A note on a canonical dynamical r-matrix

Abstract

It is well known that a classical dynamical $r$-matrix can be associated with every finite-dimensional self-dual Lie algebra $\G$ by the definition $R(\omega ):=f(\mathrm{ad}\omega )$, where $ω\in \G$ and $f$ is the holomorphic function given by $f(z)=1/2\coth \frac{z}{2}-\frac{1}{z}$ for $z\in \C\setminus 2πi \Z^*$. We present a new, direct proof of the statement that this canonical $r$-matrix satisfies the modified classical dynamical Yang-Baxter equation on $\G$.

17 pages, LaTeX2e

References in corpus (2)

• On dynamical r-matrices obtained from Dirac reduction and their generalizations to affine Lie algebras
• Dynamical r-matrices and the chiral WZNW phase space

Cited by in corpus (3)

• Generalizations of Felder's elliptic dynamical r-matrices associated with twisted loop algebras of self-dual Lie algebras
• On a Poisson-Lie analogue of the classical dynamical Yang-Baxter equation for self-dual Lie algebras
• The non-Abelian momentum map for Poisson-Lie symmetries on the chiral WZNW phase space