Reductive Lie algebra

In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, whence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: ${\displaystyle {\mathfrak {g}}={\mathfrak {s}}\oplus {\mathfrak {a}};}$ there are alternative characterizations, given below.

The most basic example is the Lie algebra ${\displaystyle {\mathfrak {gl}}_{n}}$ of ${\displaystyle n\times n}$ matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an n-dimensional vector space, ${\displaystyle {\mathfrak {gl}}(V).}$ This is the Lie algebra of the general linear group GL(n), and is reductive as it decomposes as ${\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl}}_{n}\oplus {\mathfrak {k}},}$ corresponding to traceless matrices and scalar matrices.

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