Generalized Onsager algebras

TL;DR

This paper provides an explicit presentation of the fixed-point subalgebra of Kac-Moody algebras under the Chevalley involution, generalizing Onsager algebra structures and computing structure constants in specific cases.

Contribution

It introduces a new presentation of the fix-point Lie subalgebra involving inhomogeneous Serre relations, extending the Onsager algebra framework.

Findings

Explicit presentation of $rak{k}(A)$ for symmetrizable Kac-Moody algebras.

Computed structure constants for finite and untwisted affine cases.

Described one-dimensional representations for symplectic Lie algebra cases.

Abstract

Let $\mathfrak{g}(A)$ 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 $\mathfrak{k}(A)$ of $\mathfrak{g}(A)$ with respect to the Chevalley involution. It is a presentation of $\mathfrak{k}(A)$ 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 $\mathfrak{k}(A)$ in terms of a Chevalley type basis of $\mathfrak{k}(A)$. For the symplectic Lie algebra and its untwisted affine extension we explicitly describe the one-dimensional representations of $\mathfrak{k}(A)$.