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Ideals of Lie algebras

DEFINITION 5.1   For a linear subspace $ S,T$ of a Lie algebra $ L$ , let us define denote by $ [S,T]$ the following linear subspace of $ L$ .
  1. $\displaystyle [S,T]=($linear span of $\displaystyle \{[x,y]; x\in S, y\in T\})
$

DEFINITION 5.2   Let $ L$ be a Lie algebra over a field $ k$ . A $ k$ -linear subspace $ \mathfrak{a}$ of $ L$ is said to be an ideal of $ L$ if

$\displaystyle [x,y]\in \mathfrak{a}\qquad(\forall x\in L,\forall y\in \mathfrak{a})
$

holds. This clearly is equivalent to saying that

$\displaystyle [L,\mathfrak{a}]\subset L
$

holds.

PROPOSITION 5.3   Let $ \mathfrak{a}$ be an ideal of a Lie algebra $ L$ . Then
  1. $ \mathfrak{a}$ is a sub $ L$ -module (sub representation) of $ L$ .
  2. $ L/\mathfrak{a}$ caries a natural structure of a Lie algebra.

PROOF.. As usual. $ \qedsymbol$



2007-12-19