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Lie Algebra Lower Central Series


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 decreasing with respect to inclusion or dimension, and becomes stable when g is finite dimensional. The notation [a,b] means the linear span of elements of the form [A,B], where A in a and B in b.

When the lower central series ends in the zero subspace, the Lie algebra is called nilpotent. For example, consider the Lie algebra of strictly upper triangular matrices, then

g_0=[0 a_(12) a_(13) a_(14) a_(15); 0 0 a_(23) a_(24) a_(25); 0 0 0 a_(34) a_(35); 0 0 0 0 a_(45); 0 0 0 0 0]
(2)
g_1=[0 0 a_(13) a_(14) a_(15); 0 0 0 a_(24) a_(25); 0 0 0 0 a_(35); 0 0 0 0 0; 0 0 0 0 0]
(3)
g_2=[0 0 0 a_(14) a_(15); 0 0 0 0 a_(25); 0 0 0 0 0; 0 0 0 0 0; 0 0 0 0 0]
(4)
g_3=[0 0 0 0 a_(15); 0 0 0 0 0; 0 0 0 0 0; 0 0 0 0 0; 0 0 0 0 0],
(5)

and g_4=0. By definition, g^k subset g_k, where g^k is the term in the Lie algebra commutator series, as can be seen by the example above.

In contrast to the nilpotent Lie algebras, the semisimple Lie algebras have a constant lower central series. Others are in between, e.g.,

 [gl_n,gl_n]=sl_n,
(6)

which is semisimple, because the matrix trace satisfies

 Tr(AB)=Tr(BA).
(7)

Here, gl_n is a general linear Lie algebra and sl_n is the special linear Lie algebra.


See also

Group Lower Central Series, Lie Algebra, Lie Algebra Commutator Series, Lie Algebra Representation, Lie Group, Nilpotent Lie Group, Nilpotent Lie Group Representation, Unipotent

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Lie Algebra Lower Central Series." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LieAlgebraLowerCentralSeries.html

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