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 \(\mathfrak {k}(A)\).

中文翻译：

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广义Onsager代数

令\（\ mathfrak {g}（A）\）是关于对称对称的广义Cartan矩阵A的Kac-Moody代数。对于Chevalley对合，我们给出\（\ mathfrak {g}（A）\）的不动点Lie子代数\（\ mathfrak {k}（A）\）。它是\（\ mathfrak {k}（A）\）的表示形式，涉及Serre关系的不均匀版本，或者从另一个角度来看，是概括Onsager代数的Dolan-Grady表示形式的表示形式。在有限和无捻仿射情况下，我们明确地计算的结构常数\（\ mathfrak {K}（A）\）中的Chevalley型物计\（\ mathfrak {K}（A）\）。对于辛李代数及其不加捻的仿射扩展，我们明确描述\（\ mathfrak {k}（A）\）的一维表示。