Root Systems for Levi Factors and Borel-de Siebenthal Theory

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dc.creatorKostant, Bertram
dc.date2007-11-18
dc.date2008-06-13
dc.date.accessioned2026-07-07T09:44:04Z
dc.date.available2026-07-07T09:44:04Z
dc.descriptionLet $\frak{m}$ be a Levi factor of a proper parabolic subalgebra $\frak{q}$ of a complex semisimple Lie algebra $\frak{g}$. Let $\frak{t} = cent \frak{m}$. A nonzero element $ν\in \frak{t}^*$ is called a $\frak {t}$-root if the corresponding adjoint weight space $\frak{g}_{nu}$ is not zero. If $ν$ is a $\frak{t}$-root, some time ago we proved that $\frak{g}_ν$ is $ad \frak{m}$ irreducible. Based on this result we develop in the present paper a theory of $\frak{t}$-roots which replicates much of the structure of classical root theory (case where $\frak{t}$ is a Cartan subalgebra). The results are applied to obtain new reults about the structure of the nilradical $\frak{n}$ of $\frak{q}$. Also applications in the case where $dim \frak{t}=1$ are used in Borel-de Siebenthal theory to determine irreducibility theorems for certain equal rank subalgebras of $\frak{g}$. In fact the irreducibility results readily yield a proof of the main assertions of the Borel-de Siebenthal theory.
dc.description28 pages, plain tex
dc.identifierhttps://arxiv.org/abs/0711.2809
dc.identifierhttp://arxiv.org/abs/0711.2809
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/162740
dc.subjectRepresentation Theory
dc.subjectGroup Theory
dc.subject20CXX, 20Gxx, 22EXX, 22E10, 22E25, 22E46
dc.titleRoot Systems for Levi Factors and Borel-de Siebenthal Theory
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