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LieAlgebras[TensorProductOfRepresentations] - form the tensor product representation for a list of representations of a Lie algebra; form various tensor product representations from a single representation of a Lie algebra

Calling Sequences

TensorProductOfRepresentations(R, W)

TensorProductOfRepresentations(${\mathbf{ρ}}$, T, W)

Parameters

R         - a list of representations of a Lie algebra on vector spaces ${V}_{1},{V}_{2}$...

W         - a Maple name or string, the name of the frame for the representation space for the tensor product representation

$\mathrm{ρ}$         - a representation of a Lie algebra $\mathrm{𝔤}$on a vector space $V$

T         - a list of linearly independent type tensors on defining a subspace of tensors invariant under the induced representation of $\mathrm{ρ}$

Description

 • Let , ... be a list of representations of a Lie algebra. Let be the tensor product of the vector spaces The tensor product of the representations is the representation defined by

where and.

 • Let be a representation. Then determines a representation of $\mathrm{𝔤}$ on ${T}_{s}^{r}\left(V\right)$, the space of type  tensors on The representation , in turn, the restricts to any $\mathrm{τ}$-invariant subspace, spanned by a list T of  type tensors. The second calling sequence returns this $p-$dimensional representation of $\mathrm{ρ}$.

Examples

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

Example 1.

Define the standard representation and the adjoint representation for $\mathrm{sl}\left(2\right)$. Then form the tensor product representation. First, set up the representation spaces.

 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2}\right],\mathrm{V1}\right):$
 V1 > $\mathrm{DGsetup}\left(\left[\mathrm{y1},\mathrm{y2},\mathrm{y2}\right],\mathrm{V2}\right):$

Define the standard representation.

 V2 > $\mathrm{M1}≔\left[\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[1,0\right]\right]\right)\right]$
 V2 > $L≔\mathrm{LieAlgebraData}\left(\mathrm{M1},\mathrm{sl2}\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e3}}\right]$ (2.1)
 V2 > $\mathrm{DGsetup}\left(L\right):$
 sl2 > $\mathrm{ρ1}≔\mathrm{Representation}\left(\mathrm{sl2},\mathrm{V1},\mathrm{M1}\right)$

Define the adjoint representation using the Adjoint command.

 sl2 > $\mathrm{ρ2}≔\mathrm{Representation}\left(\mathrm{sl2},\mathrm{V2},\mathrm{Adjoint}\left(\right)\right)$

We will need a 6-dimensional vector space to represent the tensor product of rho1 and rho2.

 sl2 > $\mathrm{DGsetup}\left(\left[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4},\mathrm{z5},\mathrm{z6}\right],\mathrm{W1}\right):$
 W1 > $\mathrm{φ1}≔\mathrm{TensorProductOfRepresentations}\left(\left[\mathrm{ρ1},\mathrm{ρ2}\right],\mathrm{W1}\right)$

Use the Query command to verify that rho1 is a representation.

 sl2 > $\mathrm{Query}\left(\mathrm{φ1},"Representation"\right)$
 ${\mathrm{true}}$ (2.2)

Example 2.

Compute the representation of rho1 (the standard representation of sl2) on the 3rd symmetric product ${\mathrm{Sym}}^{3}\left({V}_{1}\right)$of ${V}_{1}$. First, use the GenerateSymmetricTensors command to generate a basis T1 for ${\mathrm{Sym}}^{3}\left({V}_{1}\right)$.

 sl2 > $\mathrm{ChangeFrame}\left(\mathrm{V1}\right):$
 V1 > $\mathrm{T1}≔\mathrm{Tensor}:-\mathrm{GenerateSymmetricTensors}\left(\left[\mathrm{D_x1},\mathrm{D_x2}\right],3\right)$
 ${\mathrm{T1}}{:=}\left[{\mathrm{D_x1}}{}{\mathrm{D_x1}}{}{\mathrm{D_x1}}{,}\frac{{1}}{{3}}{}{\mathrm{D_x1}}{}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{+}\frac{{1}}{{3}}{}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{+}\frac{{1}}{{3}}{}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{}{\mathrm{D_x1}}{,}\frac{{1}}{{3}}{}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{}{\mathrm{D_x2}}{+}\frac{{1}}{{3}}{}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{}{\mathrm{D_x2}}{+}\frac{{1}}{{3}}{}{\mathrm{D_x2}}{}{\mathrm{D_x2}}{}{\mathrm{D_x1}}{,}{\mathrm{D_x2}}{}{\mathrm{D_x2}}{}{\mathrm{D_x2}}\right]$ (2.3)

We will need a - dimensional representation space.

 V1 > $\mathrm{DGsetup}\left(\left[\mathrm{z1},\mathrm{z2},\mathrm{z3},\mathrm{z4}\right],\mathrm{W2}\right):$
 W2 > $\mathrm{φ2}≔\mathrm{TensorProductOfRepresentations}\left(\mathrm{ρ1},\mathrm{T1},\mathrm{W2}\right)$

Example 3.

Compute the representation of rho1 (the standard representation of sl2) on the 2nd exterior product of the 3rd symmetric product ${\wedge }^{2}\left({\mathrm{Sym}}^{3}\left(\mathrm{V1}\right)\right).$

 sl2 > $\mathrm{ChangeFrame}\left(\mathrm{W2}\right):$
 W2 > $\mathrm{T3}≔\mathrm{Tools}:-\mathrm{GenerateForms}\left(\left[\mathrm{dz1},\mathrm{dz2},\mathrm{dz3},\mathrm{dz4}\right],2\right)$
 ${\mathrm{T3}}{:=}\left[{\mathrm{dz1}}{}{\bigwedge }{}{\mathrm{dz2}}{,}{\mathrm{dz1}}{}{\bigwedge }{}{\mathrm{dz3}}{,}{\mathrm{dz1}}{}{\bigwedge }{}{\mathrm{dz4}}{,}{\mathrm{dz2}}{}{\bigwedge }{}{\mathrm{dz3}}{,}{\mathrm{dz2}}{}{\bigwedge }{}{\mathrm{dz4}}{,}{\mathrm{dz3}}{}{\bigwedge }{}{\mathrm{dz4}}\right]$ (2.4)

We will need a 6-dimensional representation space.

 W2 > $\mathrm{DGsetup}\left(\left[\mathrm{p1},\mathrm{p2},\mathrm{p3},\mathrm{p4},\mathrm{p5},\mathrm{p6}\right],\mathrm{W3}\right):$
 W3 > $\mathrm{φ3}≔\mathrm{TensorProductOfRepresentations}\left(\mathrm{φ2},\mathrm{T3},\mathrm{W3}\right)$

Use the Invariants command to calculate the invariants of this representation.

 sl2 > $\mathrm{Invariants}\left(\mathrm{φ3}\right)$
 $\left[{-}{3}{}{\mathrm{D_p3}}{+}{\mathrm{D_p4}}\right]$ (2.5)