paper

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

arXiv:math/0109132 · doi:10.1016/S0550-3213(01)00609-5

Abstract

A dynamical -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 $\ad κ$ is invertible on $\A_n$ if and $κ\in \check \A_0$. Examples are furnished by taking $\A$ to be an affine Lie algebra obtained from the central extension of a twisted loop algebra $\ell(\G,μ)$ of a finite-dimensional self-dual Lie algebra $\G$. These -matrices, $R: \check \A_0 \to \mathrm{End}(\A)$, yield generalizations of the basic trigonometric dynamical -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 -matrices by evaluation homomorphisms of $\ell(\G,μ)$ into $\G$. The spectral-parameter-dependent dynamical -matrix that corresponds analogously to an arbitrary scalar-product-preserving finite order automorphism of a self-dual Lie algebra is here calculated explicitly.

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