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Journal of Zhejiang University SCIENCE A 2005 Vol.6 No.10 P.1065-1079

http://doi.org/10.1631/jzus.2005.A1065


Quantum Yang-Baxter equation and constant R-matrix over Grassmann algebra


Author(s):  DUPLIJ Steven, KOTULSKA Olga, SADOVNIKOV Alexander

Affiliation(s):  Department of Physics and Technology, V.N. Karazin Kharkov National University, Svoboda Sq. 4, Kharkov 61077, Ukraine

Corresponding email(s):   Steven.A.Duplij@univer.kharkov.ua

Key Words:  Constant solution, Grassmann algebra, Regularity, R-matrix


DUPLIJ Steven, KOTULSKA Olga, SADOVNIKOV Alexander. Quantum Yang-Baxter equation and constant R-matrix over Grassmann algebra[J]. Journal of Zhejiang University Science A, 2005, 6(10): 1065-1079.

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PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2005.A1065


Abstract: 
constant solutions to Yang-Baxter equation are investigated over grassmann algebra for the case of 6-vertex R-matrix. The general classification of all possible solutions over grassmann algebra and particular cases with 2,3,4 generators are studied. As distinct from the standard case, when R-matrix over number field can have a maximum 5 nonvanishing elements, we obtain over grassmann algebra a set of new full 6-vertex solutions. The solutions leading to regular R-matrices which appear in weak Hopf algebras are considered.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

[1] Baxter, R.J., 1972. Partition function for the eight-vertex model. Ann. Phys., 70:193-228.

[2] Baxter, R.J., 1982. Exactly Solved Models in Statistical Mechanics. Academic Press, London.

[3] Berezin, F.A., 1983. Vvedenije v Algebru i Analiz s Antikommutirujushimi Peremennymi. Izd-vo MGU, Moskva, p.208.

[4] Brylinski, J.L., Brylinski, R., 1994. Universal Quantum Gates. Mathematics of Quantum Computation. Chapman & Hall/CRC Press, Boca Raton, p.124-134.

[5] Carter, J.S., Saito, M., 1996. On formulation and solutions of simplex equations. J. Mod. Phys., A11:4453-4463.

[6] Chang, D., Phillips, I., Rozansky, I., 1992. R-matrix approach to quantum superalgebras. J. Math. Phys., 33(11):3710-3715.

[7] Chari, V., Pressley, A., 1996. A Guide to Quantum Groups. Cambridge University Press, Cambridge.

[8] Clifford, A., Preston, G., 1972. Algerbraic Theory of SemiGroups. Mir, Moskva, p.283.

[9] De Witt, B.S., 1992. Supermanifolds. 2nd Edition. Cambridge University Press, Cambridge, p.407.

[10] Demidov, E.E., 1998. Quantum Groups. Faktorial, Moskva, p.146.

[11] Di Francesco, P., Mathieu, P., Sénéchal, D., 1997. Conformal Field Theory. Springer-Verlag, Berlin, p.890.

[12] Drinfeld, V.G., 1987. Quantum Groups. Proceedings of the ICM. Phode Island. AMS, Berkeley, p.798-820.

[13] Drinfeld, V.G., 1992. On some unsolved problems in quantum group theory. Lect. Notes. Math., 1510:1-8.

[14] Duplij, S.A., 2000. Semisupermanifolds and Semigroups. Krok, Kharkov, p.220.

[15] Duplij, S.A., 2003. On supermatrix idempotent operator semigroups. Linear Algebra Appl., 360:59-81.

[16] Duplij, S.A., Li, F., 2001a. On regular solutions of quantum Yang-Baxter equation and weak Hopf algebras. Journal of Kharkov National University, ser. Nuclei, Particles and Fields, 521(2):15-30.

[17] Duplij, S.A., Li, F., 2001b. Regular solutions of quantum Yang-Baxter equation from weak Hopf algebras. Czech. J. Phys., 51(12):1306-1311.

[18] Duplij, S.A., Kotulskaja, O.I., 2002. Supermatrix structures and generalized inverses. Journal of Kharkov National University, ser. Nuclei, Particles and Fields, 548(1):3-14.

[19] Duplij, S.A., Sadovnikov, A.S., 2002. Regular supermatrix solutions for quantum Yang-Baxter equation. Journal of Kharkov National University, ser. Nuclei, Particles and Fields, 569(3):15-22.

[20] Duplij, S.A., Kalashnikov, V.V., Maslov, E.A., 2005. Quantum information, qubits and quantum algorithms. Journal of Kharkov National University, ser. Nuclei, Particles and Fields, 657(1):99-104.

[21] Dye, H.A., 2003. Unitary solutions to the Yang-Baxter equation in dimension four. Quantum Information Processing, 2:117-150.

[22] Etingof, P., Schedler, T., Soloviev, A., 1997. On Set-theoretical Solutions to the Quantum Yang-Baxter Equation. Cambridge, p.4 (Preprint /MIT, q-alg/9707027).

[23] Etingof, P., Schedler, T., Soloviev, A., 1999. Set-theoretical solutions to the quantum Yang-Baxter equation. Duke Math. J., (2):169-209.

[24] Faddeev, L.D., Reshetikhin, N.Y., Takhtajan, L.A., 1990. Quantum Lie groups and Lie algebras. Leningrad Math. J., 1:193-236.

[25] Gu, P., 1997. A set-theoretical solution of the Yang-Baxter equation and “metahomomorphisms” of groups. Chinese Sci. Bull, 42(15):1602-1606.

[26] Hietarinta, J., 1992. All solutions to the constant quantum Yang-Baxter equation in two dimensions. Phys. Lett., A165:245-251.

[27] Hietarinta, J., 1993. Solving the two-dimensional constant quantum Yang-Baxter equation. J. Math. Phys., 34:1725-1756.

[28] Hietarinta, J., 1997. Permutation-type solutions to the Yang-Baxter and other n-simplex equations. J. Phys., A30:4757-4771.

[29] Holievo, A.S., 2002. Introduction to Quantum Information Theory. MCNMO, Moskva.

[30] Kac, V.G., 1977. Lie superalgebras. Adv. Math., 26(1):8-96.

[31] Kassel, C., 1995. Quantum Groups. Springer-Verlag, New York, p.531.

[32] Kauffman, L.H., 1991. Knots and Physics. World Sci., Singapore.

[33] Kauffman, L.H., Lomonaco, S.J., 2004. Braiding operators are universal quantum gates. New J. Phys., 6:134-139.

[34] Khoroshkin, S.M., Tolstoy, V.N., 1991. Universal R-matrix for quantized (super) algebras. Comm. Math. Phys., 141(3):599-617.

[35] Kitajev, A., Shen, A., Vialyj, M., 1999. Klassi-cheskije i Kvantovyje Vychislienija. Mcnmo, Moskva.

[36] Lambe, L.A., Radford, D.E., 1997. Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: an Algebraic Approach. Kluwer, Dordrecht, p.292.

[37] Leites, D.A., 1980. Vvedenije v teoriju supermnogoobrazij. Uspiehi Mat, Nauk, 35(1):3-57.

[38] Leites, D.A., 1983. A Theory of Supermanifolds. Karelskij filial AN SSSR, Petrozavodsk, p.199.

[39] Li, F., Duplij, S., 2002. Weak Hopf algebras and singular solutions of quantum Yang-Baxter equation. Commun. Math. Phys., 225(1):191-217.

[40] Links, J., Scheunert, M., Gould, M.D., 1994. Diagonalization of the braid generator on unitary irreps of quantum supergroups. Lett. Math. Phys., 32(3):231-240.

[41] Lu, J.H., Yan, M., Zhu, Y.C., 2000. On set-theoretical Yang-Baxter equation. Duke. Math. J., 104(1):1-18.

[42] Majid, S., 1995. Foundations of Quantum Group Theory. Cambridge University Press, Cambridge.

[43] Manin, U.I., 1984. Kalibrovochnyje Polia I Kompleksnaja Geometrija. Nauka, Moskva, p.335.

[44] Manin, Y., 1989. Multiparametric quantum deformation of the general linear supergroup. Comm. Math. Phys., 123:123-135.

[45] Penrose, R., 1955. A generalized inverse for matrices. Math. Proc. Cambridge Phil. Soc., 51:406-413.

[46] Rabin, J.M., 1987. Supermanifolds and Super Riemann Surfaces. Super Field Theories. Plenum Press, New York, p.557-569.

[47] Rabin, J.M., 1991. Status of the Algebraic Approach to Super Riemann Surfaces. Physics and Geometry. Plenum Press, New York, p.653-668.

[48] Rao, C.R., Mitra, S.K., 1971. Generalized Inverse of Matrices and Its Application. Wiley, New York, p.251.

[49] Rogers, A., 1980. A global theory of supermanifolds. J. Math. Phys., 21(5):1352-1365.

[50] Shnider, S., Sternberg, S., 1993. Quantum Groups. International Press, Boston, p.371.

[51] Soloviev, A., 2002. Non-unitary set-theoretical solutions to the quantum Yang-Baxter equation. Math. Res. Lett., 7:577-596.

[52] Turaev, V.G., 1994. Quantum Invariants of Knots and 3-Manifolds. W. de Greuter, Berlin.

[53] Yang, C.N., 1967. Some exact results for the many-body problem in one dimension with repulse delta-function interaction. Phys. Rev. Lett., 19:1312-1315.

[54] Zhang, R.B., 1991. Graded representations of the Temperley-Lieb algebra, quantum supergroups, and the Jones polynomial. J. Math. Phys., 32(10):2605-2613.

[55] Zhang, R.B., Gould, M.D., 1991. Universal R-matrices and invariants of quantum supergroups. J. Math. Phys., 32(12):3261-3267.

[56] Zhang, Y., Kauffman, L.H., Ge, M.L., 2005. Yang-Baxterization, Universal Quantum Gate, and Hamiltonians. Chicago, p.36 (Preprint/Univ. Illinois, quant-ph/0502015).

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