TY - GEN

T1 - Pseudo-symmetric pairs for Kac-Moody algebras

AU - Regelskis, Vidas

AU - Vlaar, Bart

N1 - Funding Information:
17B37, 20F55. Key words and phrases. Kac-Moody algebras, automorphism group, symmetric pairs, restricted Weyl group. The second-named author was supported by the UK Engineering and Physical Sciences Research Council, grant number EP/R009465/1. 1We always work over an algebraically closed field of characteristic 0. Lie algebra automorphisms are always understood to fix this field pointwise.
Publisher Copyright:
© 2022 American Mathematical Society.

PY - 2022/8/31

Y1 - 2022/8/31

N2 - 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.

AB - 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.

KW - automorphism group

KW - Kac-Moody algebras

KW - restricted Weyl group

KW - symmetric pairs

UR - http://www.scopus.com/inward/record.url?scp=85137998601&partnerID=8YFLogxK

UR - https://arxiv.org/abs/2108.00260

U2 - 10.1090/conm/780/15690

DO - 10.1090/conm/780/15690

M3 - Conference contribution

AN - SCOPUS:85137998601

SN - 9781470465209

T3 - Contemporary Mathematics

SP - 155

EP - 203

BT - Hypergeometry, Integrability and Lie Theory - Virtual Conference Hypergeometry, Integrability and Lie Theory, 2020

A2 - Koelink, Erik

A2 - Kolb, Stefan

A2 - Reshetikhin, Nicolai

A2 - Vlaar, Bart

PB - American Mathematical Society

T2 - Virtual conference on Hypergeometry, Integrability and Lie Theory, 2020

Y2 - 7 December 2020 through 11 December 2020

ER -