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Yang-Baxter 方程式と関連した話題

Yang-Baxter , C.N. Yang R.J. Baxter にちなんで けられたもので あるが , Perk Au-Yang による Encyclopedia of Mathematical Physics [ FNT06 ] [ PAY06 ] によると , その 1970 Faddeev により 導入 されたものら しい

Jimbo による [ Jim89a ] Yang-Baxter する reprint めた もの [ Jim89b ] もある

なところに , えば , この website した げると 以下 のようになる

えられている えば , 以下 のようなものがある

  • quantum Yang-Baxter equation
  • dynamical Yang-Baxter equation
  • generalized Yang-Baxter equation (Rowell, Zhang, Wu, Ge [ RZWG10 ] )
  • Hom -Yang-Baxter equation (Yau [ Yau09 ] )

これらの monoidal category していて えば , Donin Mudrov [ DM05 ] dynamical Yang-Baxter equation から dynamical category という している Kitaev Wang [ KW12 ] によると , generalized Yang-Baxter equation ribbon fusion category あるようで ある

, した としては , rack brace とい たものがある

また , Yang-Baxter equation simplex equation という した つでもある Yang-Baxter equation Zamolodchikov tetrahedral equation である

  • tetrahedral equation
  • simplex equation

Dimakis Müller-Hoissen [ DMH15 ] では Bazhanov Stroganov [ BS82 ] されている にも Frenkel Moore [ FM91 ] Maillet Nijhoff [ MN89 ] などを している

References

[BS82]     V. V. Bazhanov and Yu. G. Stroganov. Commutativity conditions for transfer matrices on a multidimensional lattice. Teoret. Mat. Fiz. , 52(1):105–113, 1982.

[DM05]     J. Donin and A. Mudrov. Dynamical Yang-Baxter equation and quantum vector bundles. Comm. Math. Phys. , 254(3):719–760, 2005, arXiv:math/0306028 .

[DMH15]     Aristophanes Dimakis and Folkert Müller-Hoissen. Simplex and polygon equations. SIGMA Symmetry Integrability Geom. Methods Appl. , 11:Paper 042, 49, 2015, arXiv:1409.7855 .

[FM91]     Igor Frenkel and Gregory Moore. Simplex equations and their solutions. Comm. Math. Phys. , 138(2):259–271, 1991, http://projecteuclid.org/euclid.cmp/1104202944 .

[FNT06]     Jean-Pierre Françoise, Gregory L. Naber, and Tsou Sheung Tsun, editors. Encyclopedia of mathematical physics. Vol. 1, 2, 3, 4, 5 . Academic Press/Elsevier Science, Oxford, 2006.

[Jim89a]     Michio Jimbo. Introduction to the Yang-Baxter equation. In Braid group, knot theory and statistical mechanics , volume 9 of Adv. Ser. Math. Phys. , pages 111–134. World Sci. Publ., Teaneck, NJ, 1989, http://dx.doi.org/10.1142/9789812798350_0005 .

[Jim89b]     Michio Jimbo, editor. Yang-Baxter equation in integrable systems , volume 10 of Advanced Series in Mathematical Physics . World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.

[KW12]     Alexei Kitaev and Zhenghan Wang. Solutions to generalized Yang-Baxter equations via ribbon fusion categories. In Proceedings of the Freedman Fest , volume 18 of Geom. Topol. Monogr. , pages 191–197. Geom. Topol. Publ., Coventry, 2012, arXiv:1203.1063 .

[MN89]     Jean-Michel Maillet and Frank Nijhoff. On the algebraic structure of integrable systems in multidimensions. In XVIIth International Colloquium on Group Theoretical Methods in Physics (Sainte-Ad èle, PQ, 1988) , pages 504–507. World Sci. Publ., Teaneck, NJ, 1989.

[PAY06]     Jacques H. H. Perk and Helen Au-Yang. Yang-Baxter Equations. Encycl. Math. Phys. , 5:465–473, 2006, math-ph/0606053.

[RZWG10]     Eric C. Rowell, Yong Zhang, Yong-Shi Wu, and Mo-Lin Ge. Extraspecial two-groups, generalized Yang-Baxter equations and braiding quantum gates. Quantum Inf. Comput. , 10(7-8):685–702, 2010, arXiv:0706.1761 .

[Yau09]     Donald Yau. The Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras. J. Phys. A , 42(16):165202, 12, 2009, arXiv:0903.0585 .