On Stable Cartan Subgroups in Lie Algebras

Kuparala Venkata Vidyasagar; P. Mangamma

doi:https://doi.org/10.14445/22315373/IJMTT-V71I8P105

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 71 | Issue 8 | Year 2025 | Article Id. IJMTT-V71I8P105 | DOI : https://doi.org/10.14445/22315373/IJMTT-V71I8P105

On Stable Cartan Subgroups in Lie Algebras

Kuparala Venkata Vidyasagar, P. Mangamma

Received	Revised	Accepted	Published
16 Jun 2025	25 Jul 2025	13 Aug 2025	30 Aug 2025

Citation :

Kuparala Venkata Vidyasagar, P. Mangamma, "On Stable Cartan Subgroups in Lie Algebras," International Journal of Mathematics Trends and Technology (IJMTT), vol. 71, no. 8, pp. 24-33, 2025. Crossref, https://doi.org/10.14445/22315373/IJMTT-V71I8P105

Abstract

This paper investigates 𝛤-stable Cartan subgroups in connected Lie groups and their associated Lie algebras. We extend the results of Borel and Mostow on the existence of automorphism-invariant Cartan subalgebras to the group setting. For a real semisimple Lie algebra 𝔤, we prove that there exists a nonidentity automorphism that fixes representatives of all conjugacy classes of Cartan subalgebras. Explicit constructions for classical Lie algebras (𝐴𝑛, 𝐵𝑛, 𝐶𝑛, 𝐷𝑛) are provided. Applications include characterizing stable Cartan subgroups in quotients and normal subgroups.

Keywords

Cartan subgroups, Automorphism-invariant subgroups, Admissible root systems, Classical Lie algebras.

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