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Generalized Onsager Algebras

  • Jasper V. StokmanEmail author
Open Access
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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 \(\mathfrak {k}(A)\).

Keywords

Kac-Moody algebra Chevalley involution Fix-point Lie subalgebra Onsager algebra Dolan-Grady relations Inhomogeneous Serre relations Split symmetric pairs 

Mathematics Subject Classification (2010)

17B67 81R10 

Notes

Acknowledgments

I thank Stefan Kolb, Pascal Baseilhac, Samuel Belliard and Nicolai Reshetikhin for useful discussions on the topic of the paper. Shortly after the appearance on the arXiv of this paper Xinhong Chen, Ming Lu and Weiqiang Wang informed me that they have obtained an explicit presentation of Kolb’s coideal subalgebra associated to a quasi-split Kac-Moody quantum symmetric pair using different techniques. Their results have appeared now in the preprint [3].

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Copyright information

© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

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