In mathematics, a Lie algebra is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra is the subalgebra of , denoted
that consists of all linear combinations of Lie brackets of pairs of elements of . The derived series is the sequence of subalgebras
If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable.[1] The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and solvable Lie algebras are analogs of solvable groups.
Any nilpotent Lie algebra is a fortiori solvable but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition. The solvable Lie algebras are precisely those that can be obtained from semidirect products, starting from 0 and adding one dimension at a time.[2]
A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal of a Lie algebra is called the radical.
^Humphreys 1972
^Cite error: The named reference Knapp_1 was invoked but never defined (see the help page).
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