Bialgebra structures on Lie algebras with triangular decomposition

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 24 March 2011

Lie algebras with triangular decomposition

1.1 Let $I$ be a finite set. Let $Q=\mathbb{Z}\left[I\right]$, $Q^{+}=\mathbb{N}\left[I\right]$, and $Q^{-}=-Q^{+}$.
Let $𝔤$ be a Lie algebra that satisfies the following:

1. $𝔤$ is $Q$-graded, i.e., $𝔤$ is a direct sum of spaces ${𝔤}_{\alpha },\alpha \in Q$, and $$\left[{𝔤}_{\alpha },{𝔤}_{\beta }\right]\subseteq {𝔤}_{\alpha +\beta }$$ for all $\alpha ,\beta \in Q$.
2. The Lie algebra $𝔤$ has a triangular decomposition with respect to $Q$, i.e., $$𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+}$$ where $${𝔫}^{+}=\underset{\alpha \in Q^{+}\setminus \left\{0\right\}}{⨁}{𝔤}_{\alpha },{𝔫}^{-}=\underset{\alpha \in Q^{-}\setminus \left\{0\right\}}{⨁}{𝔤}_{-\alpha },𝔥={𝔤}_0.$$
3. For each $\alpha \in Q$ the vector space ${𝔤}_{\alpha }$ is finite dimensional.
4. For each $i\in I$, $dim\left({𝔤}_i\right)=1$.
5. The subspaces ${𝔤}_i$, $i\in I$ generate $𝔤$ as a lie algebra.
6. There is a nondegenerate invariant symmetric bilinear form $⟨,⟩:𝔤\hspace{0.17em}\times \hspace{0.17em}𝔤\to k$ on $𝔤$ such that
   1. The restriciton $⟨,⟩:{𝔤}_{\alpha }\hspace{0.17em}\times \hspace{0.17em}{𝔤}_{-\alpha }\to k$ is nondegenerate for each $\alpha \in Q$ such that ${𝔤}_{\alpha }\ne 0$. In particular $⟨,⟩$ is a nondegenerate form on $𝔥$.
   2. If $\alpha ,\beta \in Q$, $\alpha \ne \beta$ and $x\in {𝔤}_{\alpha }$ and $y\in {𝔤}_{\beta }$ then $⟨x,y⟩=0$.
7. There is a linear map $\theta :𝔤\to 𝔤$ called the Chevalley involution, such that
   1. $\theta$ is a Lie algebra automorphism.
   2. $\theta \left(h\right)=-h$ for all $h\in 𝔥$.
   3. $\theta \left({𝔤}_{\alpha }\right)=\theta \left({𝔤}_{-\alpha }\right)$ for all $\alpha \in Q$.
   4. $⟨\theta \left(x\right),\theta \left(y\right)⟩=⟨x,y⟩$ for all $x,y\in 𝔤$.

1.2 For each $i\in i$ let us fix an element $X_i\in g_i$. Let $Y_i\in {𝔤}_{-i}$ be dual to $X_i$ with respect to the form $⟨,⟩$ and let $H_i=⟨X_i,Y_i⟩\in 𝔥$.

1.3 We shall let ${𝔤}^{+}$ and ${𝔟}^{-}$ be the Lie subalgebras of $𝔤$ given by $$\begin{array}{c}{𝔟}^{+}=𝔥\oplus {𝔫}^{+},\\ {𝔟}^{-}=𝔥\oplus {𝔫}^{-}.\end{array}$$

1.4 Given an element $x\in 𝔤$ we write $$x=x^{-}+x_{𝔥}+x^{+},$$ where $x^{-}\in {𝔫}^{-},x_{𝔥}\in 𝔥,x^{+}\in {𝔫}^{+}$.

1.5 The finite dimensional simple Lie algebras over $ℂ$ and the Kac-Moody Lie algebras ${𝔤}^{\prime }\left(A\right)$ in [Kc] are Lie algebras that satisfy the conditions in (1.1). See Theorems 1.2 and 2.2 of [Kc].

Lie bialgebra structure on ${𝔟}^{+}$ and $D\left({𝔟}^{+}\right)$

Let $𝔤$ be a Lie algebra with invariang form ${⟨,⟩}_{𝔤}$ satisfying the conditions of Section 1. Let ${⟨,⟩}_{𝔥}$ denote the restriction of the form ${⟨,⟩}_{𝔤}$ to $𝔥$. Given an element $x\in 𝔤$ we write we write $$x=x^{-}+x_{𝔥}+x^{+},$$ where $x^{-}\in {𝔫}^{-},x_{𝔥}\in 𝔥,x^{+}\in {𝔫}^{+}$. The triple $$\left(𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥,{𝔭}_1,{𝔭}_2\right),\text{where}{𝔭}_1=\left\{\left(x,x_{𝔥}\right)\mid x\in {𝔟}^{+}\right\}\text{and}{𝔭}_2=\left\{\left(y,-y_{𝔥}\right)\mid y\in {𝔟}^{-}\right\}$$ with the scalar product and bracket on $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$ given by $$⟨(x,x_{𝔥}),(y,y_{𝔥})⟩=⟨x,y{⟩}_{𝔤}-⟨x_{𝔥},y_{𝔥}{⟩}_{𝔥},\text{and}\left[(x,x_{𝔥}),(y,y_{𝔥})\right]=\left(\right[x,y],[h_{𝔥},y_{𝔥}\left]\right)=\left(\right[x,y],0)$$ respectively, is a Manin triple.

		Proof:
	We must show that ${𝔭}_1$ and ${𝔭}_2$ are isotropic Lie subalgebras of $𝔤\otimes 𝔥$. $𝔤\otimes 𝔥={𝔭}_1\oplus {𝔭}_2$. The form $⟨,⟩$ on $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$ is invariant. The form $⟨,⟩$ on $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$ is nondegenerate. The proofs of these facts follow in 2.2-2.5. $\square$

2.2 If $x,y\in {𝔟}^{+}$ then $\left[x,y\right]\in {𝔫}^{+}$ and $$\left[\right(x,x_{𝔥}),(y,y_{𝔥}\left)\right]=\left(\right[x,y],0)\in {𝔭}_1$$ So ${𝔭}_1$ is a Lie subalgebra of $𝔤\otimes 𝔥$. Using the fact that $x,y\in {𝔟}^{+}$, so that $⟨x,y⟩=⟨x_{𝔥},y_{𝔥}⟩$, We have that $$⟨(x,x_{𝔥}),(y,y_{𝔥})⟩=0.$$ Thus ${𝔭}_1$ is isotropic.
Similarly one shows taht ${𝔭}_2$ is an isotropic Lie algebra of $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$.

2.3 If $(x,y)\in 𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$ then $$(a,h)=\left(a^{+}+\frac{a_{𝔥}+h}{2},\frac{a_{𝔥}+h}{2}\right)+\left(a^{-}+\frac{a_{𝔥}-h}{2},\frac{-a_{𝔥}+h}{2}\right)\in {𝔭}_1+{𝔭}_2.$$ So ${𝔭}_1+{𝔭}_2=𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$.
If $(a,h)\in {𝔭}_1\cap {𝔭}_2$ then $a=a_{𝔥}$ and $h=a_{𝔥}$ and $h=-a_{𝔥}$. It follows that $a=a_{𝔥}=h=0$. Thus ${𝔭}_1\cap {𝔭}_2=0$ and we have that $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥={𝔭}_1\oplus {𝔭}_2$.

2.4 Recall that for $(x,h),(y,h^{\prime })\in 𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$, $⟨(x,h),(y,h^{\prime })⟩={⟨x,y⟩}_{𝔤}-⟨h,h^{\prime }{⟩}_{𝔥}$. It follows that $$\begin{array}{rcl}⟨\left[\right(x,h),(y,h^{\prime }\left)\right],(z,h^{\prime \prime })⟩& =& ⟨[x,y],z{⟩}_{𝔤}-⟨[h,h^{\prime }],h^{\prime \prime }⟩\\ & =& ⟨y,[z,x]{⟩}_{𝔤}-⟨h^{\prime },[h^{\prime \prime },h]{⟩}_{𝔥}\\ & =& ⟨(y,h^{\prime }),\left(\right[z,x],[h^{\prime \prime },h\left]\right)⟩\\ & =& ⟨(y,h^{\prime }),\left[\right(z,h^{\prime \prime }),(x,h\left)\right]⟩.\end{array}$$ It follows that the form $⟨,⟩$ is an invariant form on $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$.

2.5 Let $x,h)\in 𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$ such that . Since $⟨,{⟩}_{𝔥}$ is nondegenerate, if $h\ne 0$, there exists some $h^{\prime }\in 𝔥$ such that $⟨h,h^{\prime }{⟩}_{𝔥}\ne 0$. Thus if $h\ne 0$, $$⟨(x,h),(0,h^{\prime })⟩=-⟨h,h^{\prime }{⟩}_{𝔥}\ne 0.$$ If $h=0$ then $x\ne 0$, and by the nondegeneracy of $⟨,{⟩}_{𝔤}$ there exists some $y\in 𝔤$ such that $⟨x,y{⟩}_{𝔤}\ne 0$ . Thus, if $h=0$, $$⟨(x,0),(y,0)⟩=⟨x,y{⟩}_{𝔤}\ne 0.$$ Thus $⟨,⟩$ is a nondegenerate form on $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$.

2.6 For each $i\in I$ let us fix an element $X\in {𝔤}_i$. Let $Y_i\in {𝔤}_{-i}$ be dual to $X_i\in {𝔤}_i$ with respect to the form $⟨,⟩$ and let $H_i=[X_i,Y_i]\in 𝔥$. The elements $X_i,Y_i,H_i,i\in I$ generate $𝔤$. The manin triple in (2.1) determines a Lie algebra structure on ${𝔟}^{+}$ with cocommututator $\phi :{𝔤}_A\to {\bigwedge }^2{𝔤}_A$ given by $$\begin{array}{l}\phi \left(H_i\right)=0,\\ \phi \left(X_i\right)=\frac{1}{2}{X}_i\wedge H_i.\end{array}$$

		Proof:
	Thus the cobracket on ${𝔭}_1$ is determined by the equation $$⟨\phi \left(\right(z,z_{𝔥}\left)\right),(y,-y_{𝔥})\otimes (f,-f_{𝔥})⟩=⟨(z,z_{𝔥}),\left[\right(y,-y_{𝔥}),(f,-f_{𝔥}\left)\right]⟩$$ where $y,f\in {𝔟}^{-}$ and $z\in {𝔟}^{+}$. We have $$\begin{array}{rcl}⟨\phi \left(\right(z,z_{𝔥}\left)\right),(y,-y_{𝔥})\otimes (f,-h_{𝔥})⟩& =& ⟨(z,z_{𝔥}),\left[\right(y,-y_{𝔥}),(f,-f_{𝔥}\left)\right]⟩\\ & =& ⟨(z,z_{𝔥}),\left(\right[y,f],[-y_{𝔥},-f_{𝔥}]⟩\\ & =& ⟨z,[y,f]⟩-⟨,z_{𝔥}[-y_{𝔥},-f_{𝔥}]⟩.\end{array}$$ Let $H_i=[X_i,Y_i]\in 𝔥$. It follows that $⟨\phi \left(\left(H_i,H_i\right)\right),\left(y,x\right)\otimes \left(f,e\right)⟩=⟨H_i,[y,f]⟩-⟨X_i,[x,e]⟩=0-0=0$, since $[y,f]\in {𝔫}^{-}$ and $[x,e]\in {𝔫}^{+}$. Thus $$\phi \left(\left(H_i,H_i\right)\right).$$ $$\begin{array}{rcl}⟨\phi \left(\left(X_i,0\right)\right),\left(y,-y_{𝔥}\right)\otimes \left(f,-f_{𝔥}\right)⟩& =& ⟨X_i,[y,f]⟩-⟨0,[-y_{𝔥},-f_{𝔥}]⟩\\ & =& ⟨X_i,[y,f]⟩-⟨0,[-y_{𝔥},-f_{𝔥}]⟩\\ & =& ⟨X_i,[y,f]⟩\\ & =& ⟨[X_i,y],f⟩\\ & =& ⟨[X_i,⟨y,X_i⟩Y_i+y_{𝔥}],f⟩\\ & =& ⟨y,X_i⟩⟨[X_i,Y_i],f⟩+⟨[X_i,y_{𝔥}],f⟩\\ & =& ⟨y,X_i⟩⟨H_i,f⟩-⟨[X_i,f],y_{𝔥}⟩\\ & =& ⟨X_i,y⟩⟨H_i,f⟩-⟨[X_i,⟨f,X_i⟩Y_i+f_{𝔥}],y_{𝔥}⟩\\ & =& ⟨X_i,y⟩⟨H_i,f⟩-⟨f,X_i⟩⟨[X_{},Y_i],y_{𝔥}⟩+⟨[X_i,f_{𝔥}],y_{𝔥}⟩\\ & =& ⟨X_i,y⟩⟨H_i,f⟩-⟨f,X_i⟩⟨H_i,y⟩+0.\end{array}$$ On the other hand, $$\begin{array}{rcl}⟨\left(X_i,0\right)\wedge \left(H_i,H_i\right),\left(y,-y_{𝔥}\right)⟩& =& ⟨\left(X_i,0\right)\otimes \left(H_i,H_i\right)-\left(H_i,H_i\right)\otimes \left(X_i,0\right),\left(y,-y_{𝔥}\right)\otimes \left(f,-f_{𝔥}\right)⟩\\ & =& ⟨X_i,y⟩⟨H_i,f_{𝔥}+f_{𝔥}⟩-⟨H_i,y+y_{𝔥}⟩⟨X_i,f⟩\\ & =& ⟨X_i,y⟩⟨H_i,f_{𝔥}+f_{𝔥}⟩-⟨H_i,y_{𝔥}+y_{𝔥}⟩⟨X_i,f⟩\\ & =& 2\left(⟨X_i,y⟩⟨H_i,f_{𝔥}⟩-⟨H_i,y_{𝔥}⟩⟨X_i,f⟩\right).\end{array}$$ It follows that $$\phi \left(\left(X_i,0\right)\right)={\displaystyle \frac{1}{2}}\left(\left(X_i,0\right)\wedge \left(H_i,H_i\right)\right).$$ $\square$

The double $D\left({𝔟}^{+}\right)\cong 𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$.

		Proof:
	This is clear from the form of the Manin triple in (2.1). $\square$

Quasitriangular Lie bialgebra structure on 𝔤

Let $𝔤$ be a Lie algebra with invariant form $⟨,⟩$ satisfying the conditions of Section 1.

1. There is a Lie bialgebras on $𝔤$ determined by the Manin triple $\left({𝔭}_1,{𝔭}_1,{𝔭}_2\right)$ given by $$\begin{array}{rcl}𝔭& =& 𝔤\hspace{0.17em}\times \hspace{0.17em}𝔤,\\ {𝔭}_1& =& \left\{\left(x,y\right)\in 𝔤\hspace{0.17em}\times \hspace{0.17em}{𝔤}_A\mid x=y\right\}\cong 𝔤\\ {𝔭}_2& =& \left\{\left(x,y\right)\in {𝔟}_{-}\hspace{0.17em}\times \hspace{0.17em}{𝔟}_{+}\mid x_{𝔥}+y_{𝔥}=0\right\},\end{array}$$ and Lie bracket and invariant scalar product $$\begin{array}{rcl}[\left(x_1,y_1\right),\left(x_2,y_2\right)]& =& \left([x_1,x_2],[y_1,y_2]\right),\\ ⟨\left(x_1,y_1\right),\left(x_2,y_2\right)⟩& =& ⟨x_1,x_2⟩-⟨y_1,y_2⟩,\end{array}$$ for all $x_1,x_2,y_1,y_2\in {𝔤}_A$ respectively.
2. The Lie bialgebra structure on $𝔤$ determined by the Manin triple in 1) is given by the cocommutator $\phi :{𝔤}_A\to {\bigwedge }^2{𝔤}_A$ determined by $$\begin{array}{ccc}\phi \left(H_i\right)& =& 0,\\ \phi \left(X_i\right)& =& \frac{1}{2}{X}_i\wedge H_i,\\ \phi \left(Y_i\right)& =& \frac{1}{2}{Y}_i\wedge H_i,\end{array}$$ for all $i\in I$.
3. For each $\alpha \in Q^{+}\setminus \left\{0\right\}$, let $$\begin{array}{l}\left\{X_{\alpha \left(j\right)}\mid 1\le j\le dim\left({𝔤}_{\alpha }\right)\right\}\\ \left\{X_{-\alpha \left(j\right)}\mid 1\le j\le dim\left({𝔤}_{\alpha }\right)\right\}\end{array}$$ be a basis of ${𝔤}_{\alpha }$ and a dual basis in ${𝔤}_{-\alpha }$. Let $\left\{{\tilde{H}}_i\right\}$ be an orthonormal basis of $𝔥$. The Lie algebra $𝔤$ is a quasitriangular Lie bialgebra with $r$-matrix given by $$r=\sum _{\alpha \in Q^{+}\setminus \left\{0\right\}}\sum _{j=1}^{dim\left({𝔤}_{\alpha }\right)}{X}_{\alpha \left(j\right)}\otimes X_{-\alpha \left(j\right)}+\frac{1}{2}\sum {\tilde{H}}_i\otimes {\tilde{H}}_i.$$

		Proof:
	We must show that ${𝔭}_1$ is an isotropic Lie subalgebra of $𝔤$. ${𝔭}_2$ is an isotropic Lie subalgebra of $𝔤$. $𝔭=𝔤\hspace{0.17em}\times \hspace{0.17em}𝔤={𝔭}_1\oplus {𝔭}_2$. The bilinear form $⟨,⟩$ is invariant. The bilinear form $⟨,⟩$ is nondegenerate. These are proved in (3.2)-(3.6). This is proved in (3.7). This follows in (3.8). $\square$

3.3 ${𝔭}_1$ is an isotropic Lie subalgebra of $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔤$. THe fact that ${𝔭}_1$ is an isotropic Lie subalgebra of $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔤$ follows from the following computations. $$\begin{array}{c}⟨\left(x,x\right),\left(y,y\right)⟩⟨x,y⟩-⟨x,y⟩=0\\ [\left(x,x\right),\left(y,y\right)]=\left([x,y],[x,y]\right)\in {𝔭}_1.\end{array}$$

3.3 Let $\left(y,x\right),\left(f,e\right)\in {𝔭}_2$ so that $$\begin{array}{c}y=y^{-}+y_{𝔥},\\ f=f^{-}+f_{𝔥},\end{array}\text{and}\begin{array}{c}x=x^{+}+x_{𝔥},\\ e=e^{+}+e_{𝔥},\end{array}$$ and $x_{𝔥}=-y_{𝔥}$ and $f_{𝔥}=e_{𝔥}$. Then $$⟨\left(y,x\right),\left(f,e\right)⟩=⟨y,f⟩-⟨x,e⟩=⟨y_{𝔥},f_{𝔥}⟩-⟨x_{𝔥},e_{𝔥}⟩=⟨y_{𝔥},f_{𝔥}⟩-⟨-y_{𝔥},-f_{𝔥}⟩=0.$$ Thus ${𝔭}_2$ is isotropic. Note that since $[y,f]\in {𝔫}^{-}$, and $[x,e]\in {𝔫}^{+}$, it follows that $[y,f{]}_{𝔥}=[x,e{]}_{𝔥}=0$, and thus $$[\left(y,x\right),\left(f,e\right)]=\left([y,f],[x,e]\right)\in {𝔭}_2.$$ Thus ${𝔭}_2$ is a Lie subalgebra.

3.4 Let $\left(y,x\right)\in {𝔭}_1\cap {𝔭}_2$. It follows that $y\in {𝔟}^{-},x\in {𝔟}^{+},y=x$ and $x_{𝔥}=y_{𝔥}$ and $x_{𝔥}+y_{𝔥}=0$. Thus $x^{+}=0$ and $y^{-}=0$ giving also that $x_{𝔥}=y_{𝔥}$. Thus if $char\left(k\right)\ne 2$ we have that $x_{𝔥}=y_{𝔥}=x=y=0$. So ${𝔭}_1\cap {𝔭}_2=0$. Given two elements $x,y\in 𝔤$ we may write $$\left(x,y\right)=\left(x^{+}-y^{+}+\frac{x_{𝔥}-y_{𝔥}}{2},-x^{-}+y^{-}-\frac{x_{𝔥}-y_{𝔥}}{2}\right)+\left(x^{-}+y^{+}+\frac{x_{𝔥}+y_{𝔥}}{2},x^{-}+y^{+}-\frac{x_{𝔥}+y_{𝔥}}{2}\right)$$ to see that $\left(x,y\right)\in {𝔭}_1+{𝔭}_2$. It follows that $𝔤\times 𝔤={𝔭}_1+{𝔭}_2$.

3.5 Recall that $⟨\left(x_1,x_2\right),\left(x_1,y_2\right)⟩=⟨x_1,y_1⟩-⟨x_2,y_2⟩$. It follows that $$\begin{array}{rcl}⟨[\left(x_1,y_1\right),\left(x_2,y_2\right)],\left(x_3,y_3\right)⟩& =& ⟨[x_1,x_2],x_3⟩-⟨[y_1,y_2],y_3⟩\\ & =& ⟨x_2,[x_3,x_1]⟩-⟨y_2,[y_3,y_1]⟩\\ & =& ⟨\left(x_2,y_2\right),\left([x_3,x_1],[y_3,y_1]\right)⟩\\ & =& ⟨\left(x_2,y_2\right),[\left(x_3,y_3\right),\left(x_1,y_1\right)]⟩.\end{array}$$ Thus, the form $⟨,⟩$ is invariant.

3.6 Let $\left(x,y\right)\in 𝔤\hspace{0.17em}\times \hspace{0.17em}𝔤$ such that $\left(x,y\right)\ne \left(0,0\right)$. If $x\ne 0$ then, since the form on $𝔤$ is nondegenerate there exists some $z\in 𝔤$ such that $\left(x,z\right)\ne 0$. Thus $$⟨\left(x,y\right),\left(z,0\right)⟩=⟨x,0⟩-⟨y,0⟩=⟨x,z⟩\ne 0.$$ If $x=0$ then $y\ne 0$ and there exists $w\in 𝔤$ such that $⟨y,w⟩\ne 0$. It follows that $$⟨\left(x,y\right),\left(0,w\right)⟩=⟨x,0⟩-⟨y,w⟩=-⟨y,w⟩\ne 0.$$ Thus, the form on $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔤$ is nondegenerate.

3.7 The cobracket on ${𝔭}_1$ is determined by $$⟨\phi \left(\left(z,z\right)\right),\left(y,x\right)\otimes \left(f,e\right)⟩=⟨\left(z,z\right),[\left(y,x\right),\left(f,e\right)]⟩$$ where $y,f\in {𝔟}^{-}$ and $x,e\in {𝔟}^{+}$. We have $$⟨\phi \left(\left(z,z\right)\right),\left(y,x\right)\otimes \left(f,e\right)⟩=⟨\left(z,z\right),[\left(y,x\right),\left(f,e\right)]⟩=⟨\left(z,z\right),\left([y,f],[x,e]\right)⟩=⟨z,[y,f]⟩-⟨z,[x,e]⟩.$$ Let $H_i\in 𝔥$. Then, since $[y,f]\in {𝔫}^{-}$ and $[x,e]\in {𝔫}^{+}$, it follows that $⟨\phi \left(\left(H_i,H_i\right)\right),\left(y,x\right)\otimes \left(f,e\right)⟩=⟨H_i,[y,f]⟩-⟨H_i,[x,e]⟩=0-0=0$. Thus, $$\phi \left(\left(H_i,H_i\right)\right)=0.$$ The computations to determine $\phi \left(X_i\right)$ are as follows. $$\begin{array}{rcl}⟨\phi \left(\left(X_i,X_i\right)\right),\left(y,x\right)\otimes \left(f,e\right)⟩& =& ⟨X_i,[y,f]⟩-⟨X_i,[x,e]⟩\\ & =& ⟨X_i,[y,f]⟩-0\\ & =& ⟨[X_i,y],f⟩\\ & =& ⟨[X_i,⟨y,X_i⟩Y_i]+y_{𝔥}⟩\\ & =& ⟨y,X_i⟩⟨[X_i,Y_i],f⟩+⟨[X_i,y_{𝔥}],f⟩\\ & =& ⟨y,X_i⟩⟨H_i,f⟩-⟨[X_i,f],y_{𝔥}⟩\\ & =& ⟨X_i,y⟩⟨H_i,f⟩-⟨[X_i,⟨f,X_i⟩Y_i+f_{𝔥}],y_{𝔥}⟩\\ & =& ⟨X_i,y⟩⟨H_i,f⟩-⟨f,X_i⟩⟨[X_i,Y_i],y_{𝔥}⟩+⟨[X_i,f_{𝔥}],y_{𝔥}⟩\\ & =& ⟨X_i,y⟩⟨H_i,f⟩-⟨f,X_i⟩⟨H_i,y⟩+0.\end{array}$$ On the other hand, since $y_{𝔥}=-x_{𝔥}$ and $f_{𝔥}=-e_{𝔥}$ it follows that $$⟨\left(X_i,X_i\right)\wedge \left(H_i,H_i\right),\left(y,x\right)\otimes \left(f,e\right)⟩& =& ⟨\left(X_i,X_i\right)\otimes \left(H_i,H_i\right)-\left(H_i,H_i\right)\otimes \left(X_i,X_i\right)\otimes ,\left(y,x\right)\otimes \left(f,e\right)⟩\\ & =& ⟨X_i,y-x⟩⟨H_i,f-e⟩-⟨H_i,y-x⟩⟨X_i,f-e⟩\\ & =& ⟨X_i,y-x⟩⟨H_i,f_{𝔥}+f_{𝔥}⟩-⟨H_i,y_{𝔥}+y_{𝔥}⟩⟨X_i,f-e⟩\\ & =& 2\left(⟨X_i,y-x⟩⟨H_i,f⟩-⟨H_i,y⟩⟨X_i,f-e⟩\right).$$ Since $X_i\in {𝔫}^{+}$ and $x\in {𝔟}^{+}$ it follows that $⟨X_i,x⟩=0$. Similarly, $⟨X_i,e⟩=0$. Thus $$⟨\left(X_i,X_i\right)\wedge \left(H_i,H_i\right),\left(y,x\right)\otimes \left(f,e\right)⟩=2\left(⟨X_i,y⟩⟨H_i,f⟩-⟨H_i,y⟩⟨X_i,f⟩\right).$$ It follows that $$\phi \left(\left(X_i,X_i\right)\right)={\displaystyle \frac{1}{2}}\left(\left(X_i,X_i\right)\wedge \left(H_i,H_i\right)\right).$$ The formula $$\phi \left(\left(Y_i,Y_i\right)\right)={\displaystyle \frac{1}{2}}\left(\left(Y_i,Y_i\right)\wedge \left(H_i,H_i\right)\right).$$ is proved similarly.

3.8 We know that $D\left({𝔟}^{+}\right)\cong 𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$ from Corollary (2.7). The set $$\left\{\left(X_{\alpha \left(j\right)},0\right)\mid \alpha \in Q^{+}\setminus \left\{0\right\},1\le j\le dim\left({𝔤}_{\alpha }\right)\right\}\cup \left\{\frac{1}{\sqrt[\mathrm{}]{2}}\left({\tilde{H}}_i,{\tilde{H}}_i\right)\right\}$$ is a basis of ${𝔭}_1$ with dual basis (with respect to the form on $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$) $$\left\{\left(X_{-\alpha \left(j\right)},0\right)\mid \alpha \in Q^{+}\setminus \left\{0\right\},1\le j\le dim\left({𝔤}_{\alpha }\right)\right\}\cup \left\{\frac{1}{\sqrt[\mathrm{}]{2}}\left({\tilde{H}}_i,-{\tilde{H}}_i\right)\right\}$$ in ${𝔭}_2$. The $r$-matrix $$\hat{r}=\sum _{\alpha \in Q^{+}\setminus \left\{0\right\}}\sum _{j=1}^{dim\left({𝔤}_{\alpha }\right)}\left(X_{\alpha \left(j\right)},0\right)\otimes \left(X_{-\alpha \left(j\right)},0\right)+\frac{1}{2}\sum {\tilde{H}}_i\otimes {\tilde{H}}_i$$ gives a quasitriangular Lie bialgebra structure on $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$. The subspace $0\hspace{0.17em}\times \hspace{0.17em}𝔥$ of $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$ is an ideal of $𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥$ and $𝔤\cong \left(𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥\right)/\left(0\hspace{0.17em}\times \hspace{0.17em}𝔥\right)$. Thus the projection $$\begin{array}{c}r\in \left(𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥\right)/\left(0\hspace{0.17em}\times \hspace{0.17em}𝔥\right)\otimes \left(𝔤\hspace{0.17em}\times \hspace{0.17em}𝔥\right)/\left(0\hspace{0.17em}\times \hspace{0.17em}𝔥\right)\\ r==\sum _{\alpha \in Q^{+}\setminus \left\{0\right\}}\sum _{j=1}^{dim\left({𝔤}_{\alpha }\right)}\left(X_{\alpha \left(j\right)},0\right)\otimes \left(X_{-\alpha \left(j\right)},0\right)+\frac{1}{2}\sum \left(H_i,0\right)\otimes \left(H_i,0\right)\end{array}$$ of the $r$-matrix $\hat{r}$ gives a quasitriangular Lie bialgebra structure on $𝔤$.
It remains to check that htis bialgebra structure is the same as that given by the Manin triple in 1). By (2.6) we know that the cocommutator determined by $r$ satisfies $$\begin{array}{c}\phi \left(H_i\right)=0\\ \phi \left(X_i\right)=\frac{1}{2}{X}_i\wedge H_i,\end{array}$$ on the subalgebra ${𝔟}^{+}$. We can calculate $\phi \left(Y_i\right)$ by using the Cartan involution $\theta$. $$\begin{array}{rcl}\left({ad}^{\otimes 2}\left(Y_i\right)\right)\left(\theta \otimes \theta \right)\left(r\right)& =& \left(\theta \otimes \theta \right)\left({ad}^{\otimes 2}\left(X_i\right)\right)\left(r\right)\\ & =& \left(\theta \hspace{0.17em}\times \hspace{0.17em}\theta \right)\left(\frac{1}{2}{X}_i\wedge H_i\right)\\ & =& -\frac{1}{2}{Y}_i\wedge H_i.\end{array}$$ Since $$\begin{array}{rcl}\left(\theta \otimes \theta \right)\left(r\right)& =& \sum _{\alpha \in Q^{+}}\sum _{j=1}^{dim\left({𝔤}_{\alpha }\right)}\theta \left(X_{\alpha \left(j\right)}\right)\otimes \theta \left(X_{-\alpha \left(j\right)}\right)+\frac{1}{2}\sum \theta \left(H_i\right)\otimes \theta \left(H_i\right)\\ & =& \sum _{\alpha \in Q^{+}\setminus \left\{0\right\}}\sum _{j=1}^{dim\left({𝔤}_{\alpha }\right)}{X}_{\alpha \left(j\right)}\otimes X_{-\alpha \left(j\right)}+\frac{1}{2}\sum H_i\otimes H_i\\ & =& r^{21},\end{array}$$ and $r^{21}+r^{12}$ is invariant, it follows that $$\left({ad}^{\otimes 2}\left(Y_i\right)\right)\left(r\right)=-\left({ad}^{\otimes 2}\left(Y_i\right)\right)\left(r^{21}\right)=\frac{1}{2}{Y}_i\wedge H_i.$$ Thus we have shown that the bialgbera structure on $𝔤$ determined by the $r$-matrix and the bialgebra structure on $𝔤$ determined by the Manin triple given in 1) are identical.

References

The definitions and proofs of the fact that Kac-Moody Lie algebras satisfy the properties given in Section 1 are contained in the first few pages of the following standard book.

[Kc] V. Kac, Infinite dimensional Lie algebras, Third Ed., Cambridge University Press 1990. MR1104219

The examples of bialgebra structures geven here appear in example 3.2 of the following paper by Drinfel'd.

[D] V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820. MR0934283

[DHL] H.-D. Doebner, Hennig, J. D. and W. Lücke, Mathematical guide to quantum groups, Quantum groups (Clausthal, 1989), Lecture Notes in Phys., 370, Springer, Berlin, 1990, pp. 29–63. MR1201823

[J] N. Jacobson, Lie algebras, Interscience Publishers, New York, 1962.

page history