Regular decompositions of finite root systems and simple Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic 0. In this paper we classify all regular decompositions of $\mathfrak{g}$ and its irreducible root system $\mathrm{\Delta }$. A regular decomposition is a decomposition $\mathfrak{g}={\mathfrak{g}}_1\oplus \cdots \oplus {\mathfrak{g}}_m$, where each ${\mathfrak{g}}_i$ and ${\mathfrak{g}}_i\oplus {\mathfrak{g}}_j$ are regular subalgebras. Such a decomposition induces a partition of the corresponding root system, i.e. $\mathrm{\Delta }={\mathrm{\Delta }}_1\bigsqcup \cdots \bigsqcup {\mathrm{\Delta }}_m$, such that all ${\mathrm{\Delta }}_i$ and ${\mathrm{\Delta }}_i\bigsqcup {\mathrm{\Delta }}_j$ are closed. Partitions of $\mathrm{\Delta }$ with $m=2$ were known before. In this paper we prove that the case $m\ge 3$ is possible only for systems of type $A_n$ and describe all such partitions in terms of $m$-partitions of $(n+1)$. These results are then extended to a classification of regular decompositions of $\mathfrak{g}$.