Dynamical r-matrices on the affinizations of arbitrary self-dual lie algebras*)


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Abstract

We associate a dynamical r-matrix with any such subalgebra ℒ of a finite dimensional self-dual Lie algebra 𝒜 for which the scalar product of 𝒜 remains nondegenerate on ℒ and there exists a nonempty open subset Symbol ⊂ ℒ so that the restriction of (ad λ) ∈ End (𝒜) to ℒ is invertible ∀λ ∈ Symbol. This r-matrix is also well-defined if ℒ is the grade zero subalgebra of an affine Lie algebra 𝒜 obtained from a twisted loop algebra based on a finite dimensional self-dual Lie algebra 𝒢. Application of evaluation homomorphisms to the twisted loop algebras yields spectral parameter dependent 𝒢 ⊗ 𝒢-valued dynamical r-matrices that are generalizations of Felder's elliptic r-matrices.

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