| Solvable Lie Group |
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Information AboutSolvable Lie Group |
| CATEGORIES ABOUT SOLVABLE LIE ALGEBRA | |
| lie algebras | |
| lie groups | |
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: for the ''derived Lie algebra'' of ''g'', generated by the : {Link without Title} for ''x'' and ''y'' in ''g'', the derived series : becomes constant eventually at 0. Any Nilpotent Lie Algebra is solvable, ''a fortiori'', but the converse is not true. The solvable Lie algebras and the Semisimple Lie Algebra s form two large and generally complementary classes, as is shown by the Levi Decomposition . A maximal solvable subalgebra is called a '' Borel Subalgebra ''. The largest solvable Ideal is called the '' Radical ''. SOLVABLE LIE GROUPS The terminology arises from the Solvable Group s of abstract Group Theory . There are several possible definitions of solvable Lie group. For a Lie Group ''G'', there is
To have equivalence one needs to assume ''G'' connected. For connected Lie groups, these definitions are the same, and the derived series of Lie algebras are the Lie algebra of the derived series of (closed) subgroups. SEE ALSO EXTERNAL LINKS |
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