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Linear Lie algebra — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Linear_Lie_algebra
License (as archived): CC-BY-SA-4.0
Archived: 2026-10-07T23:32:20.814Z
SHA-256 of the archived text: fcab0a6d0c5e303e021d0d2b1722861e403b010e4ad56f66027b3770b8821248
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Linear Lie algebra — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Linear_Lie_algebra (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_bafa7ee29bfd (Metraipolis archive dc_bafa7ee29bfd, sha256 fcab0a6d0c5e303e021d0d2b1722861e403b010e4ad56f66027b3770b8821248)

BibTeX

@misc{dcbafa7ee29bfd,
  title = {Linear Lie algebra — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Linear_Lie_algebra},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_bafa7ee29bfd; sha256 fcab0a6d0c5e303e021d0d2b1722861e403b010e4ad56f66027b3770b8821248},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Linear_Lie_algebra. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

In algebra, a linear Lie algebra is a subalgebra g {\displaystyle {\mathfrak {g}}} of the Lie algebra g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} consisting of endomorphisms of a vector space V. In other words, a linear Lie algebra is the image of a Lie algebra representation. Any Lie algebra is a linear Lie algebra in the sense that there is always a faithful representation of g {\displaystyle {\mathfrak {g}}} (in fact, on a finite-dimensional vector space by Ado's theorem if g {\displaystyle {\mathfrak {g}}} is itself finite-dimensional.) Let V be a finite-dimensional vector space over a field of characteristic zero and g {\displaystyle {\mathfrak {g}}} a subalgebra of g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} . Then V is semisimple as a module over g {\displaystyle {\mathfrak {g}}} if and only if (i) it is a direct sum of the center and a semisimple ideal and (ii) the elements of the center are diagonalizable (over some extension field).

Tema degli esploratori: matematica e informatica teorica (deposito sistematico; ricerca «linear algebra»).
Fonte: Wikipedia in inglese, voce «Linear Lie algebra» — https://en.wikipedia.org/wiki/Linear_Lie_algebra
Wikidata Q16977077 (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q16977077
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Linear_Lie_algebra?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1348071401, consultata il 2026-10-07T23:11:27.335Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: 40f1b08db4e70cae6d13d8fd04a39d0295f1154b03e94453f45e7434025039d6