apply a Lie algebra homomorphism to a vector, form or tensor - Maple Help

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LieAlgebras[ApplyHomomorphism] - apply a Lie algebra homomorphism to a vector, form or tensor

Calling Sequences

ApplyHomomorphism(${\mathbf{φ}}$, T, keyword)

Parameters

- a linear transformation from a Lie algebra $\mathrm{𝔤}$ to another Lie algebra $\mathrm{𝔥}$

T        - a vector, a form, or a tensor defined on either the domain Lie algebra $\mathrm{𝔤}$ or the range Lie algebra $\mathrm{𝔥}$

keyword  - (optional) string keyword, either "domain" or "range"

Description

 • ApplyHomomorphism(${\mathbf{φ}}$, T) will apply the transformation Phi to the vector, form or tensor T and return an object of the same type. The precise evaluation rules for ApplyHomomorphism depend upon the specific properties of T and whether or not Phi is invertible. The details are as follows.
 • Applied to tensors, the command ApplyHomomorphism acts as a ring homomorphism, that is, ApplyHomomorphism(${\mathbf{φ}}$, T${\mathbf{\otimes }}$S) = ApplyHomomorphism(${\mathbf{φ}}$, T)${\mathbf{\otimes }}$ApplyHomomorphism(${\mathbf{φ}}$, S).
 • CASE 1. T is a vector in the domain algebra of ${\mathbf{φ}}$. In this case ApplyHomomorphism(Phi, T) simply applies the linear transformation ${\mathbf{φ}}$ to the vector T and the result is a vector in the range algebra of the transformation ${\mathbf{φ}}$.
 • CASE 2. T is a $p$-form on the range algebra $\mathrm{𝔥}$ of transformation ${\mathbf{φ}}$.In this case ApplyHomomorphism(${\mathbf{φ}}$, T) simply applies the pullback of the linear transformation ${\mathbf{φ}}$ to the $p$-form T and the result is a $p$-form in the domain $\mathrm{𝔤}$ of ${\mathbf{φ}}$.
 • CASE 3. T is a tensor on $\mathrm{𝔤}$ and ${\mathbf{φ}}$ is an invertible linear transformation. Then ApplyHomomorphism(${\mathbf{φ}}$, T) is the tensor on the range algebra $\mathrm{𝔥}$ obtained by the pushforward by of the contravariant components of T and the pullback of the covariant components of T by the inverse of ${\mathbf{φ}}$.
 • CASE 4. T is a tensor on  and ${\mathbf{φ}}$ is an invertible linear transformation. Then ApplyHomomorphism(${\mathbf{φ}}$, T) is the tensor on the domain algebra obtained by the pushforward of the contravariant components of T by the inverse of ${\mathbf{φ}}$ and the pullback of the covariant components of T by ${\mathbf{φ}}$.
 • CASE 5. T is a tensor on and ${\mathbf{φ}}$ is not invertible. Then T must be a contravariant tensor (that is, a tensor products of vectors) in which case ApplyHomomorphism(${\mathbf{φ}}$, T) is the contravariant tensor defined on the range algebra $\mathrm{𝔥}$ and obtained by the pushforward of Phi acting on vectors in $\mathrm{𝔤}$.
 • When , Case 4 takes precedence over Case 5. Alternatively ApplyHomomorphism can be forced to use Case 4 or Case 5 with the third optional argument "domain" or "range".
 • CASE 6. T is a tensor on $\mathrm{𝔥}$ and ${\mathbf{φ}}$ is not invertible.  Then T must be a covariant tensor (that is, a tensor product of 1-forms) in which case ApplyHomomorphism(${\mathbf{φ}}$, T) is the covariant tensor defined on the domain algebra g and obtained by the pullback of ${\mathbf{φ}}$ acting on 1-forms in $h$.
 • The command ApplyHomomorphism is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form ApplyHomomorphism(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-ApplyHomomorphism(...).

Examples

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

Example 1.

First initialize two copies of a Lie algebra, called Alg1 and Alg2, and display the Lie bracket multiplication tables.

 > $\mathrm{LC1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[4\right]\right],\left[\left[\left[1,4,1\right],1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,4,2\right],1\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LC1},\left[x\right],\left[\mathrm{α}\right]\right):$
 Alg1 > $\mathrm{LC2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg2},\left[4\right]\right],\left[\left[\left[1,4,1\right],1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,4,2\right],1\right]\right]\right]\right):$
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{LC2},\left[y\right],\left[\mathrm{β}\right]\right):$
 Alg2 > $\mathrm{print}\left(\mathrm{MultiplicationTable}\left(\mathrm{Alg1},"LieBracket"\right),\mathrm{MultiplicationTable}\left(\mathrm{Alg2},"LieBracket"\right)\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]{,}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.1)

We use AdjointExp to construct a linear transformation (in fact, an isomorphism) from Alg1 to Alg2.

 Alg2 > $A≔\mathrm{AdjointExp}\left(r\mathrm{x1}+s\mathrm{x2}+t\mathrm{x3}\right):$
 Alg1 > $\mathrm{Φ}≔\mathrm{Transformation}\left(\mathrm{Alg1},\mathrm{Alg2},A\right)$
 ${\mathrm{Φ}}{:=}\left[\left[{\mathrm{x1}}{,}{\mathrm{y1}}\right]{,}\left[{\mathrm{x2}}{,}{-}\left({t}{}{\mathrm{y1}}\right){+}{\mathrm{y2}}\right]{,}\left[{\mathrm{x3}}{,}{s}{}{\mathrm{y1}}{+}{\mathrm{y3}}\right]{,}\left[{\mathrm{x4}}{,}\left({-}\frac{{t}{}{s}}{{2}}{+}{r}\right){}{\mathrm{y1}}{+}{s}{}{\mathrm{y2}}{+}{\mathrm{y4}}\right]\right]$ (2.2)

We calculate the effects of the command ApplyHomomorphism in each of the following cases.

CASE 1: vectors in the domain algebra Alg1.

CASE 2: 1-forms on the range algebra Alg2.

CASE 3: rank 1 covariant tensors on the domain algebra Alg1.

CASE 4: rank 1 contravariant vectors on the range algebra Alg2.

In each case we show the matrix which defines the transformation.

CASE 1: vectors in the domain algebra Alg1.

 Alg2 > $\mathrm{Vectors}≔\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right]:$
 Alg2 > $A,\mathrm{map2}\left(\mathrm{ApplyHomomorphism},\mathrm{Φ},\mathrm{Vectors}\right)$
 $\left[\begin{array}{cccc}{1}& {-}{t}& {s}& {-}\frac{{1}}{{2}}{}{t}{}{s}{+}{r}\\ {0}& {1}& {0}& {s}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\end{array}\right]{,}\left[{\mathrm{y1}}{,}{-}\left({t}{}{\mathrm{y1}}\right){+}{\mathrm{y2}}{,}{s}{}{\mathrm{y1}}{+}{\mathrm{y3}}{,}\left({-}\frac{{t}{}{s}}{{2}}{+}{r}\right){}{\mathrm{y1}}{+}{s}{}{\mathrm{y2}}{+}{\mathrm{y4}}\right]$ (2.3)

CASE 2: 1-forms on the range algebra Alg2.

 Alg2 > $\mathrm{Forms}≔\left[\mathrm{β1},\mathrm{β2},\mathrm{β3},\mathrm{β4}\right]:$
 Alg2 > $\mathrm{Atr}≔\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(A\right):$
 Alg2 > $\mathrm{Atr},\mathrm{map2}\left(\mathrm{ApplyHomomorphism},\mathrm{Φ},\mathrm{Forms}\right)$
 $\left[\begin{array}{cccc}{1}& {0}& {0}& {0}\\ {-}{t}& {1}& {0}& {0}\\ {s}& {0}& {1}& {0}\\ {-}\frac{{1}}{{2}}{}{t}{}{s}{+}{r}& {s}& {0}& {1}\end{array}\right]{,}\left[{\mathrm{α1}}{-}\left({t}{}{\mathrm{α2}}\right){+}{s}{}{\mathrm{α3}}{+}\left({-}\frac{{t}{}{s}}{{2}}{+}{r}\right){}{\mathrm{α4}}{,}{\mathrm{α2}}{+}{s}{}{\mathrm{α4}}{,}{\mathrm{α3}}{,}{\mathrm{α4}}\right]$ (2.4)

CASE 3. rank 1 covariant tensors on the domain algebra Alg1.

 Alg1 > $\mathrm{CovariantTensors}≔\mathrm{map}\left(\mathrm{convert},\left[\mathrm{α1},\mathrm{α2},\mathrm{α3},\mathrm{α4}\right],\mathrm{DGtensor}\right)$
 ${\mathrm{CovariantTensors}}{:=}\left[{\mathrm{α1}}{,}{\mathrm{α2}}{,}{\mathrm{α3}}{,}{\mathrm{α4}}\right]$ (2.5)
 Alg1 > $\mathrm{Aintr}≔\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(A\right)\right):$
 Alg1 > $\mathrm{Aintr},\mathrm{map2}\left(\mathrm{ApplyHomomorphism},\mathrm{Φ},\mathrm{CovariantTensors}\right)$
 $\left[\begin{array}{cccc}{1}& {0}& {0}& {0}\\ {t}& {1}& {0}& {0}\\ {-}{s}& {0}& {1}& {0}\\ {-}\frac{{1}}{{2}}{}{t}{}{s}{-}{r}& {-}{s}& {0}& {1}\end{array}\right]{,}\left[{\mathrm{β1}}{+}{t}{}{\mathrm{β2}}{-}\left({s}{}{\mathrm{β3}}\right){-}\left(\left(\frac{{t}{}{s}}{{2}}{+}{r}\right){}{\mathrm{β4}}\right){,}{\mathrm{β2}}{-}\left({s}{}{\mathrm{β4}}\right){,}{\mathrm{β3}}{,}{\mathrm{β4}}\right]$ (2.6)

CASE 4. rank 1 contravariant vectors on the range algebra Alg2.

 Alg2 > $\mathrm{ContravariantTensors}≔\mathrm{map}\left(\mathrm{convert},\left[\mathrm{y1},\mathrm{y2},\mathrm{y3},\mathrm{y4}\right],\mathrm{DGtensor}\right):$
 Alg2 > $\mathrm{Ain}≔\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(A\right):$
 Alg2 > $\mathrm{Ain},\mathrm{map2}\left(\mathrm{ApplyHomomorphism},\mathrm{Φ},\mathrm{ContravariantTensors}\right)$
 $\left[\begin{array}{cccc}{1}& {t}& {-}{s}& {-}\frac{{1}}{{2}}{}{t}{}{s}{-}{r}\\ {0}& {1}& {0}& {-}{s}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\end{array}\right]{,}\left[{\mathrm{x1}}{,}{t}{}{\mathrm{x1}}{+}{\mathrm{x2}}{,}{-}\left({s}{}{\mathrm{x1}}\right){+}{\mathrm{x3}}{,}{-}\left(\left(\frac{{t}{}{s}}{{2}}{+}{r}\right){}{\mathrm{x1}}\right){-}\left({s}{}{\mathrm{x2}}\right){+}{\mathrm{x4}}\right]$ (2.7)

We show, by way of a simple example, the extensions of the mappings in CASE 1 and CASE 3 form a mixed tensor on the range Alg2.

 Alg1 > $T≔\mathrm{α2}&tensor\mathrm{x3}$
 ${T}{:=}{\mathrm{α2}}{}{\mathrm{x3}}$ (2.8)
 Alg1 > $\mathrm{ApplyHomomorphism}\left(\mathrm{Φ},T\right)$
 $\left({s}{}{\mathrm{β2}}\right){}{\mathrm{y1}}{+}{\mathrm{β2}}{}{\mathrm{y3}}{-}\left(\left({{s}}^{{2}}{}{\mathrm{β4}}\right){}{\mathrm{y1}}\right){-}\left(\left({s}{}{\mathrm{β4}}\right){}{\mathrm{y3}}\right)$ (2.9)

We show, by way of a simple example, the extensions of the mappings in CASE 2 and CASE 4 form a mixed tensor on the domain Alg1.

 Alg2 > $T≔\mathrm{β2}&tensor\mathrm{y3}$
 ${T}{:=}{\mathrm{β2}}{}{\mathrm{y3}}$ (2.10)
 Alg2 > $\mathrm{ApplyHomomorphism}\left(\mathrm{Φ},T\right)$
 ${-}\left(\left({s}{}{\mathrm{α2}}\right){}{\mathrm{x1}}\right){+}{\mathrm{α2}}{}{\mathrm{x3}}{-}\left(\left({{s}}^{{2}}{}{\mathrm{α4}}\right){}{\mathrm{x1}}\right){+}\left({s}{}{\mathrm{α4}}\right){}{\mathrm{x3}}$ (2.11)