Geodesic
Pentagon Relations in Direct Sums and Grassmann Algebras
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Pachner moves; pentagon relations; Grassmann algebras. 
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     author = {Igor G. Korepanov and Nurlan M. Sadykov},
     title = {Pentagon {Relations} in {Direct} {Sums} and {Grassmann} {Algebras}},
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Igor G. Korepanov; Nurlan M. Sadykov. Pentagon Relations in Direct Sums and Grassmann Algebras. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a29/

[1] Berezin F. A., The method of second quantization, Pure and Applied Physics, 24, Academic Press, New York, 1966 | MR | Zbl

[2] Doliwa A., Sergeev S. M., The pentagon relation and incidence geometry, arXiv: 1108.0944

[3] Gantmacher F. R., The theory of matrices, v. 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, arXiv: solv-int/9512005 | DOI | MR | Zbl

[5] Kashaev R. M., “On matrix generalizations of the dilogarithm”, Theoret. and Math. Phys., 118 (1999), 314–318 | DOI | MR | Zbl

[6] Kashaev R. M., Korepanov I. G., Sergeev S. M., “The functional tetrahedron equation”, Theoret. and Math. Phys., 117 (1998), 1402–1413, arXiv: solv-int/9801015 | DOI | MR | Zbl

[7] Korepanov I. G., “A dynamical system connected with inhomogeneous $6$-vertex model”, J. Math. Sci., 85 (1997), 1671–1683, arXiv: hep-th/9402043 | DOI | MR

[8] Korepanov I. G., “Relations in {G}rassmann algebra corresponding to three- and four-dimensional Pachner moves”, SIGMA, 7 (2011), 117, 23 pp., arXiv: 1105.0782 | DOI | MR | Zbl

[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 $B^{(1)}_n$, $D^{(1)}_n$ and $D^{(2)}_{n+ 1}$”, arXiv: 1203.6436

[11] Lickorish W. B.R., Simplicial moves on complexes and manifolds, Proceedings of the Kirbyfest (Berkeley, CA, 1998), Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999, 299–320, arXiv: math.GT/9911256 | DOI | MR | Zbl

[12] Pachner U., “P.L. homeomorphic manifolds are equivalent by elementary shellings”, European J. Combin., 12 (1991), 129–145 | MR | Zbl

[13] Sergeev S. M., “Quantization of three-wave equations”, J. Phys. A: Math. Theor., 40 (2007), 12709–12724, arXiv: nlin.SI/0702041 | DOI | MR | Zbl

[14] Sergeev S. M., “Quantum curve in $q$-oscillator model”, Int. J. Math. Math. Sci., 2006 (2006), 92064, 31 pp., arXiv: nlin.SI/0510048 | DOI | MR | Zbl

[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 | Zbl

[16] Takagi T., “Remarks on an algebraic problem”, Japan J. Math., 2 (1925), 13–17 | Zbl