Definitions
A Lie algebra over a field of characteristic 0 is called reductive if any of the following equivalent conditions are satisfied:
- The adjoint representation (the action by bracketing) of is completely reducible (a direct sum of irreducible representations).
- admits a faithful, completely reducible, finite-dimensional representation.
- The radical of equal the center:
- The radical always contains the center, but need not equal it.
- is the direct sum of its derived ideal and its center
- is the direct sum of a semisimple ideal and its center
- Compare to the Levi decomposition, which decomposes a Lie algebra as its radical (which is solvable, not abelian in general) and a Levi subalgebra (which is semisimple).
- is a direct sum of a semisimple Lie algebra and an abelian Lie algebra
- is a direct sum of prime ideals:
Some of these equivalences are easily seen. For example, the center and radical of is while if the radical equals the center the Levi decomposition yields a decomposition Further, simple Lie algebras and the trivial 1-dimensional Lie algebra are prime ideals.
Read more about this topic: Reductive Lie Algebra
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