In mathematics, a Lie algebra is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras
We write , and for all . If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nilpotent. The lower central series for Lie algebras is analogous to the lower central series in group theory, and nilpotent Lie algebras are analogs of nilpotent groups.
The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras, by successive central extensions.
Note that the definition means that, viewed as a non-associative non-unital algebra, a Lie algebra is nilpotent if it is nilpotent as an ideal.
and 25 Related for: Nilpotent Lie algebra information
In mathematics, a Liealgebra g {\displaystyle {\mathfrak {g}}} is nilpotent if its lower central series terminates in the zero subalgebra. The lower central...
of the algebra is zero. The concept of a nilpotentLiealgebra has a different definition, which depends upon the Lie bracket. (There is no Lie bracket...
in the classification of Lie groups. Analogous terms are used for Liealgebras (using the Lie bracket) including nilpotent, lower central series, and...
mathematics, a Liealgebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an...
In algebra, the nilradical of a Liealgebra is a nilpotent ideal, which is as large as possible. The nilradical n i l ( g ) {\displaystyle {\mathfrak...
{n}}} is the Liealgebra of the Lie group of unitriangular matrices. In fact, by Engel's theorem, any finite-dimensional nilpotentLiealgebra is conjugate...
and nilpotent and solvable subgroups of smaller dimension. The diffeomorphism group of a Lie group acts transitively on the Lie group Every Lie group...
subalgebra, often abbreviated as CSA, is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Liealgebra g {\displaystyle {\mathfrak {g}}} that...
exponential map of any nilpotentLiealgebra is a diffeomorphism between the Liealgebra and the unique associated connected, simply-connected Lie group. This discussion...
mathematical theories of Lie groups and Liealgebras. For the topics in the representation theory of Lie groups and Liealgebras, see Glossary of representation...
ring unit (ring theory), Idempotent, Nilpotent, Zero divisor Characteristic (algebra) Ring homomorphism, Algebra homomorphism Ring epimorphism Ring monomorphism...
mathematics, a Liealgebra is semisimple if it is a direct sum of simple Liealgebras. (A simple Liealgebra is a non-abelian Liealgebra without any non-zero...
In mathematics, the special linear Liealgebra of order n over a field F {\displaystyle F} , denoted s l n F {\displaystyle {\mathfrak {sl}}_{n}F} or...
and nonzero degree one component) always contains nilpotent elements. A Z-graded anticommutative algebra with the property that x2 = 0 for every element...
nilpotent ideals, and if the algebra is finite-dimensional, the radical itself is a nilpotent ideal. A finite-dimensional algebra is then said to be semisimple...
P of a Lie group as the product of semisimple, abelian, and nilpotent subgroups. The Levi decomposition writes a finite dimensional Liealgebra as a semidirect...
of representation theory, a Liealgebra representation or representation of a Liealgebra is a way of writing a Liealgebra as a set of matrices (or endomorphisms...
Equivalently, a Liealgebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every toral Liealgebra is abelian;...
mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings...
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