 Invariants - Maple Help

LieAlgebras[Invariants] - calculate the invariant vectors for a representation of a Lie algebra, calculate the invariant tensors for a tensor product representation of a Lie algebra

Calling Sequences

Invariants(${\mathbf{ρ}}$)

Invariants(${\mathbf{\rho }}$, T)

Parameters

- a representation of a Lie algebra on a vector space $V$

T         - list of tensors on $V$ defining a subspace of tensors invariant under the induced representation Description

 • Let be a representation of a Lie algebra on a vector space $V$. A vector  is an invariant vector for the representation if for all .
 • Let be the vector space of type  tensors on Then the representation defines an induced representation .
 • The procedure Invariants(${\mathbf{\rho }}$) returns a basis for the vector subspace of invariant vectors for the representation rho. An empty list is returned if the zero vector is the only invariant vector.
 • The procedure Invariants(${\mathbf{\rho }}$, T) returns a basis for the subspace of tensors which belong to T and which are invariant with respect to the representation $\stackrel{‾}{\mathrm{\rho }}$ . Examples

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

Example 1.

We define a 6-dimensional representation of $\mathrm{sl}\left(2\right)$and find the invariant vectors.

 > $L≔\mathrm{LieAlgebraData}\left(\left[\left[\mathrm{x1},\mathrm{x2}\right]=-2\mathrm{x1},\left[\mathrm{x1},\mathrm{x3}\right]=\mathrm{x2},\left[\mathrm{x2},\mathrm{x3}\right]=-2\mathrm{x3}\right],\left[\mathrm{x1},\mathrm{x2},\mathrm{x3}\right],\mathrm{sl2}\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(L\right):$
 sl2 > $\mathrm{DGsetup}\left(\left[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5},\mathrm{z6}\right],\mathrm{W1}\right):$
 W1 > $M≔\left[\mathrm{Matrix}\left(\left[\left[0,0,0,0,0,0\right],\left[-2,0,0,0,0,0\right],\left[0,-1,0,0,0,0\right],\left[0,-3,0,0,0,0\right],\left[0,0,-3,-1,0,0\right],\left[0,0,0,0,-2,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[-4,0,0,0,0,0\right],\left[0,-2,0,0,0,0\right],\left[0,0,0,0,0,0\right],\left[0,0,0,0,0,0\right],\left[0,0,0,0,2,0\right],\left[0,0,0,0,0,4\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,-2,0,0,0,0\right],\left[0,0,-3,-1,0,0\right],\left[0,0,0,0,-1,0\right],\left[0,0,0,0,-3,0\right],\left[0,0,0,0,0,-2\right],\left[0,0,0,0,0,0\right]\right]\right)\right]:$
 W1 > $\mathrm{ρ1}≔\mathrm{Representation}\left(\mathrm{sl2},\mathrm{W1},M\right)$ sl2 > $\mathrm{Inv}≔\mathrm{Invariants}\left(\mathrm{ρ1}\right)$
 ${\mathrm{Inv}}{:=}\left[{-}\frac{{1}}{{3}}{}{\mathrm{D_z3}}{+}{\mathrm{D_z4}}\right]$ (2.2)

We check this result using the ApplyRepresentation command.

 W1 > $\mathrm{map2}\left(\mathrm{ApplyRepresentation},\mathrm{ρ1},\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right],{\mathrm{Inv}}_{1}\right)$
 $\left[{0}{}{\mathrm{D_z1}}{,}{0}{}{\mathrm{D_z1}}{,}{0}{}{\mathrm{D_z1}}\right]$ (2.3)

Example 2.

In this example we calculate the invariant (1,1) tensors, the invariant (0,2) symmetric tensors and the type (1,2) invariant tensors for the adjoint representation of the Lie algebra [3,2] in the Winternitz tables of Lie algebras. We begin by using the Retrieve command to obtain the structure equations for this Lie algebra.

 W1 > $L≔\mathrm{Retrieve}\left("Winternitz",1,\left[3,2\right],\mathrm{Alg1}\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}\right]$ (2.4)
 V > $\mathrm{DGsetup}\left(L\right):$
 Alg1 > $\mathrm{DGsetup}\left(\left[x,y,z\right],V\right):$
 V > $\mathrm{ρ2}≔\mathrm{Representation}\left(\mathrm{Alg1},V,\mathrm{Adjoint}\left(\mathrm{Alg1}\right)\right)$ There are no vector invariants.

 Alg1 > $F≔\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]$
 ${F}{:=}\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{\mathrm{D_z}}\right]$ (2.5)
 V > $\mathrm{Invariants}\left(\mathrm{ρ2},F\right)$
 $\left[{}\right]$ (2.6)

There is one 1-form invariant.

 Alg1 > $\mathrm{Ω}≔\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right]$
 ${\mathrm{Ω}}{:=}\left[{\mathrm{dx}}{,}{\mathrm{dy}}{,}{\mathrm{dz}}\right]$ (2.7)
 V > $\mathrm{Invariants}\left(\mathrm{ρ2},\mathrm{Ω}\right)$
 $\left[{\mathrm{dz}}\right]$ (2.8)

There is 1 invariant type (1,1) tensor.

 V > $\mathrm{T1}≔\mathrm{Tensor}:-\mathrm{GenerateTensors}\left(\left[\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right],\left[\mathrm{D_x},\mathrm{D_y},\mathrm{D_z}\right]\right]\right)$
 ${\mathrm{T1}}{:=}\left[{\mathrm{dx}}{}{\mathrm{D_x}}{,}{\mathrm{dx}}{}{\mathrm{D_y}}{,}{\mathrm{dx}}{}{\mathrm{D_z}}{,}{\mathrm{dy}}{}{\mathrm{D_x}}{,}{\mathrm{dy}}{}{\mathrm{D_y}}{,}{\mathrm{dy}}{}{\mathrm{D_z}}{,}{\mathrm{dz}}{}{\mathrm{D_x}}{,}{\mathrm{dz}}{}{\mathrm{D_y}}{,}{\mathrm{dz}}{}{\mathrm{D_z}}\right]$ (2.9)
 V > $\mathrm{Inv1}≔\mathrm{Invariants}\left(\mathrm{ρ2},\mathrm{T1}\right)$
 ${\mathrm{Inv1}}{:=}\left[{\mathrm{dx}}{}{\mathrm{D_x}}{+}{\mathrm{dy}}{}{\mathrm{D_y}}{+}{\mathrm{dz}}{}{\mathrm{D_z}}\right]$ (2.10)

There is 1 invariant symmetric type (0,2)  tensor (but no invariant metrics).

 V > $\mathrm{T2}≔\mathrm{Tensor}:-\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right],2\right)$
 ${\mathrm{T2}}{:=}\left[{\mathrm{dx}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dx}}{,}\frac{{1}}{{2}}{}{\mathrm{dx}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{dy}}{,}\frac{{1}}{{2}}{}{\mathrm{dy}}{}{\mathrm{dz}}{+}\frac{{1}}{{2}}{}{\mathrm{dz}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.11)
 V > $\mathrm{Inv2}≔\mathrm{Invariants}\left(\mathrm{ρ2},\mathrm{T2}\right)$
 ${\mathrm{Inv2}}{:=}\left[{\mathrm{dz}}{}{\mathrm{dz}}\right]$ (2.12)

There are 3 type (1,2) invariant tensors.

 V > $\mathrm{T3}≔\mathrm{Tensor}:-\mathrm{GenerateTensors}\left(\left[\mathrm{T1},\left[\mathrm{dx},\mathrm{dy},\mathrm{dz}\right]\right]\right)$
 ${\mathrm{T3}}{:=}\left[{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{,}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{,}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{,}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{,}{\mathrm{dx}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{,}{\mathrm{dx}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{,}{\mathrm{dx}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{,}{\mathrm{dy}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{,}{\mathrm{dy}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{,}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{,}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{,}{\mathrm{dy}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{,}{\mathrm{dy}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{,}{\mathrm{dy}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{,}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{,}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{,}{\mathrm{dz}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{,}{\mathrm{dz}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{,}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dx}}{,}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dy}}{,}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dz}}\right]$ (2.13)
 V > $\mathrm{Inv3}≔\mathrm{Invariants}\left(\mathrm{ρ2},\mathrm{T3}\right)$
 ${\mathrm{Inv3}}{:=}\left[{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{+}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{+}{\mathrm{dz}}{}{\mathrm{D_z}}{}{\mathrm{dz}}{,}{-}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{-}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dz}}{+}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{+}{\mathrm{dz}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{,}{-}{\mathrm{dy}}{}{\mathrm{D_x}}{}{\mathrm{dz}}{+}{\mathrm{dz}}{}{\mathrm{D_x}}{}{\mathrm{dy}}\right]$ (2.14)

We can check the validity of the these calculations in two steps. First we use the matrices for the representation  to construct linear vector fields on the representation space $V$. This gives a vector field realization of our Lie algebra. The invariance of the tensors Inv1, Inv2, Inv3 means that the Lie derivatives of these tensors with respect to the vector fields in $\mathrm{Γ}$ vanishes.

 V > $A≔\mathrm{map2}\left(\mathrm{ApplyRepresentation},\mathrm{ρ2},\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right]\right)$ Alg1 > $\mathrm{ChangeFrame}\left(V\right)$
 ${\mathrm{Alg1}}$ (2.15)
 V > $\mathrm{Gamma}≔\mathrm{map}\left(\mathrm{convert},A,\mathrm{DGvector}\right)$
 ${\mathrm{Γ}}{:=}\left[{z}{}{\mathrm{D_x}}{,}{z}{}{\mathrm{D_x}}{+}{z}{}{\mathrm{D_y}}{,}{-}\left({x}{+}{y}\right){}{\mathrm{D_x}}{-}{y}{}{\mathrm{D_y}}\right]$ (2.16)

Use the LieDerivative command to verify the invariance of the tensors calculated by the Invariants command.

 V > $\mathrm{Matrix}\left(1,3,\left(i,j\right)→\mathrm{LieDerivative}\left({\mathrm{Gamma}}_{j},{\mathrm{Inv1}}_{i}\right)\right)$ V > $\mathrm{Matrix}\left(1,3,\left(i,j\right)→\mathrm{LieDerivative}\left({\mathrm{Gamma}}_{j},{\mathrm{Inv2}}_{i}\right)\right)$ V > $\mathrm{Matrix}\left(3,3,\left(i,j\right)→\mathrm{LieDerivative}\left({\mathrm{Gamma}}_{j},{\mathrm{Inv3}}_{i}\right)\right)$ 