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Let $r \in \bigwedge^2 \mathfrak {g}$ be a skew-symmetric solution of the CYBE (classical Yang-Baxter equation) so that it gives rise to a non-degenrate triangular structure on $\mathfrak {g}$ i.e. $r$ determines a non-degenerate coboundary structure on $\mathfrak {g}$ with $CYB\ (r) = 0.$ Then the bivector field $\Pi_r^{L}$ defined on the associated simply connected Lie group $G$ defined by $x \mapsto (d_e \lambda_x)^{\otimes 2} (r)$ is a left-invariant non-degenerate Poisson structure on $G$ (where $\lambda_x$ is the left translation by $x \in G$).

I have proved that it is a left-invariant Poisson structure on $G$ but couldn't able to show that the Poisson structure is non-degenerate. Could anyone please help me in this regard?

Thanks for your time.

Source $:$ Lectures on Quantum Groups by Pavel Etingof and Oliver Schiffmann.

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  • $\begingroup$ $r=0$ seems to satisfy your assumptions, but clearly $\Pi^L_r=0$ is not non-degenerate. $\endgroup$
    – Pavel Safronov
    Commented Jan 17, 2023 at 14:05
  • $\begingroup$ @PavelSafronov$:$ Sorry, I forgot to mention that $r$ is itself non-degenerate i.e. we assume that the map $\xi \mapsto (\xi \otimes \text {id}) (r)$ is an isomorphism from $\mathfrak {g}^{\ast} \longrightarrow \mathfrak {g}$ which clearly discards the case $r = 0.$ $\endgroup$
    – Anil Bagchi.
    Commented Jan 17, 2023 at 14:36
  • $\begingroup$ @PavelSafronov$:$ If $r = \sum\limits_{s,t} r^{st} X_s \otimes X_t$ then $r$ is non-degenerate if and only if the matrix $(r^{st})$ is invertible. $\endgroup$
    – Anil Bagchi.
    Commented Jan 17, 2023 at 15:06
  • $\begingroup$ As you've probably already understood, $\Pi^L_r$ is non-degenerate iff it is non-degenerate at the unit (by left invariance) iff $r$ is non-degenerate. $\endgroup$
    – Pavel Safronov
    Commented Jan 18, 2023 at 9:37
  • $\begingroup$ @PavelSafronov$:$ Nice point. Another thing is that let $G$ be a Lie group with Lie algebra $\mathfrak {g}.$ Let $\omega$ be a left-invariant non-degenerate closed $2$-form on $G.$ Then how to show that $\omega (e)$ is a $2$-cocycle on $\mathfrak {g}$ i.e. $$\omega (e) ([a,b], c) + \omega(e) ([b,c], a) + \omega (e) ([c,a], b) = 0$$ for all $a,b,c \in \mathfrak {g},$ where $[\cdot, \cdot]$ is the Lie bracket on $\mathfrak {g}.$ Thanks for your time. $\endgroup$
    – Anil Bagchi.
    Commented Jan 18, 2023 at 18:38

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