Generalizations of Felder's elliptic dynamical r-matrices associated with twisted loop algebras of self-dual Lie algebras

Abstract

A dynamical $r$-matrix is associated with every self-dual Lie algebra $\A$ which is graded by finite-dimensional subspaces as $\A=\oplus_{n \in \cZ} \A_n$, where $\A_n$ is dual to $\A_{-n}$ with respect to the invariant scalar product on $\A$, and $\A_0$ admits a nonempty open subset $\check \A_0$ for which $\ad \kappa$ is invertible on $\A_n$ if $n\neq 0$ and $\kappa \in \check \A_0$. Examples are furnished by taking $\A$ to be an affine Lie algebra obtained from the central extension of a twisted loop algebra $\ell(\G,\mu)$ of a finite-dimensional self-dual Lie algebra $\G$. These $r$-matrices, $R: \check \A_0 \to \mathrm{End}(\A)$, yield generalizations of the basic trigonometric dynamical $r$-matrices that, according to Etingof and Varchenko, are associated with the Coxeter automorphisms of the simple Lie algebras, and are related to Felder's elliptic $r$-matrices by evaluation homomorphisms of $\ell(\G,\mu)$ into $\G$. The spectral-parameter-dependent dynamical $r$-matrix that corresponds analogously to an arbitrary scalar-product-preserving finite order automorphism of a self-dual Lie algebra is here calculated explicitly.