Regular subalgebras and nilpotent orbits of real graded Lie algebras

TL;DR

This paper develops algorithms to classify regular semisimple subalgebras and nilpotent orbits in real graded semisimple Lie algebras, extending complex case methods and implementing them in GAP for practical computations.

Contribution

It introduces new algorithms for classifying regular semisimple subalgebras and carrier algebras in real graded semisimple Lie algebras, with implementation details.

Findings

Algorithms successfully classify subalgebras and carrier algebras in real cases.

Implementation in GAP enables practical computations and examples.

Potential applications in differential geometry and physics.

Abstract

For a semisimple Lie algebra over the complex numbers, Dynkin (1952) developed an algorithm to classify the regular semisimple subalgebras, up to conjugacy by the inner automorphism group. For a graded semisimple Lie algebra over the complex numbers, Vinberg (1979) showed that a classification of a certain type of regular subalgebras (called carrier algebras) yields a classification of the nilpotent orbits in a homogeneous component of that Lie algebra. Here we consider these problems for (graded) semisimple Lie algebras over the real numbers. First, we describe an algorithm to classify the regular semisimple subalgebras of a real semisimple Lie algebra. This also yields an algorithm for listing, up to conjugacy, the carrier algebras in a real graded semisimple real algebra. We then discuss what needs to be done to obtain a classification of the nilpotent orbits from that; such classifications have applications in differential geometry and theoretical physics. Our algorithms are implemented in the language of the computer algebra system GAP, using our package CoReLG; we report on example computations.