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We know that the classical Yang-Baxter equation is $$[r_{12}, r_{13}]+[r_{12}, r_{23}]+[r_{13}, r_{23}]=0,\quad(*)$$ and we have the following Theorem.

Theorem: Let $\mathcal{G}$ be a Lie algebra and $r\in \mathcal{G}\otimes \mathcal{G}.$ Then the map $\delta: \mathcal{G}\rightarrow \mathcal{G}\otimes \mathcal{G}$ defined by Eq. $\delta(X)=X.r$ induces a Lie bialgebra structure on $\mathcal{G}$ if and only if the following two conditions are satisfied (for any $x\in \mathcal{G}$).

  1. $(\mathrm{ad}(x)\otimes \mathrm{id} +\mathrm{id}\otimes \mathrm{ad}(x))(r+r^{21})=0;$
  2. $(\mathrm{ad}(x)\otimes \mathrm{id} \otimes \mathrm{id} +\mathrm{id} \otimes \mathrm{ad}(x)\otimes \mathrm{id}+\mathrm{id} \otimes \mathrm{id} \otimes \mathrm{ad}(x))([r_{12}, r_{13}]+[r_{12}, r_{23}]+[r_{13}, r_{23}])=0.$

The theorem shows that when $r\in \mathcal{G}\otimes \mathcal{G},$ induces a Lie bialgebra structure if and only if satisfying two conditions. My question is that how to induce the lie bialgebra when $r\in \mathcal{G}\otimes \mathcal{G}$ is the solution of the equation (*), can you give me some references? Thank you very much.

Cameron Williams
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yuanyuanzhang
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1 Answers1

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Simply take $\delta(x)=ad_x(r)$. First condition says that the image of $\delta$ is contained in the subspace of anti-symmetric tensor.

The cocycle condition is trivial because $\delta$ is a coboundary.

Condition 2 translates into co-Jacobi. This requires some pacient, but no trick. Write $r=r_1+r_2$ with $r_1$ Anti symmetric and $r_2$ symmetric. Condition 2 is equivalent to the fact that $ [r_1,r_1]\in \Lambda^3g$ is an invariant tensor. When $\delta(x)=ad_x(r)$ (this is equal to $ad_x(r_1)$ because $r_2$ is ad invariant), the invariance of this element is tautologically co-Jacobi condition.

Marco Farinati
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