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 A0 for which ad κ 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 Felder's 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.
- Classical Yang-Baxter equation
- Dynamical r-matrix
- Self-dual Lie algebra
ASJC Scopus subject areas
- Nuclear and High Energy Physics