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Quasi-split group

Linear algebraic group

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Record originEnglish Wikipedia
Text licenseCC BY-SA 4.0
Source revisionMay 17, 2023
Entity authorityQ7269474 ↗
Source-derived summary

In mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field correspond to actions of the absolute Galois group on a Dynkin diagram.

Examples

All split groups (those with a split maximal torus) are quasi-split. These correspond to quasi-split groups where the action of the Galois group on the Dynkin diagram is trivial.

Lang (1956) showed that all simple algebraic groups over finite fields are quasi-split.

Over the real numbers, the quasi-split groups include the split groups and the complex groups, together with the orthogonal groups On,n+2, the unitary groups SUn,n and SUn,n+1, and the form of E6 with signature 2.

Editorial summary

Begin with the source’s own compact description: “Quasi-split group” is linear algebraic group. The dossier treats that line as a proposition to test through Quasi-split, group and Linear, not as a finished interpretation.

Editorial reviewA practical starting point whose main value is the path it opens into stronger specialist and primary sources. The current lead gives the account dated anchors—1956—that can be checked directly. The selected authority fields contribute no independent date. For this dossier, Quasi-split, group and Linear is the immediate research focus.
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The phrase “linear algebraic group” supplies a clear boundary for inquiry. It also exposes the unanswered questions: who defined that boundary, when it became stable and which sources sit outside it.

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Named sources, stable identifiers and responsible institutions provide the strongest route from overview to verifiable evidence. The source revision retrieved here is dated May 17, 2023. The linked authority identifier is Q7269474. None of the 0 selected statements returned an explicit reference. The first chronological checks are 1956.

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Source & attribution

This entry incorporates text from “Quasi-split group” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.