## Lie bialgebras

1.1 A Lie bialgebra is a Lie algebra with bracket $\left[,\right]$ and cobracket $\phi :𝔤\to 𝔤\otimes 𝔤$ such that
1. ${𝔤}^{\ast }$ with bracket ${\phi }^{\ast }:{𝔤}^{\ast }\otimes {𝔤}^{\ast }\to {𝔤}^{\ast }$ is a Lie algebra, and
2. $\phi$ satisfies the $1$-cocycle condition,
In 1), ${\phi }^{\ast }:{\left(𝔤\otimes 𝔤\right)}^{\ast }\to {𝔤}^{\ast }$ is the induced mapping on the dual spaces. Given an element ${x}^{\ast }\in {𝔤}^{\ast }$ let us let $⟨{x}^{\ast },a⟩={x}^{\ast }\left(a\right)$ denote the evaluation of ${x}^{\ast }$ on the element $a\in 𝔤$. Then ${\phi }^{\ast }:{\left(𝔤\otimes 𝔤\right)}^{\ast }\to {𝔤}^{\ast }$ is given explicitly by $⟨φ∗x∗⊗y∗,a⊗b⟩= ⟨x∗,a⟩ ⟨y∗,b⟩,$ for all ${x}^{\ast },{y}^{\ast }\in {𝔤}^{\ast }$. Note that ${\phi }^{\ast }:{𝔤}^{\ast }\otimes {𝔤}^{\ast }\to {𝔤}^{\ast }$ is a well defined map since ${𝔤}^{\ast }\otimes {𝔤}^{\ast }\subseteq {\left(𝔤\otimes 𝔤\right)}^{\ast }$ (even if $𝔤$ is infinite dimensional).

1.2 An element $r\in 𝔤\otimes 𝔤=V={C}^{0}\left(𝔤,𝔤\otimes 𝔤\right)$ determines a $1$-coboundary $dr\in {B}^{1}\left(𝔤,𝔤\otimes 𝔤\right)$ given by $dr\left(x\right)=\left[1\otimes x+x\otimes 1,r\right]$, for all $x\in 𝔤$. Since ${d}^{2}=0$ we know that every $1$-coboundary is a $1$-cocycle. Thus $r\in 𝔤\otimes 𝔤$ determines a 1-cocycle and posibly a bialgebra structure on $𝔤$. Thus it is a natural question:

1. Given an element $r\in 𝔤\otimes 𝔤$, what conditions should we place on $r$ to guarantee that the map $\delta :𝔤\to 𝔤\otimes 𝔤$ determined by $\delta \left(x\right)=\left[1\otimes x+x\otimes 1,r\right]$, determines a Lie bialgebra structure on $𝔤$?
This question is answered by the following proposition:

Let $𝔤$ be a Lie algebra, let $r\in 𝔤\otimes 𝔤$ and define a map $dr:𝔤\to 𝔤\otimes 𝔤$ by $dr\left(x\right)=\left[1\otimes x+x\otimes 1,r\right]$. Let $\rho =\frac{1}{2}\left({r}^{12}-{r}^{21}\right)$ and let $P=\frac{1}{2}\left({r}^{12}+{r}^{21}\right)$ so that $r=P+\rho$.

1. The map ${dr}^{\ast }:{𝔤}^{\ast }\otimes {𝔤}^{\ast }\to {𝔤}^{\ast }$ satisfies the skew-symmetric condition if and only if $ad⊗2 r12+r21 =0$ for all $x\in 𝔤$.
2. Assume that $P$ is ${ad}^{\otimes 2}$ invariant. Then ${dr}^{\ast }$ satisfies the Jacobi identity if and only if $ad⊗3 r12 ,r13+ r12r23+ r13r23 = 0$ for all $x\in 𝔤$.

If $r=\sum _{i}{a}_{i}\otimes {b}_{i}$ then the notation above is $r12+r21= ∑ i ai⊗bi+ bi⊗ai , r12r13+ r12r23+ r13r23= ∑ i,j aiaj⊗ bi⊗bj+ ai⊗biaj ⊗bj+ ai⊗aj⊗ bibj .$ Condition a) states that ${r}^{\mathrm{12}}+{r}^{\mathrm{21}}$ is $𝔤$-invariant and condition b) states that $\left[{r}^{12},{r}^{13}\right]+\left[{r}^{12},{r}^{23}\right]+\left[{r}^{13},{r}^{23}\right]$ is $𝔤$-invariant.

1.4 A quasitraingular Lie bialgebra is a pair $\left(𝔤,r\right)$ such that $𝔤$ is a Lie bialgebra, $r\in 𝔤\otimes 𝔤$, the cobraket in $𝔤$ is equal to $dr$ and $r$ satisfies $r12+r21=0,$ i.e, $r\in {\bigwedge }^{2}𝔤$.

A triangular Lie bialgebra is a pair $\left(𝔤,r\right)$ such that $𝔤$ is a Lie bialgebra, $r\in 𝔤\otimes 𝔤$, the cobracket in $𝔤$ is equal to $dr$ and $r$ satisfies $r12+r21=0,and r12r13+ r12r23+ r13r23=0.$ The equation $r12r13+ r12r23+ r13r23 =0$ is the classical Yang-Baxter equation (CYBE).

1.5 If $𝔤$ is a semisimple Lie algebra over a field of characteristic $0$, then it follows from Whitehead's lemma ([J] III §7 Lemma 3 p.77 and Thm. 13 p.95) that ${H}^{1}\left(𝔤,r\right)$=0 for all finite dimensional $𝔤$-modules $V$. Thus all $1$-cocycles are coboundaries; in this case, if $\phi :𝔤\to 𝔤\otimes 𝔤$ is any linear map which satisfies the $1$-cocycle condition then there is an element $r\in 𝔤\otimes 𝔤$ such that $φx= 1⊗x+x⊗1,r$ for all $x\in 𝔤$.

## Manin triples and the double

2.1 Let $⟨,⟩:𝔭\otimes 𝔭\to 𝔭$ be a symmetric bilinear form on a vector space $𝔭$, i.e. $⟨x,y⟩=⟨y,x⟩$ for all $x,y\in 𝔭$.

The form $⟨,⟩$ is nondegenerate if the map given by $∗: 𝔭 → 𝔭∗ x ↦ ⟨x,⋅⟩$ is an isomorphism. Alternatively, the form $⟨,⟩$ is invariant if for every $x\in 𝔭$, $x\ne 0$ there is a $y\in 𝔭$ such that $⟨x,y⟩\ne 0$. A third way to say it is that $⟨,⟩$ is invariant if the null space of the form is $0$. A fourht way to say it is that the matrix of the form (with respect to any fixed basis of $𝔭$) has nonzero determininant.

A subspace ${𝔭}^{\prime }$ of $𝔭$ is isotropic if $⟨x,x⟩=0$ for all $x\in {𝔭}^{\prime }$.

Given vector spaces ${𝔭}_{1}$ and ${𝔭}_{2}$ with symmetric bilinear forms ${⟨,⟩}_{1}$ and ${⟨,⟩}_{2}$ respectively then the vector space ${𝔭}_{1}\otimes {𝔭}_{2}$ has a bilinear form $⟨,⟩$ given by $⟨x\otimes y,z\otimes w⟩={⟨x,z⟩}_{1}{⟨y,w⟩}_{2}$ for all $x,z\in {𝔭}_{1}$ and $y,w\in {𝔭}_{2}$. In particular, if $𝔭$ is a vector space with a bilinear form $⟨,⟩$ then $𝔭\otimes 𝔭$ has a bilinear form given by $⟨x⊗y,z⊗w⟩= ⟨x,z⟩⟨y,w⟩$ for all $x,y,z,w\in 𝔭$.

Suppose that the vector space $𝔭$ is a Lie algebra. The form $⟨,⟩$ is invariant if $⟨adzx,y⟩= -⟨x,adzy⟩,$ for all $x,y,z\in 𝔭.$

2.2 A Manin triple is a triple $\left(𝔭,{𝔭}_{1},{𝔭}_{2}\right)$ such that

1. $𝔭$ is a Lie algebra with a nondegenerat invariant symmetric bilinear form $⟨,⟩$ and
2. ${𝔭}_{1}$ and ${𝔭}_{2}$ are isotropic Lie subalgebra of $𝔭$.
3. $𝔭={𝔭}_{1}\oplus {𝔭}_{2}$ as vector spaces.

Let $\left(𝔭,{𝔭}_{1},{𝔭}_{2}\right)$ be a Manin triple. Then ${𝔭}_{1}$ is a Lie algebra with cobracket $\delta :{𝔭}_{1}\to {𝔭}_{1}\oplus {𝔭}_{2}$ determined by the equation $⟨δx ,y1⊗y2⟩= ⟨x,y1,y2⟩,$ for all $x\in {𝔭}_{1}$ and all ${y}_{1},{y}_{2}\in {𝔭}_{2}$. In other words, $\delta$ is the adjoint of the bracket on ${𝔭}_{2}$.

Let $\left(𝔤,\delta \right)$ be a Lie bialgebra. Then the triple $\left(𝔤\oplus {𝔤}^{\ast },𝔤,{𝔤}^{\ast }\right)$ is a Manin triple where

1. The bilinear form $⟨,⟩$ on $𝔤\oplus {𝔤}^{\ast }$ is given by $⟨x1+ y1∗ ,x2+ y2∗ ⟩ = ⟨x1, y2∗ ⟩+ ⟨x2, y1∗ ⟩ = y2∗ x1+ y1∗ x2,$ for all ${x}_{1},{x}_{2}\in 𝔤$ and all ${y}_{1}^{\ast },{y}_{2}^{\ast }\in {𝔤}^{\ast }$, and
2. The bracket on $𝔤\oplus {𝔤}^{\ast }$ is determined by the formulas where $x\in 𝔤$ and $y,{y}_{1},{y}_{2}^{\ast }\in {𝔤}^{\ast }$.

This is essentially the only way to define things so that the bracket on ${𝔤}^{\ast }$ is the dual of the cobracket on $𝔤$ and so that the bilinear form is invariant. The Lie algebra $𝔤\oplus 𝔤=D\left(g\right)$ constructed from the the Lie bialgebra $𝔤$ is called the double corresponding to the Lie algebra $𝔤$.

The previous two propositions show that there is a one-to-one correspondence between Lie bialgebras and Manin triples.

Let $𝔤$ be a Lie bialgebra and $D\left(𝔤\right)=𝔤\oplus {𝔤}^{\ast }$ be the double of $𝔤$. Let $\left\{{a}_{i}\right\}$ be a basis of $𝔤$ and let $\left\{{a}^{i}\right\}$ be the dual basis in ${𝔤}^{\ast }$. Define $r= ∑ i ai⊗ai∈ 𝔤⊗𝔤∗ ⊆ D𝔤⊗D𝔤.$ Then

1. The element $r$ does not depend on the choice of basis ${a}_{i}$ of $𝔤$.
2. $r$ satisfies the CYBE.
3. $\left(D\left(𝔤\right),r\right)$ is a quasitriangular Lie bialgebra.

## Proofs

Let $𝔤$ be a Lie algebra, let $r\in 𝔤\otimes 𝔤$ and define a map $dr:𝔤\to 𝔤\otimes 𝔤$ by $dr\left(x\right)=\left[1\otimes x+x\otimes 1,r\right]$. Let $\rho =\frac{1}{2}\left({r}^{12}-{r}^{21}\right)$ and let $P=\frac{1}{2}\left({r}^{12}-{r}^{21}\right)$ so that $r=P+\rho$.

1. The map ${dr}^{\ast }:{𝔤}^{\ast }\otimes {𝔤}^{\ast }\to {𝔤}^{\ast }$ satisfies the skew-symmetric condition if and only if $ad⊗2 r12+r21 =0$ for all $x\in 𝔤$.
2. Assume that $P$ is ${ad}^{\otimes 2}$ invariant. Then ${dr}^{\ast }$ satisfies the Jacobi identity if and only if $ad⊗3 r12 ,r13+ r12r23+ r13r23 = 0$ for all $x\in 𝔤$.

 Proof: The following computation shows that $\left({ad}^{\otimes 2}z\right)\left({r}^{12}+{r}^{21}\right)=0$ if and only if ${\delta }^{\ast }\left({x}^{\ast }\otimes {y}^{\ast }\right)+{\delta }^{\ast }\left({y}^{\ast }\otimes {x}^{\ast }\right)=0$ for all ${x}^{\ast },{y}^{\ast }\in {𝔤}^{\ast }$. $0 = ⟨ δ∗ x∗⊗y∗ +δ∗y∗⊗x∗ ,z⟩ = ⟨x∗⊗y∗+ y∗⊗x∗, δz⟩ = ⟨x∗⊗y∗ +y∗⊗x∗, ad⊗2z ∑ i ai⊗bi⟩ = ⟨x∗⊗y∗, ad⊗2z ∑ i ai⊗bi +bi⊗ai ⟩ = ⟨x∗⊗y∗, ad⊗2z r12+r21 ⟩.$ The proof will be done in several steps. Step 1. $dr=d\rho$. Step 2. $C\left(r\right)=C\left(\rho \right)+C\left(P\right)$. Step 3. If $P$ is ${ad}^{\otimes 2}$ invariant then $C\left(P\right)$ is ${ad}^{\otimes 3}$ invariant. Step 4. If $r\in {\bigwedge }^{2}𝔤$ then ${dr}^{\ast }$ satisfies the Jacobi identity if and only if $C\left(r\right)$ is ${ad}^{\otimes 3}$ invariant. Punchline. Let us show that these 4 steps are sufficient to prove the result. By step 3, $C\left(P\right)$ is ${ad}^{\otimes 3}$ invariant. Thus by step 2, $C\left(r\right)$ is ${ad}^{\otimes 3}$ invariant if and only if $C\left(\rho \right)$ is. Now $\rho \in {\bigwedge }^{2}$ and so by step 4 ${d\rho }^{\ast }$ satisfies the Jacobi identity if and only if $C\left(\rho \right)$ is ${ad}^{\otimes 3}$ invariant. Since $d\rho =dr$, it follows that ${dr}^{\ast }$ satifies the Jacobi identity if and only if $C\left(r\right)$ is ${ad}^{\otimes 3}$ invariant. Step 1. Since $P$ is ${ad}^{\otimes 2}$ invariant we have that $dρx = ad⊗2xρ = ad⊗2x r- 12 P = ad⊗2x r- 12 ad⊗2xP = ad⊗2xr = drx.$ Step 2. Using the fact that $P$ is ${ad}^{\otimes 2}$ invariant and the fact that $\rho =\sum {\rho }^{\left(1\right)}\otimes {\rho }^{\left(2\right)}=-\sum {\rho }^{\left(2\right)}\otimes {\rho }^{\left(1\right)}$, we get $Cr = r12r13+ r12r23+ r13r23 = ρ12+P12,ρ13+P13+ ρ12+P12,ρ23+P23+ ρ13+P13,ρ23+P23 = Cρ+CP+ ρ12,P13+ ρ12,P23+ ρ13,P23 P12,ρ13+ P13,ρ23 = Cρ+CP+ ∑ ρ(1),P(1) ⊗ρ(2)⊗P(2)+ ρ(1)⊗ ρ(2),P(1) ⊗P(2)+ ρ(1)⊗P(1)⊗ ρ(2),P(2) + P(1),ρ(1) ⊗P(2)⊗ρ(2)+ P(1)⊗ P(2),ρ(1) ⊗ρ(2)+ P(1)⊗ρ(1)⊗ P(2),ρ(2) = Cρ+CP+ ∑ ρ(1),P(1) ⊗ ρ(2)⊗P(2) = Cρ+CP+ ∑ ρ(1),P(1) ⊗ρ(2)⊗P(2) +ρ(1)⊗ ρ(2),P(1) ⊗P(2)+ ρ(1)⊗P(1)⊗ ρ(2),P(2) + P(1),ρ(1) ⊗P(2)⊗ρ(2)+ P(1)⊗ P(2),ρ(1) ⊗ρ(2) + P(1)⊗ρ(1)⊗ P(2),ρ(2) = Cρ+CP+∑ ρ1,P1⊗ ρ2⊗P2+0+0- P1⊗ρ2⊗ P2,ρ1 = Cρ+CP+∑ ρ(1),P(1) ρ(2)⊗P(2)+ P(1)⊗ρ(2)⊗ ρ(1),P(2) = Cρ+CP.$ Step 3. Suppose that $P=\sum {P}_{\left(1\right)}\otimes {P}_{\left(2\right)}={\sum }_{j}{P}^{\left(1\right)}\otimes {P}^{\left(2\right)}$. Then $ad⊗3x P12 , P13 = ad⊗3x ∑ P(1) , P(1) ⊗P(2)⊗P(2) = ∑ x , P(1) , P(1) ⊗ P(2)⊗P(2)+ P(1) , P(1) ⊗ x , P(2) ⊗P(2)+ P(1) , P(1) ⊗ P(2)⊗ x , P(2) = ∑- P(1) , x , P(1) ⊗P(2) ⊗P(2)+ P(1) , P(1) ⊗ x , P(2) ⊗P(2)- P(1) , P(1) , x ⊗P(2)⊗P(2) + P(1) , P(1) ⊗P(2)⊗ x , P(2) = ∑ x , P(1) , P(1) ⊗P(2)⊗ P(2)+ P(1) , P(1) ⊗ x , P(2) ⊗P(2)+ P(1) , x , P(1) ⊗ P(2)⊗P(2) + P(1) , P(1) ⊗ P(2)⊗ x , P(2) = 0.$ By arguing similarly for the other terms one shows that $C\left(P\right)=\left[{P}^{12},{P}^{13}\right]+\left[{P}^{12},{P}^{23}\right]+\left[{P}^{13},{P}^{23}\right]$ is an invariant element of ${𝔤}^{\otimes 3}$. Step 4. Assume that $r\in {\bigwedge }^{2}𝔤$ so that ${r}^{12}+{r}^{21}=0$ or equivalently that $r=\sum {a}_{i}\otimes {b}_{i}=-\sum {b}_{i}\otimes {a}_{i}$. Let us first calculate $⟨δ∗ x∗⊗δ∗ x∗⊗z∗ ,w⟩ = ⟨ x∗⊗δ∗ y∗⊗z∗ ,δw⟩ = ⟨x∗ δ∗y∗⊗z∗ , ∑w,ai⊗bi +ai⊗w,bi ⟩ = ⟨x∗⊗y∗⊗z∗, ∑ i w,ai⊗ δbi+ ai⊗δ w,bi ⟩ = ⟨ x∗⊗y∗⊗z∗ , ∑ i w,ai⊗ ∑ j bi,aj⊗ bj +aj⊗b, +ai⊗ ∑ j w,bi, aj ⊗bj+aj⊗ w,bi, bj ⟩ = ⟨ x∗⊗y∗⊗z∗ , ∑ i,j w,ai⊗ bi,aj⊗ bj+ w,ai⊗aj ⊗bi,bj +ai⊗ w,bi ,aj⊗bj+ ai⊗aj⊗ w,bi ,bj ⟩.$ Now consider the sum $⟨ δ∗ x∗⊗δ∗ y∗⊗z∗ ,w⟩+ ⟨δ∗ z∗⊗δ∗ x∗⊗y∗ ,w⟩+ ⟨ δ∗ y∗⊗δ∗ z∗⊗x∗ ,w⟩.$ For simplicity, write only the terms in this sum which have a $w$ in the first tensor slot, $w,ai⊗ bi,aj⊗ bj, w,ai⊗aj ⊗bi,bj, w,ai ,aj⊗bj ⊗ai, w,bi ,bj ⊗ai⊗aj.$ The first term is equal to $\left(adw\otimes 1\otimes 1\right)\left[{r}^{12},{r}^{23}\right]$ and the second term is equal to $\left(adw\otimes 1\otimes 1\right)\left[{r}^{13},{r}^{23}\right]$. For the third term, applying the skew symmetry relation and then switching the indices $i$ and $j$, $w,bi ,bj ⊗ai⊗aj= - w,ai ,aj⊗bj⊗bj = - w,aj, ai⊗bi⊗bj = aj,w, ai⊗bi⊗bj.$ For the fourth term, apllying the skew-symmetry relation twice gives $w,bi ,bj⊗ai⊗ aj= - w,ai ,bj⊗bi ⊗aj= w,ai ,aj ⊗bi⊗bj.$ Now it is clear that the sum of the third and fourth terms is, by the Jacobi identity $-\left[\left[{a}_{i},{a}_{j}\right],w\right]\otimes {b}_{i}\otimes {b}_{j}=\left[w,\left[{a}_{i},{a}_{j}\right]\right]\otimes {b}_{i}\otimes {b}_{j}=\left(adw\otimes 1\otimes 1\right)\left[{r}^{12},{r}^{13}\right]$. The result now follows by symmetry. This completes the proof of Proposition (1.1). $\square$

Let $𝔭,{𝔭}_{1},{𝔭}_{2}$ be a Manin triple. Then ${𝔭}_{1}$ is a Lie bialgebra with cobracket $\delta :{𝔭}_{1}\to {𝔭}_{1}\otimes {𝔭}_{1}$ determined by the equation $⟨δx, y1⊗y2 ⟩= ⟨x, y1,y2⟩$ for all $x\in {𝔭}_{1}$ and all ${y}_{1},{y}_{2}\in {𝔭}_{2}$. In other words, $\delta$ is the adjoint of the bracket on ${𝔭}_{2}$.

 Proof: It is clear from the fact that delta is the adjoint of the bracket on ${𝔭}_{2}$ that $\delta$ is a cobracket on ${𝔭}_{1}$. It remains to check the cocycle condition. If $z\in 𝔭$ then ${z}_{1}$ and ${z}_{2}$ shall denote the elements of ${𝔭}_{1}$ and ${𝔭}_{2}$ respectively such that $z={z}_{1}+{z}_{2}$. Then we have that, for any ${x}_{1}\in {𝔭}_{1},{x}_{2}\in {𝔭}_{2}$, $⟨ ad⊗2x1 δy1 ,x1⊗y2⟩ = - ⟨δy1, ad x1⊗1+1 ⊗ad x1 x1⊗y2 ⟩ = -⟨ δy1 , x1,x2 2 ⊗y2+x2⊗ x1,y2 2 ⟩ = -⟨ y1 , x1,x22,y2+ x2, x1,y22 ⟩ = ⟨y1,y21, x1,x22 ⟩+ ⟨ x2,y11 , x1,y22 ⟩$ Similarly, $⟨ ad⊗2y1 δx1 , x2⊗y2 ⟩ = -⟨ δx1 , ad y1⊗1+ 1⊗ad y1 x2⊗y2 ⟩ = -⟨ δx1 , y1,x22 ⊗y2+x2⊗ y1,y22 ⟩ = -⟨x1, y1,x22 ,y2+ x2, y1,y22 ⟩ = ⟨ x1,y21 , y1,y22 ⟩+ ⟨ x2,x11 , y1,y22 ⟩ = - ⟨ x1,x21 , y1,y22 ⟩+ ⟨ x2,y21 , x2,y12 ⟩.$ Thus $⟨ ad⊗2x1 δy1 - ad⊗2y1 δx1 , x2⊗y2 ⟩ = ⟨ y1,y21 , x1,x22 ⟩+ ⟨ x2,y11 , x1,y22 ⟩ + ⟨ x1,x21 , y1,y22 ⟩ + ⟨ x2,y21 , x2,y12 ⟩ = ⟨ x1,y2 , x2,y1 ⟩+ ⟨ x1,x2 , y1,y2 ⟩.$ On the other hand we have that for any ${x}_{1},{y}_{1}\in {𝔭}_{1}$ and ${x}_{2},{y}_{2}\in {𝔭}_{2}$ $⟨ δ x1,y1 , x2⊗y2 ⟩ = ⟨ x1,y1 , x2,y2 ⟩ = - ⟨ x1,y1 ,y2 , x2 ⟩ = ⟨ y1 , x1 , y1 , x2 ⟩+ ⟨ y1 , y2 , x1 , x2 ⟩ = - ⟨ y2 , x1 , x2 , y1 ⟩ + ⟨ y1 , y2 , x1 , x2 ⟩ = ⟨ x1 , y2 , x2 , y1 ⟩+ ⟨ x1 , x2 , y1 , y2 ⟩.$ It follows from this computation oand the previous one that $δ x1 , y1 = ad⊗2x1 δy1- ad⊗2y1 δx1,$ for all ${x}_{1},{y}_{1}\in {𝔭}_{1}$. It follows that ${𝔭}_{1}$ is a Lie bialgebra. $\square$

Let $\left(𝔤,\delta \right)$ be a Lie bialgebra. Then the triple $\left(𝔤\oplus {𝔤}^{\ast },𝔤,{𝔤}^{\ast }\right)$ is a Manin triple where

1. The bilinear form $⟨,⟩$ on $𝔤\oplus {𝔤}^{\ast }$ is given by $⟨ x1+ y1∗ , x2+ y2∗ ⟩ = ⟨ x1 , y2∗ ⟩ + ⟨ x2 , y1∗ ⟩ = y2∗ x1+ y1∗ x2,$ for all ${x}_{1},{x}_{2}\in 𝔤$ and all ${y}_{1}^{\ast },{y}_{2}^{\ast }\in {𝔤}^{\ast }$
2. The bracket on $𝔤\oplus {𝔤}^{\ast }$ is determined by the formulas where $x\in 𝔤$ and $y,{y}_{1},{y}_{2}^{\ast }\in {𝔤}^{\ast }$.

 Proof: The facts, ${𝔤}^{\ast }$ is Lie subalgebra of $𝔤\oplus {𝔤}^{\ast }$, the bilinear form is invariant, the bracket of $𝔤\oplus {𝔤}^{\ast }$ satisfies the skew-symmetry condition, all follow directly form the definitions. Only the Jacobi identity really needs a proof. If all three elements in the Jacobi sum are in $𝔤$ or all three are in ${𝔤}^{\ast }$ then the Jacobi identity follows immediately from the Jacobi identity for these Lie subalgebras. Suppose two indices are in ${𝔤}^{\ast }$ and the third is in $𝔤$. THen we have that, for any $x\in 𝔤$ and ${y}_{1}^{\ast },{y}_{2}^{\ast },{z}^{\ast }\in {𝔤}^{\ast }$, $⟨ y2∗ , x , y1∗ + y1∗ , y2∗ , x + x , y1∗ , y2∗ , z∗ ⟩ = ⟨ x , y1∗ , z∗ , y2∗ + z∗ , y1∗ , y2∗ + y1∗ , y2∗ , z∗ ⟩ = ⟨ x , y2∗ , z∗ , y1∗ + z∗ , y1∗ , y2∗ + y1∗ , y2∗ , z∗ ⟩ = 0,$ by the Jacobi identity in ${𝔤}^{\ast }$. In the other case we have, for any $x,z\in 𝔤$ and ${y}_{1},{y}_{2}\in {𝔤}^{\ast }$, $⟨ y2∗ , x , y1∗ + y1∗ , y2∗ , x + x , y1∗ , y2∗ , z∗ ⟩ = ⟨ x , y1∗ , z , y2∗ ⟩ + ⟨ y2∗ , x , z , y2∗ ⟩ + ⟨ y1∗ , y2∗ , z , x ⟩ = - ⟨ ad⊗2x δx - ad⊗2z δx , y1∗ ⊗ y2∗ ⟩ + ⟨ y1∗ ⊗ y2∗ , δ z , x ⟩ = 0.$ In this computation we are using the $1$-cocycle condition on $𝔤$ and the fact that $⟨ x , y1∗ , z , y2∗ ⟩ + ⟨ y2∗ , x , z , y1∗ ⟩= ⟨ ad⊗2x δx- ad⊗2z δx , y1∗ ⊗ y2∗ ⟩$ which was proved in the proof of Proposition (2.3). The case of the Jacobi identity where two of the indices are in $𝔤$ and one is in ${𝔤}^{\ast }$ is proved similarly. Thus the bracket on $𝔤\oplus {𝔤}^{\ast }$ satisfies the Jacobi identity and $𝔤\oplus {𝔤}^{\ast }$ is a Lie algebra with an invariant bilinear form. $\square$

Let $𝔤$ be a Lie bialgebra and let $D\left(𝔤\right)=𝔤\oplus {𝔤}^{\ast }$ be the double of $𝔤$. Let $\left\{{a}_{i}\right\}$ be a basis of $𝔤$ and $\left\{{a}^{i}\right\}$ be the dual basis in ${𝔤}^{\ast }$. Define $r= ∑ i ai⊗ai∈ 𝔤⊗𝔤∗⊆ D𝔤⊗D𝔤.$ Then

1. The element $r$ does not depend on the choice of basis $\left\{{a}_{i}\right\}$ of $𝔤$.
2. $r$ satisfies the CYBE.
3. $\left(D\left(𝔤\right),r\right)$ is a quasitriangular Lie bialgebra.

 Proof: Suppose that $\left\{{b}_{j}\right\}$ is another basis of $𝔤$ and that $\left\{{b}^{j}\right\}$ is the dual basis in ${𝔤}^{\ast }$. Then $r = ∑ i ai⊗ai = ∑ i,j ⟨ ai , bj ⟩ bj⊗ai = ∑ i,j bj⊗ai ⟨ ai , bj ⟩ = ∑ j bj⊗bj.$ Thus the element $r$ is independent of the choice of basis. Let us check that $r$ satisfies the CYBE. $r12 , r13 + r12 , r23 + r13 , r23 = ∑ i,j ai , aj ⊗bi⊗bj + ai⊗ bi , aj ⊗bj +ai⊗aj⊗ bi , bj = ∑ i,j ai , aj ⊗ai⊗aj+ ai⊗ ai , aj ⊗ aj+ ai⊗aj⊗ ai , aj = ∑ i,j,s ⟨ ai , aj , as ⟩ as ⊗ai ⊗aj + ⟨ ai , aj , as ⟩ ai⊗ as⊗ aj + ⟨ ai , aj , as ⟩ ai⊗ as⊗ aj+ ⟨ ai , aj , as ⟩ ai⊗ aj⊗ as = ∑ i,j,s ⟨ ai , aj , as ⟩ ⊗as ⊗ai ⊗aj - ⟨ ai , as , aj ⟩ ai⊗ as⊗ aj - ⟨ ai , as , aj ⟩ aa⊗ as⊗ aj+ ⟨ ai , aj , as ⟩ ai⊗ aj⊗ as = 0.$ In view of Proposition () we only need to show that $\left({ad}^{\otimes 2}x\right)\left({r}^{12}+{r}^{21}\right)=0$ for all $x\in 𝔤\oplus {𝔤}^{\ast }$. Let $x\in 𝔤$. Then $x⊗1+1⊗x , r12+r21 = ∑ i x , ai ⊗ai+ x , ai ⊗ai+ ai⊗ x , ai = ∑ i,k ⟨ x , ai , ak ⟩ ak⊗ai+ ⟨ x , ai , ak ⟩ ak⊗ai + ⟨ x , ai , ak ⟩ ak⊗ai + ⟨ x , ai , ak ⟩ ai⊗ak + ⟨ x , ai , ak ⟩ ai⊗ak + ⟨ x , ai , ak ⟩ ak⊗ai.$ After reindexing we have $x⊗1+1⊗x , r12+r21 = ∑ i,k ⟨ x , ai , ak ⟩ ak⊗ai+ ⟨ x , ai , ak ⟩ ak⊗ai+ ⟨ x , ai , ak ⟩ ak⊗ai + ⟨ x , ak , ai ⟩ ak⊗ai+ ⟨ x , ak , ai ⟩ ak⊗ai+ ⟨ x , ak , ai ⟩ ak⊗ai.$ Now note that, by invariance of the inner product we have that $⟨ x , ai , ak ⟩= - ⟨ x , ak , ai ⟩, ⟨ x , ai , ak ⟩= - ⟨ x , ak , ai ⟩, ⟨ x , ai , ak ⟩=- ⟨ x , ak , ai ⟩.$ It follows that $\left[x\otimes 1+1\otimes x,{r}^{12}+{r}^{21}\right]=0$. The case when $x\in {𝔤}^{\ast }$ is proved similarly. The result follows. $\square$

## References

The motivating reference is

[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

There is a detailed exposition of Lie bialgebras in the following article

[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.