 MatrixAlgebras - Maple Help

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

 • 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

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$

Example 1.

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

 > $\mathrm{L1}≔\mathrm{MatrixAlgebras}\left("Full",2,\mathrm{gl2}\right)$
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}\right]{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}{,}{\mathrm{e21}}{,}{\mathrm{e22}}\right]{,}\left[{\mathrm{ε11}}{,}{\mathrm{ε12}}{,}{\mathrm{ε21}}{,}{\mathrm{ε22}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

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

 gl2 > $M≔\left[\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[1,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[0,1\right]\right]\right)\right]$
 ${M}{≔}\left[\left[\begin{array}{rr}{1}& {0}\\ {0}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {1}\\ {0}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {0}\\ {1}& {0}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {0}\\ {0}& {1}\end{array}\right]\right]$ (2.2)
 gl2 > $\mathrm{L2}≔\mathrm{LieAlgebraData}\left(M,\mathrm{newgl2}\right)$
 ${\mathrm{L2}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}\right]$ (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 > $\mathrm{L3}≔\mathrm{MatrixAlgebras}\left("Upper",3,\mathrm{T3}\right)$
 ${\mathrm{L3}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}\right]{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}{,}{\mathrm{e13}}{,}{\mathrm{e22}}{,}{\mathrm{e23}}{,}{\mathrm{e33}}\right]{,}\left[{\mathrm{ε11}}{,}{\mathrm{ε12}}{,}{\mathrm{ε13}}{,}{\mathrm{ε22}}{,}{\mathrm{ε23}}{,}{\mathrm{ε33}}\right]$ (2.4)
 gl2 > $\mathrm{DGsetup}\left(\mathrm{L3}\right):$
 T3 > $\mathrm{MultiplicationTable}\left("LieTable"\right)$
 $\left[\begin{array}{cccccccc}{}& {\mathrm{|}}& {\mathrm{e11}}& {\mathrm{e12}}& {\mathrm{e13}}& {\mathrm{e22}}& {\mathrm{e23}}& {\mathrm{e33}}\\ {}& {\mathrm{---}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e11}}& {\mathrm{|}}& {0}& {\mathrm{e12}}& {\mathrm{e13}}& {0}& {0}& {0}\\ {\mathrm{e12}}& {\mathrm{|}}& {-}{\mathrm{e12}}& {0}& {0}& {\mathrm{e12}}& {\mathrm{e13}}& {0}\\ {\mathrm{e13}}& {\mathrm{|}}& {-}{\mathrm{e13}}& {0}& {0}& {0}& {0}& {\mathrm{e13}}\\ {\mathrm{e22}}& {\mathrm{|}}& {0}& {-}{\mathrm{e12}}& {0}& {0}& {\mathrm{e23}}& {0}\\ {\mathrm{e23}}& {\mathrm{|}}& {0}& {-}{\mathrm{e13}}& {0}& {-}{\mathrm{e23}}& {0}& {\mathrm{e23}}\\ {\mathrm{e33}}& {\mathrm{|}}& {0}& {0}& {-}{\mathrm{e13}}& {0}& {-}{\mathrm{e23}}& {0}\end{array}\right]$ (2.5)
 T3 > $\mathrm{LieAlgebraCheck}\left(\mathrm{T3},"Solvable"\right)$
 ${\mathrm{LieAlgebraCheck}}{}\left({\mathrm{T3}}{,}{"Solvable"}\right)$ (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 > $\mathrm{L3}≔\mathrm{MatrixAlgebras}\left("Full",3,\mathrm{gl3}\right)$
 ${\mathrm{L3}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}\right]{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}{,}{\mathrm{e13}}{,}{\mathrm{e21}}{,}{\mathrm{e22}}{,}{\mathrm{e23}}{,}{\mathrm{e31}}{,}{\mathrm{e32}}{,}{\mathrm{e33}}\right]{,}\left[{\mathrm{ε11}}{,}{\mathrm{ε12}}{,}{\mathrm{ε13}}{,}{\mathrm{ε21}}{,}{\mathrm{ε22}}{,}{\mathrm{ε23}}{,}{\mathrm{ε31}}{,}{\mathrm{ε32}}{,}{\mathrm{ε33}}\right]$ (2.7)
 T3 > $\mathrm{DGsetup}\left(\mathrm{L3}\right):$

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

 gl3 > $\mathrm{DGsetup}\left(\left[x,y,z\right],\mathrm{R3}\right):$

Define the standard volume form on R3.

 R3 > $\mathrm{\nu }≔\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&wedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&wedge\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dz}$
 ${\mathrm{ν}}{≔}{\mathrm{dx}}{}{\mathrm{^}}{}{\mathrm{dy}}{}{\mathrm{^}}{}{\mathrm{dz}}$ (2.8)

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

 R3 > $\mathrm{SL3}≔\mathrm{MatrixAlgebras}\left("Subalgebra",\mathrm{gl3},\left[\mathrm{\nu }\right],\mathrm{sl3}\right)$
 ${\mathrm{SL3}}{≔}\left[{\mathrm{e11}}{-}{\mathrm{e33}}{,}{\mathrm{e12}}{,}{\mathrm{e13}}{,}{\mathrm{e21}}{,}{\mathrm{e22}}{-}{\mathrm{e33}}{,}{\mathrm{e23}}{,}{\mathrm{e31}}{,}{\mathrm{e32}}\right]$ (2.9)

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

 gl3 > $\mathrm{L4}≔\mathrm{LieAlgebraData}\left(\mathrm{SL3},\mathrm{sl3}\right)$
 ${\mathrm{L4}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e5}}\right]$ (2.10)
 gl3 > $\mathrm{DGsetup}\left(\mathrm{L4}\right)$
 ${\mathrm{Lie algebra: sl3}}$ (2.11)
 sl3 > $\mathrm{Query}\left(\mathrm{sl3},"Semisimple"\right),\mathrm{Query}\left(\mathrm{sl3},"Indecomposable"\right)$
 ${\mathrm{true}}{,}{\mathrm{true}}$ (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 > $\mathrm{L5}≔\mathrm{MatrixAlgebras}\left("Full",4,\mathrm{gl4}\right):$
 sl3 > $\mathrm{DGsetup}\left(\mathrm{L5}\right):$

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

 gl4 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],\mathrm{R5}\right):$

Define the standard Euclidean metric tensor on R4.

 R5 > $g≔\mathrm{evalDG}\left(\mathrm{dx1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx1}+\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx2}+\mathrm{dx3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx3}+\mathrm{dx4}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4}\right)$
 ${g}{≔}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}{+}{\mathrm{dx4}}{}{\mathrm{dx4}}$ (2.13)

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

 R5 > $\mathrm{SO4}≔\mathrm{MatrixAlgebras}\left("Subalgebra",\mathrm{gl4},\left[g\right],\mathrm{sl3}\right)$
 ${\mathrm{SO4}}{≔}\left[{\mathrm{e12}}{-}{\mathrm{e21}}{,}{\mathrm{e13}}{-}{\mathrm{e31}}{,}{\mathrm{e14}}{-}{\mathrm{e41}}{,}{\mathrm{e23}}{-}{\mathrm{e32}}{,}{\mathrm{e24}}{-}{\mathrm{e42}}{,}{\mathrm{e34}}{-}{\mathrm{e43}}\right]$ (2.14)

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

 gl4 > $\mathrm{L6}≔\mathrm{LieAlgebraData}\left(\mathrm{SO4},\mathrm{so4}\right)$
 ${\mathrm{L6}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e4}}\right]$ (2.15)
 gl4 > $\mathrm{DGsetup}\left(\mathrm{L6}\right):$
 so4 > $\mathrm{MultiplicationTable}\left(\mathrm{so4},"LieBracket"\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e4}}\right]$ (2.16)
 so4 > $\mathrm{Query}\left(\mathrm{so4},"Semisimple"\right),\mathrm{Query}\left(\mathrm{so4},"Indecomposable"\right)$
 ${\mathrm{true}}{,}{\mathrm{false}}$ (2.17)