Homomorphism - Maple Help

Query[Homomorphism] - check if a matrix defines a Lie algebra homomorphism between two Lie algebras

Calling Sequences

Query(Alg1, Alg2, A, "Homomorphism")

Query(Alg1, Alg2, phi, "Homomorphism")

Query(Alg1, Alg2, A, parm, "Homomorphism")

Parameters

Alg1     - the name of an initialized Lie algebra $\mathrm{𝔤}$, the domain algebra for the homomorphism defined by A

Alg2     - the name of an initialized Lie algebra $\mathrm{𝔥}$, the range algebra for the homomorphism defined by A

A        - an matrix,where $n$ is the dimension of the Lie algebra and $m$ is the dimension of $\mathrm{𝔥}$

phi      -  a transformation from Alg1 to Alg2

parm     - a set of parameters appearing in the matrix A or in the Lie algebras and $k$

Description

 • A matrix $A$ defines a Lie algebra homomorphism from a Lie algebra $\mathrm{𝔤}$ to a Lie algebra $\mathrm{𝔥}$ if the linear transformation ${{L}_{A}}_{}$ satisfies  for all .
 • Query(Alg1, Alg2, A, "Homomorphism") returns true if the matrix A defines a Lie algebra homomorphism from $\mathrm{𝔤}$ to $\mathrm{𝔥}$ and false otherwise.
 • Query(Alg1, Alg2, phi, "Homomorphism") returns true if the transformation $\mathrm{φ}$ defines a Lie algebra homomorphism  -> $\mathrm{𝔥}$ and false otherwise.
 • Query(Alg1, Alg2, parm, "Homomorphism") returns a 4-tuple TF, Eq, Soln, B.  Here TF is true if Maple finds a set of values for the parameters for which the Matrix A is a homomorphism; Eq is the defining set of equations for the parameters parm in order that the matrix A be a homomorphism; Soln is a list of solutions to the equations Eq; and B is the list of Matrices obtained by evaluating A on the solutions in the list Soln.
 • The command Query is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).

Examples

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

Example 1.

First initialize a Lie algebra.  We illustrate the fact that the Adjoint matrix Ad($x)$, for any $x$ in the Lie algebra, is always a Lie algebra homomorphism (in fact, an isomorphism).

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[4\right]\right],\left[\left[\left[1,4,1\right],1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,4,2\right],1\right]\right]\right]\right)$
 ${\mathrm{L1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$
 Alg1 > $\mathrm{A1}≔\mathrm{AdjointExp}\left(r\mathrm{e1}+s\mathrm{e2}+t\mathrm{e3}\right)$
 ${\mathrm{A1}}{:=}\left[\begin{array}{cccc}{1}& {-}{t}& {s}& {-}\frac{{1}}{{2}}{}{t}{}{s}{+}{r}\\ {0}& {1}& {0}& {s}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\end{array}\right]$ (2.2)
 Alg1 > $\mathrm{Query}\left(\mathrm{Alg1},\mathrm{Alg1},\mathrm{A1},"Homomorphism"\right)$
 ${\mathrm{true}}$ (2.3)

Example 2.

The matrix exponential of any outer derivation is also a Lie algebra homomorphism (isomorphism).

 Alg1 > $\mathrm{Outer}≔\mathrm{Derivations}\left("Outer"\right)$
 ${\mathrm{Outer}}{:=}\left[\left[\begin{array}{rrrr}{1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]\right]$ (2.4)
 Alg1 > $\mathrm{A2}≔\mathrm{LinearAlgebra}:-\mathrm{MatrixExponential}\left(t\mathrm{Outer}\left[1\right]\right)$
 ${\mathrm{A2}}{:=}\left[\begin{array}{cccc}{{ⅇ}}^{{t}}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {{ⅇ}}^{{t}}& {0}\\ {0}& {0}& {0}& {1}\end{array}\right]$ (2.5)
 Alg1 > $T≔\mathrm{Transformation}\left(\mathrm{Alg1},\mathrm{Alg1},\mathrm{A2}\right)$
 ${T}{:=}\left[\left[{\mathrm{e1}}{,}{{ⅇ}}^{{t}}{}{\mathrm{e1}}\right]{,}\left[{\mathrm{e2}}{,}{\mathrm{e2}}\right]{,}\left[{\mathrm{e3}}{,}{{ⅇ}}^{{t}}{}{\mathrm{e3}}\right]{,}\left[{\mathrm{e4}}{,}{\mathrm{e4}}\right]\right]$ (2.6)
 Alg1 > $\mathrm{Query}\left(\mathrm{Alg1},\mathrm{Alg1},T,"Homomorphism"\right)$
 ${\mathrm{true}}$ (2.7)

Example 3.

In this example we construct the quotient algebra of Alg1 by the ideal $\left[{e}_{1}\right]$  Call the quotient Alg2.  We check that the canonical projection map from Alg1 to Alg2 is a Lie algebra homomorphism.

 Alg1 > $\mathrm{L2}≔\mathrm{QuotientAlgebra}\left(\left[\mathrm{e1}\right],\left[\mathrm{e2},\mathrm{e3},\mathrm{e4}\right],\mathrm{Alg2}\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}\right]$ (2.8)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2},\left[x\right],\left[\mathrm{\alpha }\right]\right):$
 $\left[\begin{array}{ccccc}{}& {\mathrm{|}}& {{{\mathrm{x1}}}_{{}}}_{{}}& {{{\mathrm{x2}}}_{{}}}_{{}}& {{{\mathrm{x3}}}_{{}}}_{{}}\\ {}& {\mathrm{---}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {{{\mathrm{x1}}}_{{}}}_{{}}& {\mathrm{|}}& {0}& {0}& {{\mathrm{x1}}}_{{}}\\ {{{\mathrm{x2}}}_{{}}}_{{}}& {\mathrm{|}}& {0}& {0}& {0}\\ {{{\mathrm{x3}}}_{{}}}_{{}}& {\mathrm{|}}& {-}{{\mathrm{x1}}}_{{}}& {0}& {0}\end{array}\right]$ (2.9)

The following matrix ${A}_{3}$ maps  .

 Alg2 > $\mathrm{A3}≔\mathrm{Matrix}\left(\left[\left[0,1,0,0\right],\left[0,0,1,0\right],\left[0,0,0,1\right]\right]\right)$
 ${\mathrm{A3}}{:=}\left[\begin{array}{rrrr}{0}& {1}& {0}& {0}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\end{array}\right]$ (2.10)
 Alg2 > $\mathrm{Query}\left(\mathrm{Alg1},\mathrm{Alg2},\mathrm{A3},"Homomorphism"\right)$
 ${\mathrm{true}}$ (2.11)

Example 4.

In this example we shall find all the monomorphisms from the 2-dimensional solvable Lie algebra into Alg1. This effectively computes all the 2-dimensional non-Abelian subalgebras of Alg1. First initialize the 2-dimensional solvable algebra and call it Alg3.

 Alg2 > $\mathrm{L2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg3},\left[2\right]\right],\left[\left[\left[1,2,1\right],1\right]\right]\right]\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}\right]$ (2.12)
 Alg2 > $\mathrm{DGsetup}\left(\mathrm{L2}\right):$

Define a matrix representing an arbitrary linear transformation from Alg1 to Alg2.

 Alg3 > $\mathrm{A4}≔\mathrm{Matrix}\left(\left[\left[\mathrm{a1},\mathrm{a2}\right],\left[\mathrm{a3},\mathrm{a4}\right],\left[\mathrm{a5},\mathrm{a6}\right],\left[\mathrm{a7},\mathrm{a8}\right]\right]\right)$
 ${\mathrm{A4}}{:=}\left[\begin{array}{cc}{\mathrm{a1}}& {\mathrm{a2}}\\ {\mathrm{a3}}& {\mathrm{a4}}\\ {\mathrm{a5}}& {\mathrm{a6}}\\ {\mathrm{a7}}& {\mathrm{a8}}\end{array}\right]$ (2.13)

Determine the parameter values for which  is a Lie algebra homomorphism.

 Alg3 > $\mathrm{TF},\mathrm{EQ},\mathrm{SOLN},B≔\mathrm{Query}\left(\mathrm{Alg3},\mathrm{Alg1},\mathrm{A4},\left\{\mathrm{a1},\mathrm{a2},\mathrm{a3},\mathrm{a4},\mathrm{a5},\mathrm{a6},\mathrm{a7},\mathrm{a8}\right\},"Homomorphism"\right)$
 ${\mathrm{TF}}{,}{\mathrm{EQ}}{,}{\mathrm{SOLN}}{,}{B}{:=}{\mathrm{true}}{,}\left\{{0}{,}{\mathrm{a5}}{,}{\mathrm{a7}}{,}{-}{\mathrm{a5}}{,}{-}{\mathrm{a7}}{,}{-}{\mathrm{a3}}{}{\mathrm{a8}}{+}{\mathrm{a4}}{}{\mathrm{a7}}{+}{\mathrm{a3}}{,}{\mathrm{a3}}{}{\mathrm{a8}}{-}{\mathrm{a4}}{}{\mathrm{a7}}{-}{\mathrm{a3}}{,}{-}{\mathrm{a1}}{}{\mathrm{a8}}{+}{\mathrm{a2}}{}{\mathrm{a7}}{-}{\mathrm{a3}}{}{\mathrm{a6}}{+}{\mathrm{a4}}{}{\mathrm{a5}}{+}{\mathrm{a1}}{,}{\mathrm{a1}}{}{\mathrm{a8}}{-}{\mathrm{a2}}{}{\mathrm{a7}}{+}{\mathrm{a3}}{}{\mathrm{a6}}{-}{\mathrm{a4}}{}{\mathrm{a5}}{-}{\mathrm{a1}}\right\}{,}\left[\left\{{\mathrm{a1}}{=}{\mathrm{a1}}{,}{\mathrm{a2}}{=}{\mathrm{a2}}{,}{\mathrm{a3}}{=}{\mathrm{a3}}{,}{\mathrm{a4}}{=}{\mathrm{a4}}{,}{\mathrm{a5}}{=}{0}{,}{\mathrm{a6}}{=}{0}{,}{\mathrm{a7}}{=}{0}{,}{\mathrm{a8}}{=}{1}\right\}{,}\left\{{\mathrm{a1}}{=}{0}{,}{\mathrm{a2}}{=}{\mathrm{a2}}{,}{\mathrm{a3}}{=}{0}{,}{\mathrm{a4}}{=}{\mathrm{a4}}{,}{\mathrm{a5}}{=}{0}{,}{\mathrm{a6}}{=}{\mathrm{a6}}{,}{\mathrm{a7}}{=}{0}{,}{\mathrm{a8}}{=}{\mathrm{a8}}\right\}{,}\left\{{\mathrm{a1}}{=}{\mathrm{a1}}{,}{\mathrm{a2}}{=}{\mathrm{a2}}{,}{\mathrm{a3}}{=}{0}{,}{\mathrm{a4}}{=}{\mathrm{a4}}{,}{\mathrm{a5}}{=}{0}{,}{\mathrm{a6}}{=}{\mathrm{a6}}{,}{\mathrm{a7}}{=}{0}{,}{\mathrm{a8}}{=}{1}\right\}\right]{,}\left[\left[\begin{array}{cc}{\mathrm{a1}}& {\mathrm{a2}}\\ {\mathrm{a3}}& {\mathrm{a4}}\\ {0}& {0}\\ {0}& {1}\end{array}\right]{,}\left[\begin{array}{cc}{0}& {\mathrm{a2}}\\ {0}& {\mathrm{a4}}\\ {0}& {\mathrm{a6}}\\ {0}& {\mathrm{a8}}\end{array}\right]{,}\left[\begin{array}{cc}{\mathrm{a1}}& {\mathrm{a2}}\\ {0}& {\mathrm{a4}}\\ {0}& {\mathrm{a6}}\\ {0}& {1}\end{array}\right]\right]$ (2.14)

The equations that must hold for ${A}_{4}$ to define a Lie algebra homomorphism are given by EQ.

 Alg1 > $\mathrm{EQ}$
 $\left\{{0}{,}{\mathrm{a5}}{,}{\mathrm{a7}}{,}{-}{\mathrm{a5}}{,}{-}{\mathrm{a7}}{,}{-}{\mathrm{a3}}{}{\mathrm{a8}}{+}{\mathrm{a4}}{}{\mathrm{a7}}{+}{\mathrm{a3}}{,}{\mathrm{a3}}{}{\mathrm{a8}}{-}{\mathrm{a4}}{}{\mathrm{a7}}{-}{\mathrm{a3}}{,}{-}{\mathrm{a1}}{}{\mathrm{a8}}{+}{\mathrm{a2}}{}{\mathrm{a7}}{-}{\mathrm{a3}}{}{\mathrm{a6}}{+}{\mathrm{a4}}{}{\mathrm{a5}}{+}{\mathrm{a1}}{,}{\mathrm{a1}}{}{\mathrm{a8}}{-}{\mathrm{a2}}{}{\mathrm{a7}}{+}{\mathrm{a3}}{}{\mathrm{a6}}{-}{\mathrm{a4}}{}{\mathrm{a5}}{-}{\mathrm{a1}}\right\}$ (2.15)

The possible Lie algebra homomorphisms are given by $B$.  Note that can be chosen to be full rank and therefore define Lie algebra isomorphisms.

 Alg1 > $B$
 $\left[\left[\begin{array}{cc}{\mathrm{a1}}& {\mathrm{a2}}\\ {\mathrm{a3}}& {\mathrm{a4}}\\ {0}& {0}\\ {0}& {1}\end{array}\right]{,}\left[\begin{array}{cc}{0}& {\mathrm{a2}}\\ {0}& {\mathrm{a4}}\\ {0}& {\mathrm{a6}}\\ {0}& {\mathrm{a8}}\end{array}\right]{,}\left[\begin{array}{cc}{\mathrm{a1}}& {\mathrm{a2}}\\ {0}& {\mathrm{a4}}\\ {0}& {\mathrm{a6}}\\ {0}& {1}\end{array}\right]\right]$ (2.16)