Generalizations of Felderʹs elliptic dynamical r-matrices associated with twisted loop algebras of self-dual Lie algebras Original Research Article

A dynamical r-matrix is associated with every self-dual Lie algebra A which is graded by finite-dimensional subspaces as A=⊕n∈ZAn, where An is dual to A−n with respect to the invariant scalar product on A, and A0 admits a nonempty open subset Ǎ0 for which adκ is invertible on An if n≠0 and κ∈Ǎ0. Examples are furnished by taking A to be an affine Lie algebra obtained from the central extension of a twisted loop algebra ℓ(G,μ) of a finite-dimensional self-dual Lie algebra G. These r-matrices, R :Ǎ0→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 Felderʹs elliptic r-matrices by evaluation homomorphisms of ℓ(G,μ) 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.