杨代数

在表示论中，杨代数（或杨子、Yangian）是一种无限维的霍普夫代数和量子群。这是以杨振宁命名的。俄罗斯物理学家弗拉基米尔·德林费尔德和路德维希·法捷耶夫首先研究了杨子。

定义

若a是半单李代数，Y(a)是无限维的霍普夫代数（跟泛包络代数有关系）。Y(a)是杨代数。

有些方程

$[t_{ij}^{(p+1)},t_{kl}^{(q)}]-[t_{ij}^{(p)},t_{kl}^{(q+1)}]=-(t_{kj}^{(p)}t_{il}^{(q)}-t_{kj}^{(q)}t_{il}^{(p)}).$

若$t_{ij}^{(-1)}=\delta _{ij}$

$T(z)=\sum _{p\geq -1}t_{ij}^{(p)}z^{-p+1}$

若R-矩阵是 R(z)= I+ z^-1 P

$\displaystyle {R_{12}(z-w)T_{1}(z)T_{2}(w)=T_{2}(w)T_{1}(z)R_{12}(z-w).}$

$(\Delta \otimes \mathrm {id} )T(z)=T_{12}(z)T_{13}(z),\,\,(\varepsilon \otimes \mathrm {id} )T(z)=I,\,\,(s\otimes \mathrm {id} )T(z)=T(z)^{-1}.$

$\displaystyle {S(z)=T(z)\sigma T(-z),}$

$\displaystyle {\sigma (E_{ij})=(-1)^{i+j}E_{2N-j+1,2N-i+1}.}$

应用

• 杨–巴克斯特方程
• 超对称杨–米尔斯理论

阅读

• Chari, Vyjayanthi; Andrew Pressley. A Guide to Quantum Groups. Cambridge, U.K.: Cambridge University Press. 1994. ISBN 0-521-55884-0. 
• Drinfeld, Vladimir Gershonovich. Алгебры Хопфа и квантовое уравнение Янга-Бакстера [Hopf algebras and the quantum Yang–Baxter equation]. Doklady Akademii Nauk SSSR. 1985, 283 (5): 1060–1064 （俄语）. 
• Drinfeld, V. G. A new realization of Yangians and of quantum affine algebras. Doklady Akademii Nauk SSSR. 1987, 296 (1): 13–17 （俄语）.  翻译在 Soviet Mathematics - Doklady. 1988, 36 (2): 212–216. 
• Drinfeld, V. G. Вырожденные аффинные алгебры Гекке и янгианы [Degenerate affine Hecke algebras and Yangians]. Funktsional'nyi Analiz I Ego Prilozheniya. 1986, 20 (1): 69–70. MR 0831053. Zbl 0599.20049 （俄语）.  翻译在 Drinfeld, V. G. Degenerate affine hecke algebras and Yangians. Functional Analysis and Its Applications. 1986, 20 (1): 58–60. doi:10.1007/BF01077318. 
• Molev, Alexander Ivanovich. Yangians and Classical Lie Algebras. Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society. 2007. ISBN 978-0-8218-4374-1. 
• Bernard, Denis. An Introduction to Yangian Symmetries. NATO ASI Series. 1993, 310 (5): 39–52. ISBN 978-1-4899-1518-4. arXiv:hep-th/9211133 . doi:10.1007/978-1-4899-1516-0_4. 
• MacKay, Niall. Introduction to Yangian Symmetry in Integrable Field Theory. International Journal of Modern Physics A. 2005, 20 (30): 7189–7217. Bibcode:2005IJMPA..20.7189M. arXiv:hep-th/0409183 . doi:10.1142/s0217751x05022317. 
• Drummond, James; Henn, Johannes; Plefka, Jan. Yangian Symmetry of Scattering Amplitudes in N = 4 super Yang-Mills Theory. Journal of High Energy Physics. 2009, 2009 (5): 046. Bibcode:2009JHEP...05..046D. arXiv:0902.2987 . doi:10.1088/1126-6708/2009/05/046.