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# Nilradical of a Lie algebra

In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. [1]

## Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

## Exact sequence

An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry.

## Levi decomposition

In Lie theory and representation theory, the Levi decomposition, conjectured by Killing and Cartan and proved by, states that any finite-dimensional real Lie algebra g is the semidirect product of a solvable ideal and a semisimple subalgebra.

## Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

## Nilpotent ideal

In mathematics, more specifically ring theory, an ideal, I, of a ring is said to be a nilpotent ideal, if there exists a natural number k such that Ik.

In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements of the ring.

## Radical of a Lie algebra

In the mathematical field of Lie theory, the radical of a Lie algebra \mathfrak is the largest solvable ideal of \mathfrak.

## Reductive Lie algebra

In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name.

## References

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