Lie Algebra Lower Central Series

Author

Todd Rowland
Wolfram Research, Inc.
July 12, 2000

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/LieAlgebraLowerCentralSeries.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LieAlgebraLowerCentralSeries.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Example

Here are some Mathematica functions for determining the lower central series, when given a list of matrices which is a basis for 𝔤.

MatrixBasis[a_List]:=Partition[#1,Length[a[[1]]]]&/@LatticeReduce[Flatten/@a]
LieCommutator[a_,b_]:=a.b-b.a

NextLCS[gold_List,{}]={};
NextLCS[gold_List,g_List]:=MatrixBasis[Flatten[Outer[LieCommutator,gold,g,1],1]]

kthLCS[g_List,k_Integer]:=Nest[NextLCS[g,#1]&,g,k]

For example,

gl5=Flatten[Table[ReplacePart[Table[0,{i,5},{j,5}],1,{k,l}],{k,5},{l,5}],1];
sl5=kthLCS[gl5,1];

MatrixForm/@sl5