Subalgebra - Maple Help

LieAlgebraData[Subalgebra] - convert a list of vectors defining a Lie subalgebra to a Lie algebra data structure

Calling Sequence

Parameters

Subalgebra        - a list of linearly independent vectors defining a subalgebra of a Lie algebra $\mathrm{𝔤}$

SubSubalgebras    - (optional) a list of lists of linearly independent vectors defining subalgebras of the subalgebra $h$.

AlgName           - a name or string, the name to be assigned to the Lie algebra

"Matrix"          - (optional) keyword

Description

 • In the LieAlgebras package, the command DGsetup is used to initialize a Lie algebra, that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory.  The first argument for DGsetup is a Lie algebra data structure which contains the structure constants in a standard format used by the LieAlgebras package.
 • Let be a subalgebra. The command LieAlgebraData(Subalgebra) will return the Lie algebra data structure for the subalgebra  defined by the vectors Subalgebra which can then be initialized. In this way the subalgebra can be studied as a Lie algebra in its own right, independent of the original ambient algebra $\mathrm{𝔤}$.
 • With the optional argument "Matrix", LieAlgebraData(Subalgebra, "Matrix") will return the Lie algebra data structure for the subalgebra  defined by the vectors Subalgebra and a matrix which defines the inclusion map from the subalgebra $h$ to the Lie algebra $\mathrm{𝔤}$.
 • Let ... be subalgebras of $h$ (and hence subalgebras of $\mathrm{𝔤}$).  The command LieAlgebraData(Subalgebra, SubSubalgebras) will return the Lie algebra data structure for the subalgebra  defined by the vectors Subalgebra and the list of lists of components for the vectors in SubSubalgebras with respect to the given basis for the subalgebra $h$.  With this information and the DGzip program ...  can then be converted to subalgebras of the Lie algebra $h$.
 • The command LieAlgebraData is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form LieAlgebraData(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-LieAlgebraData(...).

Examples

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

Example 1.

We first initialize a 5-dimensional Lie algebra called Alg1 and display its multiplication table.

 > $L≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[5\right]\right],\left[\left[\left[1,4,1\right],1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,4,2\right],1\right],\left[\left[2,5,2\right],-1\right],\left[\left[3,5,3\right],1\right]\right]\right]\right)$
 ${L}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(L,\left[x\right],\left[a\right]\right):$

Let's check that is a subalgebra.

 Alg1 > $\mathrm{S1}≔\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right]:$
 Alg1 > $\mathrm{Query}\left(\mathrm{S1},"Subalgebra"\right)$
 ${\mathrm{true}}$ (2.2)

Now use LieAlgebraData to obtain the Lie algebra data structure for ${S}_{1}$.  Denote the Lie algebra obtained from ${S}_{1}$by Alg2.

 Alg1 > $\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{S1},\mathrm{Alg2}\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]$ (2.3)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2},\left[y\right],\left[b\right]\right):$
 Alg2 > $\mathrm{MultiplicationTable}\left("LieBracket"\right)$
 $\left[\left[{\mathrm{y1}}{,}{\mathrm{y4}}\right]{=}{\mathrm{y1}}{,}\left[{\mathrm{y2}}{,}{\mathrm{y3}}\right]{=}{\mathrm{y1}}{,}\left[{\mathrm{y2}}{,}{\mathrm{y4}}\right]{=}{\mathrm{y2}}\right]$ (2.4)

This is the same multiplication as before.

Example 2.

We continue with Example 1 but this time use LieAlgebraData with the keyword "Matrix".

 Alg2 > $\mathrm{L2},M≔\mathrm{LieAlgebraData}\left(\mathrm{S1},\mathrm{Alg2},"Matrix"\right)$
 ${\mathrm{L2}}{,}{M}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]{,}\left[\begin{array}{rrrr}{1}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}\end{array}\right]$ (2.5)

We can use the matrix M to define a Lie algebra homomorphism from Alg2 to Alg1 which maps  , ${y}_{2}\to {x}_{2}$, etc.

 Alg1 > $\mathrm{\phi }≔\mathrm{Transformation}\left(\mathrm{Alg2},\mathrm{Alg1},M\right)$
 ${\mathrm{φ}}{:=}\left[\left[{\mathrm{y1}}{,}{\mathrm{x1}}\right]{,}\left[{\mathrm{y2}}{,}{\mathrm{x2}}\right]{,}\left[{\mathrm{y3}}{,}{\mathrm{x3}}\right]{,}\left[{\mathrm{y4}}{,}{\mathrm{x4}}\right]\right]$ (2.6)

Example 3.

We continue with Example 1.  The subalgebras  and    are subalgebras of  .  We use LieAlgebraData to obtain ${\mathrm{SS}}_{1}$ and ${\mathrm{SS}}_{2}$ as subalgebras of Alg2.

 Alg1 > $\mathrm{SS1}≔\left[\mathrm{x1},\mathrm{x2}\right]:$
 Alg1 > $\mathrm{SS2}≔\left[\mathrm{x3},\mathrm{x4}\right]:$
 Alg1 > $\mathrm{L2},\mathrm{SS}≔\mathrm{LieAlgebraData}\left(\mathrm{S1},\left[\mathrm{SS1},\mathrm{SS2}\right],\mathrm{Alg2}\right)$
 ${\mathrm{L2}}{,}{\mathrm{SS}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}\right]{,}\left[\left[\left[{1}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{1}{,}{0}{,}{0}\right]\right]{,}\left[\left[{0}{,}{0}{,}{1}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{1}\right]\right]\right]$ (2.7)
 Alg1 > $\mathrm{Basis}≔\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{Alg2},"FrameBaseVectors"\right)$
 ${\mathrm{Basis}}{:=}\left[{\mathrm{y1}}{,}{\mathrm{y2}}{,}{\mathrm{y3}}{,}{\mathrm{y4}}\right]$ (2.8)
 Alg2 > $\mathrm{newSS1}≔\mathrm{map}\left(\mathrm{DGzip},\mathrm{SS}\left[1\right],\mathrm{Basis},"plus"\right)$
 ${\mathrm{newSS1}}{:=}\left[{\mathrm{y1}}{,}{\mathrm{y2}}\right]$ (2.9)
 Alg2 > $\mathrm{newSS2}≔\mathrm{map}\left(\mathrm{DGzip},\mathrm{SS}\left[2\right],\mathrm{Basis},"plus"\right)$
 ${\mathrm{newSS2}}{:=}\left[{\mathrm{y3}}{,}{\mathrm{y4}}\right]$ (2.10)

Example 4.

The program LieAlgebraData can often be used to simplify the structure equations of a Lie algebra. In this example we initialize a Lie algebra Alg4 and compute its derived series.  We then use elements of the derived series to define a new basis for the Lie algebra.  In this new basis the structure equations of the Lie algebra become much simpler.

 Alg2 > $\mathrm{L3}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg3},\left[5\right]\right],\left[\left[\left[1,2,1\right],2\right],\left[\left[1,2,2\right],5\right],\left[\left[1,2,3\right],1\right],\left[\left[1,2,4\right],2\right],\left[\left[1,2,5\right],-5\right],\left[\left[1,3,2\right],-2\right],\left[\left[1,3,4\right],-1\right],\left[\left[1,3,5\right],2\right],\left[\left[1,4,2\right],-1\right],\left[\left[1,4,4\right],-1\right],\left[\left[1,4,5\right],1\right],\left[\left[1,5,1\right],2\right],\left[\left[1,5,2\right],4\right],\left[\left[1,5,3\right],1\right],\left[\left[1,5,4\right],1\right],\left[\left[1,5,5\right],-4\right],\left[\left[2,3,1\right],2\right],\left[\left[2,3,2\right],6\right],\left[\left[2,3,3\right],1\right],\left[\left[2,3,4\right],1\right],\left[\left[2,3,5\right],-6\right],\left[\left[2,4,4\right],-1\right],\left[\left[2,5,2\right],2\right],\left[\left[2,5,4\right],1\right],\left[\left[2,5,5\right],-2\right],\left[\left[3,4,2\right],1\right],\left[\left[3,4,4\right],1\right],\left[\left[3,4,5\right],-1\right],\left[\left[3,5,1\right],-2\right],\left[\left[3,5,2\right],-5\right],\left[\left[3,5,3\right],-1\right],\left[\left[3,5,5\right],5\right],\left[\left[4,5,4\right],1\right]\right]\right]\right):$
 Alg2 > $\mathrm{DGsetup}\left(\mathrm{L3}\right):$
 Alg3 > $\mathrm{MultiplicationTable}\left("LieBracket"\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{2}{}{\mathrm{e1}}{+}{5}{}{\mathrm{e2}}{+}{\mathrm{e3}}{+}{2}{}{\mathrm{e4}}{-}{5}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e1}}{+}{4}{}{\mathrm{e2}}{+}{\mathrm{e3}}{+}{\mathrm{e4}}{-}{4}{}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e1}}{+}{6}{}{\mathrm{e2}}{+}{\mathrm{e3}}{+}{\mathrm{e4}}{-}{6}{}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e2}}{+}{\mathrm{e4}}{-}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e4}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e1}}{-}{5}{}{\mathrm{e2}}{-}{\mathrm{e3}}{+}{5}{}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}\right]$ (2.11)
 Alg3 > $\mathrm{DS}≔\mathrm{Series}\left("Derived"\right)$
 ${\mathrm{DS}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]{,}\left[{2}{}{\mathrm{e1}}{+}{5}{}{\mathrm{e2}}{+}{\mathrm{e3}}{+}{2}{}{\mathrm{e4}}{-}{5}{}{\mathrm{e5}}{,}{-}{2}{}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{2}{}{\mathrm{e5}}{,}{-}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}\right]{,}\left[{-}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}\right]{,}\left[{}\right]\right]$ (2.12)
 Alg3 > $\mathrm{DS}\left[3\right]$
 $\left[{-}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}\right]$ (2.13)
 Alg3 > $\mathrm{DS}\left[2\right]$
 $\left[{2}{}{\mathrm{e1}}{+}{5}{}{\mathrm{e2}}{+}{\mathrm{e3}}{+}{2}{}{\mathrm{e4}}{-}{5}{}{\mathrm{e5}}{,}{-}{2}{}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{2}{}{\mathrm{e5}}{,}{-}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}\right]$ (2.14)
 Alg3 > $\mathrm{DS}\left[1\right]$
 $\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}\right]$ (2.15)

For the new basis we take the vector in DS[3], 2 vectors in DS[2] independent of DS[3], and 3 vectors in DS[1] independent of the 3 vectors in DS[2].

 Alg3 > $\mathrm{newBasis}≔\left[-\mathrm{e2}-\mathrm{e4}+\mathrm{e5},-2\mathrm{e2}-\mathrm{e4}+2\mathrm{e5},2\mathrm{e1}+5\mathrm{e2}+\mathrm{e3}+2\mathrm{e4}-5\mathrm{e5},\mathrm{e1},\mathrm{e5}\right]$
 ${\mathrm{newBasis}}{:=}\left[{-}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{\mathrm{e5}}{,}{-}{2}{}{\mathrm{e2}}{-}{\mathrm{e4}}{+}{2}{}{\mathrm{e5}}{,}{2}{}{\mathrm{e1}}{+}{5}{}{\mathrm{e2}}{+}{\mathrm{e3}}{+}{2}{}{\mathrm{e4}}{-}{5}{}{\mathrm{e5}}{,}{\mathrm{e1}}{,}{\mathrm{e5}}\right]$ (2.16)

Now we create a Lie algebra from this adapted basis.

 Alg3 > $\mathrm{L4},N≔\mathrm{LieAlgebraData}\left(\mathrm{newBasis},\mathrm{Alg4},"Matrix"\right)$
 ${\mathrm{L4}}{,}{N}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e1}}{+}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{3}{}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{6}{}{\mathrm{e1}}{-}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e3}}\right]{,}\left[\begin{array}{rrrrr}{0}& {0}& {2}& {1}& {0}\\ {-}{1}& {-}{2}& {5}& {0}& {0}\\ {0}& {0}& {1}& {0}& {0}\\ {-}{1}& {-}{1}& {2}& {0}& {0}\\ {1}& {2}& {-}{5}& {0}& {1}\end{array}\right]$ (2.17)
 Alg3 > $\mathrm{DGsetup}\left(\mathrm{L4}\right):$

The structure equations are much simpler.

 Alg4 > $\mathrm{MultiplicationTable}\left("LieBracket"\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e1}}{+}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{3}{}{\mathrm{e1}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{6}{}{\mathrm{e1}}{-}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e3}}\right]$ (2.18)

The Matrix N defines an isomorphism from Alg4 to Alg3.

 Alg4 > $\mathrm{LinearAlgebra}:-\mathrm{Determinant}\left(N\right)$
 ${-}{1}$ (2.19)
 Alg4 > $\mathrm{Query}\left(\mathrm{Alg4},\mathrm{Alg3},N,"Homomorphism"\right)$
 ${\mathrm{true}}$ (2.20)