Pseudo-symmetric pairs for Kac-Moody algebras

Regelskis, Vidas and Vlaar, Bart (2022) Pseudo-symmetric pairs for Kac-Moody algebras. In: Contemporary Mathematics. Virtual Conference Hypergeometry, Integrability and Lie Theory, 2020 :. Contemporary Mathematics, 780 . American Mathematical Society, NLD, pp. 155-203. ISBN 978-1-4704-6520-9
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Lie algebra involutions and their fixed-point subalgebras give rise to symmetric spaces and real forms of complex Lie algebras, and are wellstudied in the context of symmetrizable Kac-Moody algebras. In this paper we study a generalization. Namely, we introduce the concept of a pseudoinvolution, an automorphism which is only required to act involutively on a stable Cartan subalgebra, and the concept of a pseudo-fixed-point subalgebra, a natural substitute for the fixed-point subalgebra. In the symmetrizable KacMoody setting, we give a comprehensive discussion of pseudo-involutions of the second kind, the associated pseudo-fixed-point subalgebras, restricted root systems and Weyl groups, in terms of generalizations of Satake diagrams.


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