查看“︁楊代數”︁的源代码

在[[表示论]]中，楊代數（或楊子、'''Yangian'''）是一種無限維的[[霍普夫代數|霍普夫代数]]和[[量子群]]。這是以[[楊振寧]]命名的。俄羅斯物理學家[[弗拉基米爾·德林費爾德]]和[[路德維希·法捷耶夫]]首先研究了楊子。

== 定義 ==
若a是[[半單李代數]]，Y(a)是无限维的[[霍普夫代數|霍普夫代数]]（跟[[泛包絡代數]]有關係）。Y(a)是楊代數。

=== 有些方程 ===
<math> [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)}).</math>

若<math>t_{ij}^{(-1)}=\delta_{ij}</math>

<math> T(z) = \sum_{p\ge -1} t_{ij}^{(p)} z^{-p+1}</math>

若[[R-矩阵]]是 ''R''(''z'')= I+ ''z''<sup>-1</sup> ''P'' 

<math>\displaystyle{ R_{12}(z-w) T_{1}(z)T_{2}(w) = T_{2}(w) T_{1}(z) R_{12}(z-w).}</math>

<math> (\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}.</math>

<math>\displaystyle{ S(z)=T(z)\sigma T(-z),}</math>

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

== 應用 ==

* [[楊-巴克斯特方程|杨–巴克斯特方程]]
* [[N=4的超级杨-米尔斯|超对称杨–米尔斯理论]]<ref>Beisert, N. (2007). The S-matrix of AdS/CFT and Yangian symmetry. arXiv preprint arXiv:0704.0400.</ref><ref>Spill, F. (2009). Weakly coupled N= 4 Super Yang-Mills and N= 6 Chern-Simons theories from u (2| 2) Yangian symmetry. Journal of High Energy Physics, 2009(03), 014, https://arxiv.org/abs/0810.3897 {{Wayback|url=https://arxiv.org/abs/0810.3897 |date=20200223203602 }}</ref> 

== 閱讀 ==

* {{Cite book|last=Chari|first=Vyjayanthi|authorlink=Vyjayanthi Chari|last2=Andrew Pressley|year=1994|title=A Guide to Quantum Groups|url=https://archive.org/details/guidetoquantumgr0000char|publisher=[[Cambridge University Press]]|location=Cambridge, U.K.|isbn=0-521-55884-0}}
* {{Cite journal|last=Drinfeld|first=Vladimir Gershonovich|authorlink=Vladimir Drinfeld|journal=[[Doklady Akademii Nauk SSSR]]|issue=5|year=1985|volume=283|pages=1060–1064|language=ru|script-title=ru:Алгебры Хопфа и квантовое уравнение Янга-Бакстера|trans-title=Hopf algebras and the quantum Yang–Baxter equation}}
* {{Cite journal|title=A new realization of Yangians and of quantum affine algebras|last=Drinfeld|first=V. G.|authorlink=Vladimir Drinfeld|journal=Doklady Akademii Nauk SSSR|issue=1|year=1987|volume=296|pages=13–17|language=ru}} 翻译在 {{Cite journal|journal=Soviet Mathematics - Doklady|issue=2|year=1988|volume=36|pages=212–216}}
* {{Cite journal|url=http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1254&volume=20&year=1986&issue=1&fpage=69&what=fullt&option_lang=eng|last=Drinfeld|first=V. G.|journal=Funktsional'nyi Analiz I Ego Prilozheniya|issue=1|year=1986|volume=20|pages=69–70|language=ru|script-title=ru:Вырожденные аффинные алгебры Гекке и янгианы|trans-title=Degenerate affine Hecke algebras and Yangians|mr=831053|zbl=0599.20049}} 翻译在 {{Cite journal|title=Degenerate affine hecke algebras and Yangians|last=Drinfeld|first=V. G.|journal=Functional Analysis and Its Applications|issue=1|doi=10.1007/BF01077318|year=1986|volume=20|pages=58–60}}
* {{Cite book|last=Molev|first=Alexander Ivanovich|authorlink=Alexander Molev|year=2007|title=Yangians and Classical Lie Algebras|series=Mathematical Surveys and Monographs|publisher=[[American Mathematical Society]]|location=Providence, RI|isbn=978-0-8218-4374-1}}
* {{Cite journal|title=An Introduction to Yangian Symmetries|last=Bernard|first=Denis|authorlink=Denis Bernard (mathematician)|journal=[[NATO ASI Series]]|issue=5|doi=10.1007/978-1-4899-1516-0_4|year=1993|volume=310|pages=39–52|arxiv=hep-th/9211133|isbn=978-1-4899-1518-4}}
* {{Cite journal|title=Introduction to Yangian Symmetry in Integrable Field Theory|last=MacKay|first=Niall|authorlink=Niall MacKay|journal=[[International Journal of Modern Physics A]]|issue=30|doi=10.1142/s0217751x05022317|year=2005|volume=20|pages=7189–7217|arxiv=hep-th/0409183|bibcode=2005IJMPA..20.7189M}}
* {{Cite journal|title=Yangian Symmetry of Scattering Amplitudes in N = 4 super Yang-Mills Theory|last=Drummond|first=James|last2=Henn|first2=Johannes|journal=[[Journal of High Energy Physics]]|issue=5|doi=10.1088/1126-6708/2009/05/046|year=2009|volume=2009|pages=046|arxiv=0902.2987|bibcode=2009JHEP...05..046D|last3=Plefka|first3=Jan}}

== 參考文獻 ==
{{Reflist}}

[[Category:表示論]]
[[Category:量子場論]]
[[Category:楊振寧]]