences, 20 Stromynka Str., Moscow 107996, Russia

Received December 19, 2012, in final form April 05, 2013; Published online April 10, 2013

We construct vast families of orthogonal operators obeying pentagon relation in a direct sum of three n-dimensional vector spaces. As a consequence, we obtain pentagon relations in Grassmann algebras, making a far reaching generalization of exotic Reidemeister torsions.

Key words: Pachner moves; pentagon relations; Grassmann algebras.

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  1. Berezin F.A., The method of second quantization, Pure and Applied Physics, Vol. 24, Academic Press, New York, 1966.
  2. Doliwa A., Sergeev S.M., The pentagon relation and incidence geometry, arXiv:1108.0944.
  3. Gantmacher F.R., The theory of matrices, Vol. II, Chelsea, New York, 1974.
  4. Kashaev R.M., On discrete three-dimensional equations associated with the local Yang-Baxter relation, Lett. Math. Phys. 38 (1996), 389-397, solv-int/9512005.
  5. Kashaev R.M., On matrix generalizations of the dilogarithm, Theoret. and Math. Phys. 118 (1999), 314-318.
  6. Kashaev R.M., Korepanov I.G., Sergeev S.M., The functional tetrahedron equation, Theoret. and Math. Phys. 117 (1998), 1402-1413, solv-int/9801015.
  7. Korepanov I.G., A dynamical system connected with inhomogeneous 6-vertex model, J. Math. Sci. 85 (1997), 1671-1683, hep-th/9402043.
  8. Korepanov I.G., Relations in Grassmann algebra corresponding to three- and four-dimensional Pachner moves, SIGMA 7 (2011), 117, 23 pages, arXiv:1105.0782.
  9. Korepanov I.G., Two deformations of a fermionic solution to pentagon equation, arXiv:1104.3487.
  10. Kuniba A., Sergeev S.M., Tetrahedron equation and quantum R matrices for spin representations of Bn(1), Dn(1) and Dn+1(2), arXiv:1203.6436.
  11. Lickorish W.B.R., Simplicial moves on complexes and manifolds, in Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., Vol. 2, Geom. Topol. Publ., Coventry, 1999, 299-320, math.GT/9911256.
  12. Pachner U., P.L. homeomorphic manifolds are equivalent by elementary shellings, European J. Combin. 12 (1991), 129-145.
  13. Sergeev S.M., Quantization of three-wave equations, J. Phys. A: Math. Theor. 40 (2007), 12709-12724, nlin.SI/0702041.
  14. Sergeev S.M., Quantum curve in q-oscillator model, Int. J. Math. Math. Sci. 2006 (2006), Art. ID 92064, 31 pages, nlin.SI/0510048.
  15. Takagi T., On an algebraic problem related to an analytic theorem of Carathéodory and Fejér and on an allied theorem of Landau, Japan J. Math. 1 (1924), 83-93.
  16. Takagi T., Remarks on an algebraic problem, Japan J. Math. 2 (1925), 13-17.

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