Regular decompositions of finite root systems and simple Lie algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916229024841728 |
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| author | Maximov, Stepan |
| author_facet | Maximov, Stepan |
| contents | 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 $Δ$.
A regular decomposition is a decomposition $\mathfrak{g} = \mathfrak{g}_1 \oplus \dots \oplus \mathfrak{g}_m$, 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. $Δ= Δ_1 \sqcup \dots \sqcup Δ_m$, such that all $Δ_i$ and $Δ_i \sqcup Δ_j$ are closed.
Partitions of $Δ$ 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 $\mathfrak{g}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03566 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regular decompositions of finite root systems and simple Lie algebras Maximov, Stepan Rings and Algebras 17B22, 17B05 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 $Δ$. A regular decomposition is a decomposition $\mathfrak{g} = \mathfrak{g}_1 \oplus \dots \oplus \mathfrak{g}_m$, 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. $Δ= Δ_1 \sqcup \dots \sqcup Δ_m$, such that all $Δ_i$ and $Δ_i \sqcup Δ_j$ are closed. Partitions of $Δ$ 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 $\mathfrak{g}$. |
| title | Regular decompositions of finite root systems and simple Lie algebras |
| topic | Rings and Algebras 17B22, 17B05 |
| url | https://arxiv.org/abs/2401.03566 |