Knowlpedia

Reductive Lie algebra

A Lie algebra that decomposes as a direct sum of its center and a semisimple ideal.

A finite-dimensional g\mathfrak{g} over a field of characteristic zero is reductive if

g=Z(g)⊕[g,g]\mathfrak{g} = Z(\mathfrak{g}) \oplus [\mathfrak{g}, \mathfrak{g}]

with Z(g)Z(\mathfrak g) its center and [g,g][\mathfrak g,\mathfrak g] a semisimple ideal.

Equivalent characterizations

Under the hypotheses above, the following are equivalent:

  1. g\mathfrak{g} is reductive.
  2. The is completely reducible.
  3. g\mathfrak{g} is a direct sum of simple and abelian Lie algebras.
Examples
  • Semisimple Lie algebras (center is trivial).
  • Abelian Lie algebras (derived algebra is trivial).
  • gln\mathfrak{gl}_n: center is scalar matrices, derived algebra is sln\mathfrak{sl}_n.
  • u(n)=Z(u(n))⊕su(n)\mathfrak{u}(n)=Z(\mathfrak u(n))\oplus\mathfrak{su}(n), where Z(u(n))=iRInZ(\mathfrak u(n))=i\mathbb R I_n.
Non-example

The Lie algebra of upper triangular matrices is not reductive (it is solvable but not semisimple).