I am trying to understand the dual Lie algebra in a Lie bialgebra.

In the above article, it is said that:

"Let ${\displaystyle {\mathfrak {g}}}$ be any semisimple Lie algebra. To specify a Lie bialgebra structure we thus need to specify a compatible Lie algebra structure on the dual vector space. Choose a Cartan subalgebra ${\displaystyle {\mathfrak {t}}\subset {\mathfrak {g}}}$ and a choice of positive roots. Let ${\displaystyle {\mathfrak {b}}_{\pm }\subset {\mathfrak {g}}}$ be the corresponding opposite Borel subalgebras, so that ${\displaystyle {\mathfrak {t}}={\mathfrak {b}}_{-}\cap {\mathfrak {b}}_{+}}$ and there is a natural projection ${\displaystyle \pi :{\mathfrak {b}}_{\pm }\to {\mathfrak {t}}}$. Then define a Lie algebra $$ {\displaystyle {\mathfrak {g'}}:=\{(X_{-},X_{+})\in {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}\ {\bigl \vert }\ \pi (X_{-})+\pi (X_{+})=0\}}$$ which is a subalgebra of the product ${\displaystyle {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}}$, and has the same dimension as ${\displaystyle {\mathfrak {g}}}$. Now identify ${\displaystyle {\mathfrak {g'}}}$ with dual of ${\displaystyle {\mathfrak {g}}}$ via the pairing $$ {\displaystyle \langle (X_{-},X_{+}),Y\rangle :=K(X_{+}-X_{-},Y)}$$ where ${\displaystyle Y\in {\mathfrak {g}}}$ and $K$ is the Killing form. This defines a Lie bialgebra structure on ${\displaystyle {\mathfrak {g}}}$, and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group. "

Where does the condition $\pi(X_-)+\pi(X_+)=0$ come from? Thank you very much.

  • $\begingroup$ 瞎猜的, 如果定义换成 $$\mathfrak {g'}:=\{(X_{-},X_{+})\in {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}\ {\bigl \vert }\ \pi (X_{-})-\pi (X_{+})=0\}$$和$$\langle (X_{-},X_{+}),Y\rangle :=K(X_{+}+X_{-},Y),$$是不是还是成立?所以这个条件只是用来限制维数? $\endgroup$ – Nirvanacs Dec 12 '17 at 0:27
  • $\begingroup$ @Nirvanacs, thank you very much for your comments. $\endgroup$ – LJR Dec 12 '17 at 1:03

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