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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

Description

• 

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(...).

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

Create the Lie algebra data structure for the Lie algebra of all 2 x 2 matrices.

L1MatrixAlgebrasFull,2,gl2

L1:=e1,e2=e2,e1,e3=e3,e2,e3=e1e4,e2,e4=e2,e3,e4=e3,e11,e12,e21,e22,ϵ11,ϵ12,ϵ21,ϵ22

(2.1)

DGsetupL1:

 

Let us check that this result agrees with the direct computation of gl(2) using LieAlgebraData.

gl2 > 

MMatrix1,0,0,0,Matrix0,1,0,0,Matrix0,0,1,0,Matrix0,0,0,1

M:=1000,0100,0010,0001

(2.2)
gl2 > 

L2LieAlgebraDataM,newgl2

L2:=e1,e2=e2,e1,e3=e3,e2,e3=e1e4,e2,e4=e2,e3,e4=e3

(2.3)

 

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.

gl2 > 

L3MatrixAlgebrasUpper,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

(2.4)
gl2 > 

DGsetupL3:

T3 > 

MultiplicationTableLieTable

`|`e11e12e13e22e23e33---------------------------e11`|`0e12e13000e12`|`e1200e12e130e13`|`e130000e13e22`|`0e1200e230e23`|`0e130e230e23e33`|`00e130e230

(2.5)
T3 > 

LieAlgebraCheckT3,Solvable

LieAlgebraCheckT3,Solvable

(2.6)

 

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.

T3 > 

L3MatrixAlgebrasFull,3,gl3

L3:=e1,e2=e2,e1,e3=e3,e1,e4=e4,e1,e7=e7,e2,e4=e1e5,e2,e5=e2,e2,e6=e3,e2,e7=e8,e3,e4=e6,e3,e7=e1e9,e3,e8=e2,e3,e9=e3,e4,e5=e4,e4,e8=e7,e5,e6=e6,e5,e8=e8,e6,e7=e4,e6,e8=e5e9,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

(2.7)
T3 > 

DGsetupL3:

 

Now define an auxiliary 3 dimensional space, call it R3.  Choose any coordinates labels.

gl3 > 

DGsetupx,y,z,R3:

 

Define the standard volume form on R3.

R3 > 

νdx&wedgedy&wedgedz

ν:=dx`^`dy`^`dz

(2.8)

 

Find the subalgebra of gl(3) which preserves this volume form.

R3 > 

SL3MatrixAlgebrasSubalgebra,gl3,ν,sl3

SL3:=e11e33,e12,e13,e21,e22e33,e23,e31,e32

(2.9)

 

Note that each of the matrices represented by the elements of the list SL3 are trace-free.

gl3 > 

L4LieAlgebraDataSL3,sl3

L4:=e1,e2=e2,e1,e3=2e3,e1,e4=e4,e1,e6=e6,e1,e7=2e7,e1,e8=e8,e2,e4=e1e5,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

(2.10)
gl3 > 

DGsetupL4

Lie algebra: sl3

(2.11)
sl3 > 

Querysl3,Semisimple,Querysl3,Indecomposable

true,true

(2.12)

 

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.

sl3 > 

L5MatrixAlgebrasFull,4,gl4:

sl3 > 

DGsetupL5:

 

Now define an auxiliary 4 dimensional space, call it R4.  Choose any coordinates labels.

gl4 > 

DGsetupx1,x2,x3,x4,R5:

 

Define the standard Euclidean metric tensor on R4.

R5 > 

gevalDGdx1&tdx1+dx2&tdx2+dx3&tdx3+dx4&tdx4

g:=dx1dx1+dx2dx2+dx3dx3+dx4dx4

(2.13)

 

Find the subalgebra of gl(3) which preserves this volume form.

R5 > 

SO4MatrixAlgebrasSubalgebra,gl4,g,sl3

SO4:=e12e21,e13e31,e14e41,e23e32,e24e42,e34e43

(2.14)

 

Note that each of the matrices represented by the elements of the list SO4 are skew-symmetric.

gl4 > 

L6LieAlgebraDataSO4,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

(2.15)
gl4 > 

DGsetupL6:

so4 > 

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

(2.16)
so4 > 

Queryso4,Semisimple,Queryso4,Indecomposable

true,false

(2.17)

See Also

DifferentialGeometry

LieAlgebras

LieAlgebraData

MultiplicationTable

Query