LieAlgebras[MatrixAlgebras] - create a Lie algebra data structure for a matrix Lie algebra
Calling Sequences
MatrixAlgebras(keyword, n, AlgName)
MatrixAlgebras("subalgebra", tensorList, AlgName)
Parameters
keyword - a keyword string, one of "Full", "Upper", "StrictlyUpper"
n - a positive integer, the dimension of the matrices for the matrix Lie algebra to be created
AlgName - a name or a string, the name of the Lie algebra to be created
tensorList - a list of vectors, differential forms or tensors defined on an n dimensional space
Description
Examples
The set of all real, n x n matrices form a Lie algebra with respect to the Lie bracket defined by the matrix commutator [a, b] = ab - ba. This Lie algebra is usually denoted by gl(n, R). A matrix Lie algebra is simply a subalgebra of gl(n, R). Examples of matrix algebras include: [i] the upper triangular n x n matrices; [ii] the strictly upper triangular n x n matrices; [iii] the trace-free n x n matrices; and [iv] the skew-symmetric n x n matrices. All of these matrix algebras, and many others, can be created with the MatrixAlgebra program.
The Lie algebras of all n x n matrices, the upper triangular n x n matrices, and the strictly upper triangular n x n matrices can be created using the first calling sequence for MatrixAlgebra. The program returns the required Lie algebra data structure and lists of labels e[i, j] for the vectors and epsilon[i, j] for the dual 1-forms for the matrix Lie algebra to be created. Here e[i, j] represents the matrix with a 1 in the i-th row and j-th column and zeros elsewhere.
Other matrix algebras are created as subalgebras of gl(n, R), which are symmetries for a list of prescribed tensors using the second calling sequence for MatrixAlgebra. For example, if T = [t^i_{jk}] is a type (1, 2) tensor on the vector space R^n, then an element a = [a^l_m] of gl(n, R) is a symmetry of T if the equation a^i_m t^m_{jk} - a^l_j t^i_{lk} - a^l_k t^i_{jl} = 0 (sum on l, m) holds. If we introduce coordinates x^i on R^n, then this symmetry condition is the same as the Lie derivative equation L_X (T) = 0, where T = t^i_{jk} partial_{x^i} dx^j dx^k and X is the linear vector field X = a^l_m x^m partial_{x^l}. The MatrixAlgebra program, with the keyword option "subalgebra", creates the matrix subalgebra of gl(n, R), which is the symmetry algebra for all the tensors in the list tensorsList.
The command MatrixAlgebras is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form MatrixAlgebras(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MatrixAlgebras(...).
withDifferentialGeometry:withLieAlgebras:
Example 1.
Create the Lie algebra data structure for the Lie algebra of all 2 x 2 matrices.
L1≔MatrixAlgebrasFull,2,gl2
L1 ≔ e1,e2=e2,e1,e3=−e3,e2,e3=e1−e4,e2,e4=e2,e3,e4=−e3,e11,e12,e21,e22,ε11,ε12,ε21,ε22
DGsetupL1:
Let us check that this result agrees with the direct computation of gl(2) using LieAlgebraData.
M≔Matrix1,0,0,0,Matrix0,1,0,0,Matrix0,0,1,0,Matrix0,0,0,1
L2≔LieAlgebraDataM,newgl2
L2 ≔ e1,e2=e2,e1,e3=−e3,e2,e3=e1−e4,e2,e4=e2,e3,e4=−e3
Example 2.
We create the 6 dimensional Lie algebra of all 3 x 3 Upper triangular matrices. This is the standard example of a solvable algebra.
L3≔MatrixAlgebrasUpper,3,T3
L3 ≔ e1,e2=e2,e1,e3=e3,e2,e4=e2,e2,e5=e3,e3,e6=e3,e4,e5=e5,e5,e6=e5,e11,e12,e13,e22,e23,e33,ε11,ε12,ε13,ε22,ε23,ε33
DGsetupL3:
MultiplicationTableLieTable
LieAlgebraCheckT3,Solvable
Example 3.
We create the 8 dimensional Lie algebra of all 3 x 3 trace-free matrices. This is the classical matrix algebra sl(3, R). It is simple (i.e. semisimple and indecomposable). First we create the Lie algebra of all 3 x 3 matrices.
L3≔MatrixAlgebrasFull,3,gl3
L3 ≔ e1,e2=e2,e1,e3=e3,e1,e4=−e4,e1,e7=−e7,e2,e4=e1−e5,e2,e5=e2,e2,e6=e3,e2,e7=−e8,e3,e4=−e6,e3,e7=e1−e9,e3,e8=e2,e3,e9=e3,e4,e5=−e4,e4,e8=−e7,e5,e6=e6,e5,e8=−e8,e6,e7=e4,e6,e8=e5−e9,e6,e9=e6,e7,e9=−e7,e8,e9=−e8,e11,e12,e13,e21,e22,e23,e31,e32,e33,ε11,ε12,ε13,ε21,ε22,ε23,ε31,ε32,ε33
Now define an auxiliary 3 dimensional space, call it R3. Choose any coordinates labels.
DGsetupx,y,z,R3:
Define the standard volume form on R3.
ν≔dx&wedgedy&wedgedz
ν ≔ dx`^`dy`^`dz
Find the subalgebra of gl(3) which preserves this volume form.
SL3≔MatrixAlgebrasSubalgebra,gl3,ν,sl3
SL3 ≔ e11−e33,e12,e13,e21,e22−e33,e23,e31,e32
Note that each of the matrices represented by the elements of the list SL3 are trace-free.
L4≔LieAlgebraDataSL3,sl3
L4 ≔ e1,e2=e2,e1,e3=2e3,e1,e4=−e4,e1,e6=e6,e1,e7=−2e7,e1,e8=−e8,e2,e4=e1−e5,e2,e5=e2,e2,e6=e3,e2,e7=−e8,e3,e4=−e6,e3,e5=−e3,e3,e7=e1,e3,e8=e2,e4,e5=−e4,e4,e8=−e7,e5,e6=2e6,e5,e7=−e7,e5,e8=−2e8,e6,e7=e4,e6,e8=e5
DGsetupL4
Lie algebra: sl3
Querysl3,Semisimple,Querysl3,Indecomposable
true,true
Example 4.
We create the 6 dimensional Lie algebra of all 4 x 4 skew-symmetric matrices. This is the classical matrix algebra so(4, R). It is semisimple but not simple (that is, it is decomposable). First we create the Lie algebra of all 4 x 4 matrices.
L5≔MatrixAlgebrasFull,4,gl4:
DGsetupL5:
Now define an auxiliary 4 dimensional space, call it R4. Choose any coordinates labels.
DGsetupx1,x2,x3,x4,R5:
Define the standard Euclidean metric tensor on R4.
g≔evalDGdx1&tdx1+dx2&tdx2+dx3&tdx3+dx4&tdx4
g ≔ dx1dx1+dx2dx2+dx3dx3+dx4dx4
SO4≔MatrixAlgebrasSubalgebra,gl4,g,sl3
SO4 ≔ e12−e21,e13−e31,e14−e41,e23−e32,e24−e42,e34−e43
Note that each of the matrices represented by the elements of the list SO4 are skew-symmetric.
L6≔LieAlgebraDataSO4,so4
L6 ≔ e1,e2=−e4,e1,e3=−e5,e1,e4=e2,e1,e5=e3,e2,e3=−e6,e2,e4=−e1,e2,e6=e3,e3,e5=−e1,e3,e6=−e2,e4,e5=−e6,e4,e6=e5,e5,e6=−e4
DGsetupL6:
MultiplicationTableso4,LieBracket
e1,e2=−e4,e1,e3=−e5,e1,e4=e2,e1,e5=e3,e2,e3=−e6,e2,e4=−e1,e2,e6=e3,e3,e5=−e1,e3,e6=−e2,e4,e5=−e6,e4,e6=e5,e5,e6=−e4
Queryso4,Semisimple,Queryso4,Indecomposable
true,false
See Also
DifferentialGeometry
LieAlgebras
LieAlgebraData
MultiplicationTable
Query
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