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Difference between revisions of "Engel algebra"

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An associative algebra or Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035670/e0356701.png" /> satisfying the Engel condition: For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035670/e0356702.png" /> the inner derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035670/e0356703.png" /> (cf. [[Derivation in a ring|Derivation in a ring]]) is nilpotent. In other words, all elements of an Engel algebra are Engel elements (cf. [[Engel element|Engel element]], see also [[Lie algebra, nil|Lie algebra, nil]]).
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An [[associative algebra]] or [[Lie algebra]] $\mathfrak{g}$ satisfying the Engel condition: For every $X \in \mathfrak{g}$ the inner derivation $\mathrm{ad}\,X$ (cf. [[Derivation in a ring]]) is nilpotent. In other words, all elements of an Engel algebra are [[Engel element]]s, see also [[Lie algebra, nil]]).
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Latest revision as of 18:39, 16 October 2016

An associative algebra or Lie algebra $\mathfrak{g}$ satisfying the Engel condition: For every $X \in \mathfrak{g}$ the inner derivation $\mathrm{ad}\,X$ (cf. Derivation in a ring) is nilpotent. In other words, all elements of an Engel algebra are Engel elements, see also Lie algebra, nil).

How to Cite This Entry:
Engel algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Engel_algebra&oldid=39421
This article was adapted from an original article by Yu.A. Bakhturin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article