Adjoint Representation

Author

Eric W. Weisstein
April 13, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/AdjointRepresentation.nb.

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

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

Additional Authors

Code written by Todd Rowland
July 25, 2000

Code

The following function gives the adjoint representation of the matrix m in the Lie algebra given by a basis, the list of matrices g.

AdjointRepresentation[g_List,m_List?MatrixQ]:=Transpose[LinearSolve[Transpose[Flatten/@g],Flatten[m.#-#.m]]&/@g]

Examples

e[1]={{1,0},{0,0}};
e[2]={{0,1},{0,0}};
e[3]={{0,0},{1,0}};
e[4]={{0,0},{0,1}};

MatrixForm/@(AdjointRepresentation[Array[e,4],#]&/@Array[e,4])