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The lower central series of a Lie algebra g is the sequence of subalgebras recursively defined by g_(k+1)=[g,g_k], (1) with g_0=g. The sequence of subspaces is always ...

A Lie algebra over a field of characteristic zero is called semisimple if its Killing form is nondegenerate. The following properties can be proved equivalent for a ...

The commutator series of a Lie algebra g, sometimes called the derived series, is the sequence of subalgebras recursively defined by g^(k+1)=[g^k,g^k], (1) with g^0=g. The ...

The word "algebra" is a distortion of the Arabic title of a treatise by al-Khwārizmī about algebraic methods. In modern usage, algebra has several meanings. One use of the ...

If g is a Lie algebra, then a subspace a of g is said to be a Lie subalgebra if it is closed under the Lie bracket. That is, a is a Lie subalgebra of g if for all x,y in a, ...

A nilpotent Lie group is a Lie group G which is connected and whose Lie algebra is a nilpotent Lie algebra g. That is, its Lie algebra lower central series ...

The infinitesimal algebraic object associated with a Lie groupoid. A Lie algebroid over a manifold B is a vector bundle A over B with a Lie algebra structure [,] (Lie ...

A solvable Lie group is a Lie group G which is connected and whose Lie algebra g is a solvable Lie algebra. That is, the Lie algebra commutator series ...

A Lie group is called semisimple if its Lie algebra is semisimple. For example, the special linear group SL(n) and special orthogonal group SO(n) (over R or C) are ...