Generalized Onsager Algebras

Let g(A) be the Kac-Moody algebra with respect to a symmetrizable generalized Cartan matrix A. We give an explicit presentation of the fix-point Lie subalgebra k(A) of g(A) with respect to the Chevalley involution. It is a presentation of k(A) involving inhomogeneous versions of the Serre relations, or, from a different perspective, a presentation generalizing the Dolan-Grady presentation of the Onsager algebra. In the finite and untwisted affine case we explicitly compute the structure constants of k(A) in terms of a Chevalley type basis of k(A). For the symplectic Lie algebra and its untwisted affine extension we explicitly describe the one-dimensional representations of k(A).