Constructing semisimple subalgebras of semisimple Lie algebras

Abstract

Algorithms are described that help with obtaining a classification of the semisimple subalgebras of a given semisimple Lie algebra, up to linear equivalence. The algorithms have been used to obtain classifications of the semisimple subalgebras of the simple Lie algebras of ranks 8. These have been made available as a database inside the SLA package of GAP4. The subalgebras in this database are explicitly given, as well as the inclusion relations among them.

Subalgebras of the rank two semisimple Lie algebras

“…Semisimple subalgebras of semisimple Lie algebras have been extensively studied [dGr11,Dyn52a,Dyn52b,LG72,Min06]. For instance, the semisimple subalgebras of the exceptional Lie algebras have been classified, up to inner automorphism [Min06].…”

“…For instance, the semisimple subalgebras of the exceptional Lie algebras have been classified, up to inner automorphism [Min06]. As another important example, de Graaf [dGr11] classified the semisimple subalgebras of the simple Lie algebras of ranks ≤ 8. In particular, the semisimple subalgebras of the rank 2 simple Lie algebras have been known for some time.…”

Levi decomposable algebras in the classical Lie algebras

“…Then [15,Theorem 3].) Let g and h be semisimple, ϕ i : h → g, i = 1, 2, two embeddings such that ϕ 1 (h) and ϕ 2 (h) are regular subalgebras of g. Then [9,10,15].) Up to inner automorphism, there is exactly one subalgebra isomorphic to…”

“…The works of Dynkin [6] and Minchenko [15] combine to classify the semisimple subalgebras of the complex exceptional Lie algebras, up to inner automorphism. Willem de Graaf [10] classified the semisimple subalgebras of all simple Lie algebras up to rank 8; this classification is up to linear equivalence, which is somewhat weaker than a classification up to inner automorphism.…”

The subalgebras of the rank two symplectic Lie algebra

“…The aim of the current paper is to complete the classification of subalgebras of the rank 2, symplectic Lie algebra sp(4, C)-the remaining rank 2, classical, semisimple Lie algebra whose subalgebras have not been classified. The semisimple subalgebras of sp(4, C) are well-known [dGr11], and the authors recently classified its Levi decomposable subalgebras [DR15].…”