# 楊代數

## 定義

• ${\displaystyle {\mathfrak {g}}_{0}}$係單李代數
• ${\displaystyle {\mathfrak {g}}={\mathfrak {g}}_{0}((t^{-1}))}$ 係Laurent級數空間，其中t係變量
• ${\displaystyle {\mathfrak {g}}_{+}={\mathfrak {g}}_{0}[t]}$
• ${\displaystyle {\mathfrak {g}}_{-}=t^{-1}{\mathfrak {g}}_{0}[[t^{-1}]]}$ 係級數空間
• ＜ , ＞ : = ${\displaystyle Res_{t=0}(a(t),b(t))}$${\displaystyle {\mathfrak {g}}}$上嘅雙線性配對

• ${\displaystyle ({\mathfrak {g}},{\mathfrak {g}}_{+},{\mathfrak {g}}_{-})}$形成一Manin三元組
• 其相應嘅李雙代數就叫Yangian

## 註

1. Beisert, Niklas (2007-08-08). "The S-Matrix of AdS/CFT and Yangian Symmetry". Proceedings of BETHE ANSATZ: 75 YEARS LATER — PoS(Solvay). Trieste, Italy: Sissa Medialab. doi:10.22323/1.038.0002.
2. Spill, Fabian (2009-03-03). "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): 014–014. doi:10.1088/1126-6708/2009/03/014. ISSN 1029-8479.

## 參攷

• Etingof/Schiffmann : "Lectures on Quantum Groups", ISBN 1-57146-063-2 pp.51-53
• V.G.Drinfeld :"Hopf algebras and the quantum Yang-Baxter equation", Soviet Mathematical Doklady, vol. 32 (1985), No.1, p.255