Let $g$ be a finite dimensional real Lie algebra and $(,)$ be a nondegenerate invariant symmetric bilinear form on $g$. Let $r\in g\bigotimes g$ be a skew-symmetric solution of the MCYBE. We may regard $r$ as a linear map $g^{\star}\mapsto g$, and hence as a linear map $\rho: g\rightarrow g$,identifying $g^{\star}$ with $g$ by using the inner product. $\mathbf{Proposition}$: With the above notation, let $r\in g\otimes g$ be a skew symmetric solution of the MCYBE and define, for $X,Y\in g$, $[X,Y]_{r}=[\rho(X),Y]+[X,\rho(Y)]$. Then $[,]_{r}$ is a Lie bracket on $g$.

I have checked it, but I have problems in proving the Jacobi identity. Can anyone tell me how to prove the Jacobi identity or if there are any useful properties of the linear map $\rho$? Thank you.

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    $\begingroup$ what is the MCYBE? $\endgroup$ – Giulio Apr 15 '16 at 13:20
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    $\begingroup$ @Giulio the MCYBE is the modified classical Yang-Baxter equation $\endgroup$ – Xiaosong Peng Apr 15 '16 at 13:26
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    $\begingroup$ could you provide a link for the definition you use for MCYBE? or, even better, include the definition? $\endgroup$ – YCor Apr 15 '16 at 15:39
  • $\begingroup$ @YCor I am reading this book "a guide to quantum groups" written by V.C and A.P. Maybe you can see page 54 and 71 for these things. $\endgroup$ – Xiaosong Peng Apr 16 '16 at 1:04

Write the modified CYBE as $B_R(x,y)+\lambda [x,y]=0$ with $$ B_R(x,y):=[R(x),R(y)]-R([R(x),y]+[x,R(y)]), $$ and $\lambda\in \mathbb{R}$. Then note that the bracket $[x,y]_R:=[R(x),y]+[x,R(y)]$ satisfies the Jacobi identity if and only if $$ [B_R(x,y),w]+[B_R(y,w),x]+[B_R(w,x),y]=0 $$ for all $x,y,w\in \mathfrak{g}$.


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