 ApplyRepresentation - Maple Help

LieAlgebras[Representation] - define a representation of a Lie algebra

LieAlgebras[ApplyRepresentation] - apply a representation of a Lie algebra to a vector in the Lie algebra

Calling Sequences

Representation(A, V, M)

ApplyRepresentation(${\mathbf{ρ}}$, x, y)

Parameters

A         - a Maple name or string, the name of a defined Lie Algebra $\mathrm{𝔤}$

V         - a Maple name or string, the name of the defined frame used as the vector space  for the representation

M         - a list of matrices  each defining a linear transformation from to $V$

$\mathrm{ρ}$         - a representation of a Lie algebra g on a vector space V, as constructed with the Representation command

x         - a vector in g

y         - (optional) a vector in V Description

 • Let g be a Lie algebra and V a vector space.  A representation of g on V is a linear transformation rho: g -> gl(V) (where gl(V) is the Lie algebra of all linear transformations on V) such that rho([x, y]) = [rho(x), rho(y)] (*) for all x, y in g.  Here [x, y] denotes the Lie bracket of two vectors in g while [rho(x), rho(y)] denotes the commutator of two linear transformations in gl(V).
 • The command Representation(A, V, M) creates a data structure which defines the representation rho : g -> gl(V) such that rho(e_i) = M_i.
 • The command Query can be used to verify that the map rho does in fact satisfy the representation property (*).
 • If  $x={c}_{1}{e}_{1}+{c}_{2}{e}_{2}+\cdot \cdot \cdot$ then ApplyRepresentation(${\mathbf{\rho }}$, x) returns the matrix $\mathrm{ρ}\left(x\right)$$={c}_{1}{e}_{1}+{c}_{2}{e}_{2}+\cdot \cdot \cdot$. If   is vector in the representation space $V$, then Representation(${\mathbf{\rho }}$$,$xy) returns the vector $\mathrm{ρ}\left(x\right)\left(y\right)\in V$. Examples

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

Example 1.

First we retrieve a Lie algebra from the DifferentialGeometry Library and initialize it.

 > $L≔\mathrm{Retrieve}\left("Winternitz",1,\left[4,7\right],\mathrm{Alg1}\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(L\right):$

The center of Alg1 is trivial and therefore the adjoint representation is faithful.  The command Adjoint will return the list of adjoint matrices for the Lie algebra Alg1.

 Alg1 > $\mathrm{Center}\left(\right)$
 $\left[{}\right]$ (2.2)
 Alg1 > $M≔\mathrm{Adjoint}\left(\right)$
 ${M}{:=}\left[\left[\begin{array}{rrrr}{0}& {0}& {0}& {2}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{-}{2}& {0}& {0}& {0}\\ {0}& {-}{1}& {-}{1}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]$ (2.3)

To define a representation of Alg1 using the matrices M, we shall need to define a representation space V.

 Alg1 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],V\right):$
 V > $\mathrm{\rho }≔\mathrm{Representation}\left(\mathrm{Alg1},V,M\right)$
 ${\mathrm{ρ}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {2}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrr}{0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrr}{-}{2}& {0}& {0}& {0}\\ {0}& {-}{1}& {-}{1}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.4)

We can verify that this is really a representation of Alg1 with the Query command.

 Alg1 > $\mathrm{Query}\left(\mathrm{\rho },"Representation"\right)$
 ${\mathrm{true}}$ (2.5)

Apply the representation to some vectors in Alg1:

 Alg1 > $\mathrm{A1}≔\mathrm{ApplyRepresentation}\left(\mathrm{\rho },\mathrm{e1}\right)$
 ${\mathrm{A1}}{:=}\left[\begin{array}{rrrr}{0}& {0}& {0}& {2}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]$ (2.6)
 Alg1 > $\mathrm{A2}≔\mathrm{ApplyRepresentation}\left(\mathrm{\rho },\mathrm{e1}+2\mathrm{e2}+3\mathrm{e4}\right)$
 ${\mathrm{A2}}{:=}\left[\begin{array}{rrrr}{-}{6}& {0}& {2}& {2}\\ {0}& {-}{3}& {-}{3}& {2}\\ {0}& {0}& {-}{3}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]$ (2.7)

Apply the linear transformations to some vectors in V:

 Alg1 > $\mathrm{ApplyRepresentation}\left(\mathrm{\rho },\mathrm{e1},\mathrm{D_x4}\right)$
 ${2}{}{\mathrm{D_x1}}$ (2.8)
 V > $\mathrm{ApplyRepresentation}\left(\mathrm{\rho },\mathrm{e1}+2\mathrm{e2}+3\mathrm{e4},\mathrm{D_x3}\right)$
 ${2}{}{\mathrm{D_x1}}{-}{3}{}{\mathrm{D_x2}}{-}{3}{}{\mathrm{D_x3}}$ (2.9)

Example 2.

The infinitesimal automorphisms or derivations of a Lie algebra g define a matrix Lie algebra which is automatically a representation.  In this case we can take g to be the representation space.

 V > $L≔\mathrm{Retrieve}\left("Winternitz",1,\left[3,2\right],\mathrm{Alg2}\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}\right]$ (2.10)
 V > $\mathrm{DGsetup}\left(L\right):$

The Derivations command calculates the derivations on the Lie algebra Alg2.

 Alg2 > $M≔\mathrm{Derivations}\left("Full"\right)$
 ${M}{:=}\left[\left[\begin{array}{rrr}{1}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {1}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]\right]$ (2.11)

We calculate the structure equations for this matrix algebra and initialize the result.

 Alg2 > $A≔\mathrm{LieAlgebraData}\left(M,\mathrm{Alg2a}\right)$
 ${A}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.12)
 Alg2 > $\mathrm{DGsetup}\left(A\right)$
 ${\mathrm{Lie algebra: Alg2a}}$ (2.13)
 Alg2a > $\mathrm{ρ2}≔\mathrm{Representation}\left(\mathrm{Alg2a},\mathrm{Alg2},M\right)$
 ${\mathrm{ρ2}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrr}{1}& {0}& {0}\\ {0}& {1}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrr}{0}& {1}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrr}{0}& {0}& {1}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.14)
 Alg2a > $\mathrm{Query}\left(\mathrm{ρ2},"Representation"\right)$
 ${\mathrm{true}}$ (2.15)

Example 3.

If Gamma is a Lie algebra of vector fields on a manifold M, then the isotropy subalgebra at a given point admits a natural representation, defined by the Lie bracket, on the tangent space of M. To illustrate the definition of this representation, we first obtain a Lie algebra of vector fields from the DifferentialGeometry Library with the Retrieve command. Then we use the IsotropySubalgebra from the GroupActions package to calculate the isotropy subalgebra and its representation.

 Alg2a > $\mathrm{DGsetup}\left(\left[x,y\right],"M"\right):$
 M > $\mathrm{Gamma}≔\mathrm{Retrieve}\left("Gonzalez-Lopez",1,\left[7\right],\mathrm{manifold}="M"\right)$
 ${\mathrm{Γ}}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{x}{}{\mathrm{D_x}}{+}{y}{}{\mathrm{D_y}}{,}{y}{}{\mathrm{D_x}}{-}{x}{}{\mathrm{D_y}}{,}\left({{x}}^{{2}}{-}{{y}}^{{2}}\right){}{\mathrm{D_x}}{+}{2}{}{x}{}{y}{}{\mathrm{D_y}}{,}{2}{}{x}{}{y}{}{\mathrm{D_x}}{+}\left({{y}}^{{2}}{-}{{x}}^{{2}}\right){}{\mathrm{D_y}}\right]$ (2.16)
 M > $\mathrm{IsoAlg},\mathrm{IsoRep}≔\mathrm{IsotropySubalgebra}\left(\mathrm{Gamma},\left[x=1,y=2\right],\mathrm{output}=\left["Vector","Representation"\right]\right)$
 ${\mathrm{IsoAlg}}{,}{\mathrm{IsoRep}}{:=}\left[{-}\left(\frac{{8}}{{25}}{}{x}{}{y}{-}\frac{{3}}{{25}}{}{{x}}^{{2}}{+}\frac{{3}}{{25}}{}{{y}}^{{2}}{-}{1}\right){}{\mathrm{D_x}}{-}\left({-}\frac{{6}}{{25}}{}{x}{}{y}{-}\frac{{4}}{{25}}{}{{x}}^{{2}}{+}\frac{{4}}{{25}}{}{{y}}^{{2}}\right){}{\mathrm{D_y}}{,}{-}\left({-}\frac{{4}}{{25}}{}{{y}}^{{2}}{+}\frac{{4}}{{25}}{}{{x}}^{{2}}{+}\frac{{6}}{{25}}{}{x}{}{y}\right){}{\mathrm{D_x}}{-}\left(\frac{{8}}{{25}}{}{x}{}{y}{-}\frac{{3}}{{25}}{}{{x}}^{{2}}{+}\frac{{3}}{{25}}{}{{y}}^{{2}}{-}{1}\right){}{\mathrm{D_y}}{,}{-}\left({-}\frac{{1}}{{5}}{}{{y}}^{{2}}{+}\frac{{1}}{{5}}{}{{x}}^{{2}}{+}\frac{{4}}{{5}}{}{x}{}{y}{-}{x}\right){}{\mathrm{D_x}}{-}\left(\frac{{2}}{{5}}{}{{y}}^{{2}}{-}\frac{{2}}{{5}}{}{{x}}^{{2}}{+}\frac{{2}}{{5}}{}{x}{}{y}{-}{y}\right){}{\mathrm{D_y}}{,}{-}\left(\frac{{2}}{{5}}{}{{y}}^{{2}}{-}\frac{{2}}{{5}}{}{{x}}^{{2}}{+}\frac{{2}}{{5}}{}{x}{}{y}{-}{y}\right){}{\mathrm{D_x}}{-}\left({-}\frac{{4}}{{5}}{}{x}{}{y}{-}\frac{{1}}{{5}}{}{{x}}^{{2}}{+}\frac{{1}}{{5}}{}{{y}}^{{2}}{+}{x}\right){}{\mathrm{D_y}}\right]{,}\left[\left[\begin{array}{cc}\frac{{2}}{{5}}& \frac{{4}}{{5}}\\ {-}\frac{{4}}{{5}}& \frac{{2}}{{5}}\end{array}\right]{,}\left[\begin{array}{cc}\frac{{4}}{{5}}& {-}\frac{{2}}{{5}}\\ \frac{{2}}{{5}}& \frac{{4}}{{5}}\end{array}\right]{,}\left[\begin{array}{rr}{1}& {0}\\ {0}& {1}\end{array}\right]{,}\left[\begin{array}{rr}{0}& {1}\\ {-}{1}& {0}\end{array}\right]\right]$ (2.17)
 M > $\mathrm{L3}≔\mathrm{LieAlgebraData}\left(\mathrm{IsoAlg},\mathrm{Alg3}\right)$
 ${\mathrm{L3}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{-}\frac{{2}}{{5}}{}{\mathrm{e3}}{-}\frac{{4}}{{5}}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{+}\frac{{4}}{{5}}{}{\mathrm{e3}}{-}\frac{{2}}{{5}}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{-}\frac{{4}}{{5}}{}{\mathrm{e3}}{+}\frac{{2}}{{5}}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{-}\frac{{2}}{{5}}{}{\mathrm{e3}}{-}\frac{{4}}{{5}}{}{\mathrm{e4}}\right]$ (2.18)
 M > $\mathrm{DGsetup}\left(\mathrm{L3}\right):$
 Alg3 > $\mathrm{ρ3}≔\mathrm{Representation}\left(\mathrm{Alg3},"M",\mathrm{IsoRep}\right)$
 ${\mathrm{ρ3}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{cc}\frac{{2}}{{5}}& \frac{{4}}{{5}}\\ {-}\frac{{4}}{{5}}& \frac{{2}}{{5}}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{cc}\frac{{4}}{{5}}& {-}\frac{{2}}{{5}}\\ \frac{{2}}{{5}}& \frac{{4}}{{5}}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rr}{1}& {0}\\ {0}& {1}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rr}{0}& {1}\\ {-}{1}& {0}\end{array}\right]\right]\right]$ (2.19)
 Alg3 > $\mathrm{Query}\left(\mathrm{ρ3},"Representation"\right)$
 ${\mathrm{true}}$ (2.20)

Example 4.

Here is 4 dimensional faithful representation of an indecomposable 4-dimensional Lie algebra with a center.

 Alg3 > $\mathrm{L4}≔\mathrm{LieAlgebraData}\left(\left[\left[\mathrm{x2},\mathrm{x4}\right]=\mathrm{x1},\left[\mathrm{x3},\mathrm{x4}\right]=\mathrm{x3}\right],\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],\mathrm{Alg4}\right)$
 ${\mathrm{L4}}{:=}\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}\right]$ (2.21)
 M > $\mathrm{DGsetup}\left(\mathrm{L4}\right)$
 ${\mathrm{Lie algebra: Alg4}}$ (2.22)
 Alg4 > $\mathrm{M4}≔\left[\mathrm{Matrix}\left(\left[\left[0,0,0,0\right],\left[0,0,0,-1\right],\left[0,0,0,0\right],\left[0,0,0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0,0,0\right],\left[0,0,0,0\right],\left[0,0,0,-1\right],\left[0,0,0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0,0,1\right],\left[0,0,0,0\right],\left[0,0,0,0\right],\left[0,0,0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[-1,0,0,0\right],\left[0,0,-1,0\right],\left[0,0,0,0\right],\left[0,0,0,0\right]\right]\right)\right]$
 ${\mathrm{M4}}{:=}\left[\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{-}{1}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]$ (2.23)
 Alg4 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],V\right):$
 V > $\mathrm{ρ4}≔\mathrm{Representation}\left(\mathrm{Alg4},V,\mathrm{M4}\right)$
 ${\mathrm{ρ4}}{:=}\left[\left[{\mathrm{e1}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e2}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e3}}{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]{,}\left[{\mathrm{e4}}{,}\left[\begin{array}{rrrr}{-}{1}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]\right]$ (2.24)
 Alg4 > $\mathrm{Query}\left(\mathrm{ρ4},"Representation"\right)$
 ${\mathrm{true}}$ (2.25)