 MinimalSubalgebra - Maple Help

LieAlgebras[MinimalSubalgebra] - find the smallest Lie subalgebra containing a given set of vectors from a Lie algebra, find the smallest matrix algebra containing a given set of matrices

Calling Sequences

MinimalSubalgebra(S)

MinimalSubalgebra(M)

Parameters

S        - a list of vectors in a Lie algebra

M        - a list of square matrices Description

 • MinimalSubalgebra(S) calculates the smallest Lie subalgebra containing the list of vectors S from a defined Lie algebra $\mathrm{𝔤}$. A list of basis vectors for the subalgebra returned.
 • MinimalSubalgebra(M) calculates the smallest matrix algebra containing the matrices in the list M.
 • The command MinimalSubalgebra is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form MinimalSubalgebra(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MinimalSubalgebra(...). Examples

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

Example 1.

First we initialize a Lie algebra and display the multiplication table.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[5\right]\right],\left[\left[\left[1,5,1\right],2\right],\left[\left[2,3,1\right],1\right],\left[\left[2,5,2\right],1\right],\left[\left[2,5,3\right],1\right],\left[\left[3,5,3\right],1\right],\left[\left[4,5,4\right],2\right]\right]\right]\right)$
 ${\mathrm{L1}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e4}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

Find the minimal subalgebra containing

 Alg1 > $\mathrm{S1}≔\left[\mathrm{e1},\mathrm{e3}\right]:$
 Alg1 > $\mathrm{A1}≔\mathrm{MinimalSubalgebra}\left(\mathrm{S1}\right)$
 ${\mathrm{A1}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]$ (2.2)

Find the minimal subalgebra containing

 Alg1 > $\mathrm{S2}≔\left[\mathrm{e2},\mathrm{e3}\right]:$
 Alg1 > $\mathrm{A2}≔\mathrm{MinimalSubalgebra}\left(\mathrm{S2}\right)$
 ${\mathrm{A2}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}\right]$ (2.3)
 Alg1 > $\mathrm{Query}\left(\mathrm{S2},"Subalgebra"\right)$
 ${\mathrm{false}}$ (2.4)
 Alg1 > $\mathrm{Query}\left(\mathrm{A2},"Subalgebra"\right)$
 ${\mathrm{true}}$ (2.5)

Find the minimal subalgebra containing

 Alg1 > $\mathrm{S3}≔\left[\mathrm{e2},\mathrm{e5}\right]:$
 Alg1 > $\mathrm{A3}≔\mathrm{MinimalSubalgebra}\left(\mathrm{S3}\right)$
 ${\mathrm{A3}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e5}}\right]$ (2.6)

Example 2.

The command MinimalSubalgebra also works with matrices.

 Alg1 > $M≔\left[\mathrm{Matrix}\left(\left[\left[1,0,0\right],\left[0,1,0\right],\left[1,0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1,0\right],\left[0,1,1\right],\left[0,1,0\right]\right]\right)\right]$ Alg1 > $N≔\mathrm{MinimalSubalgebra}\left(M\right)$ We can use the LieAlgebraData command to verify that the set of matrices N defines a 4-dimensional Lie algebra and to determine the commutator relationships.

 Alg1 > $\mathrm{LieAlgebraData}\left(N\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}{+}{\mathrm{e4}}\right]$ (2.7)
 Alg1 > 

Here denote the four matrices N, N, N, N.