Classification of Semisimple Subalgebras of Simple Lie Algebras

Abstract

An explicit classification of the semisimple complex Lie subalgebras of the simple complex Lie algebras is given for algebras up to rank 6. The notion of defining vector, introduced by Dynkin and valid for subalgebras of rank 1, has been extended to the notion of defining matrix, valid for any semisimple subalgebra. All defining matrices have been determined explicitly, which is equivalent to the determination of the embeddings of the generators of the Cartan subalgebra of a semisimple subalgebra in the Cartan subalgebra K of the simple algebra containing this subalgebra. Moreover, the embedding of the root system of the subalgebras in the dual space K* of an algebra is given for all subalgebras. For the S-subalgebras of the simple algebras (up to rank 6), the embedding of the whole subalgebra in an algebra is given explicitly. In addition, the decomposition (branching) of the defining (fundamental) and adjoint representations of an algebra with respect to the restriction to its S-subalgebras has been determined. In the first part of this article a brief review of Dynkin's theory of the classification of the semisimple Lie subalgebras of the simple Lie algebras is given. No proofs are repeated, and at places where concepts have been extended and new results derived, merely an indication for their proof is given. This part of the article will serve as a prescription for a classification of semisimple subalgebras of the simple Lie algebras of rank exceeding 6. Later in the article, explicit expressions are given for the index of an embedding of a simple Lie subalgebra in a simple Lie algebra. These expressions are valid for the classical Lie algebras of arbitrary rank as well as for the exceptional Lie algebras.