SplitAndCompactForms - Maple Help

LieAlgebras[SplitAndCompactForms] - find the real split and real compact forms of a real semi-simple Lie algebra

Calling Sequences

SplitAndCompactForms(CSA, RDS, PosRts)

Parameters

CSA      - a list of vectors, defining the Cartan subalgebra of a real semi-simple Lie algebra $\mathrm{𝔤}$

RDS      - a table, giving the root space decomposition of $𝔤$ with respect to the given Cartan subalgebra

PosRts   - a list of vectors, defining a positive set of roots for the given root space decomposition

Description

 • A real semi-simple Lie algebra $𝔤$ is called a split semi-simple Lie algebra if there exists a real root space decomposition of $𝔤$, that is to say, there is a Cartan subalgebrasuch that each of the adjoint matrices $\mathrm{ad}\left(x\right)$, for is diagonalizable over the real numbers. A real semi-simple Lie algebra $𝔤$ is called compact if the Killing form is negative-definite. This implies that the eigenvalues for the adjoint matrices $\mathrm{ad}\left(x\right)$, for are all pure imaginary for any choice of Cartan subalgebra.
 • Given any real semi-simple Lie algebra $𝔤$, then

[i] there exists a complex change of basis so that the Lie algebra $\mathrm{𝔤}'$defined by these new basis vectors is a real split Lie algebra, and

[ii] there exists another complex change of basis so that the Lie algebra $\mathrm{𝔤}''$defined by these new basis vectors is a real compact Lie algebra.

All three Lie algebras $𝔤$, $\mathrm{𝔤}\mathit{'}$ and $\mathrm{𝔤}\mathit{'}\mathit{'}$ are isomorphic as complex Lie algebras but, granted that $\mathrm{𝔤}$ is neither split nor compact, $𝔤$, $\mathrm{𝔤}\mathit{'}$ and $\mathrm{𝔤}\mathit{'}\mathit{'}$ are non-isomorphic as real Lie algebras. The Lie algebra $\mathrm{𝔤}\mathit{'}$ is called the split real form of $\mathrm{𝔤}$ and the Lie algebra $\mathrm{𝔤}\mathit{'}\mathit{'}$ is called the compact real form of $\mathrm{𝔤}$.

 • The command SplitAndCompactForms returns 2 lists of vectors which give basis for $\mathrm{𝔤}\mathit{'}$ and $\mathrm{𝔤}\mathit{'}\mathit{'}$. By applying the commands LieAlgebraData and DGsetup to these lists of vectors, one obtains the split real form and compact real form of the Lie algebra $\mathrm{𝔤}$.
 • The basis for the split real form returned by the procedure SplitAndCompactForms is a ChevalleyBasis for $\mathrm{𝔤}\mathit{'}$. From the properties of the Chevalley basis described in the help page ChevalleyBasis, it can be established that the basis

defines a real Lie algebra giving the compact real form $\mathrm{𝔤}$''.

Examples

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

Example 1.

In this example the 6-dimensional Lie algebra is created and its real and compact forms are calculated. Use the command SimpleLieAlgebraData to obtain the structure equations.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("so\left(3,1\right)",\mathrm{so31}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: so31}}$ (2.2)

To calculate the real and compact forms one needs a Cartan subalgebra, the root space decomposition and the positive roots. These data can be calculated using the commands CartanSubalgebra, RootSpaceDecomposition and PositiveRoots. Alternatively, since the Lie algebra has been created from the command SimpleLieAlgebraData, one can used the command SimpleLieAlgebraProperties to retrieve the stored values of the Cartan subalgebra, root space decomposition and positive roots.

 M > $\mathrm{Properties}≔\mathrm{SimpleLieAlgebraProperties}\left(\mathrm{so31}\right):$
 so31 > $\mathrm{CSA}≔\mathrm{Properties}\left["CartanSubalgebra"\right]$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]$ (2.3)
 so31 > $\mathrm{RSD}≔\mathrm{eval}\left(\mathrm{Properties}\left["RootSpaceDecomposition"\right]\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{-}{1}{,}{I}\right]{=}{\mathrm{e4}}{-}{I}{}{\mathrm{e5}}{,}\left[{-}{1}{,}{-}{I}\right]{=}{\mathrm{e4}}{+}{I}{}{\mathrm{e5}}{,}\left[{1}{,}{I}\right]{=}{\mathrm{e2}}{-}{I}{}{\mathrm{e3}}{,}\left[{1}{,}{-}{I}\right]{=}{\mathrm{e2}}{+}{I}{}{\mathrm{e3}}\right]\right)$ (2.4)
 so31 > $\mathrm{PosRts}≔\mathrm{Properties}\left["PositiveRoots"\right]$
 ${\mathrm{PosRts}}{:=}\left[\left[\begin{array}{c}{1}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{1}\\ {I}\end{array}\right]\right]$ (2.5)

Calculate the basis for the split and compact forms.

 so31 > $\mathrm{BasisS},\mathrm{BasisC}≔\mathrm{SplitAndCompactForms}\left(\mathrm{CSA},\mathrm{RSD},\mathrm{PosRts}\right)$
 ${\mathrm{BasisS}}{,}{\mathrm{BasisC}}{:=}\left[{\mathrm{e1}}{+}{I}{}{\mathrm{e6}}{,}{\mathrm{e1}}{-}{I}{}{\mathrm{e6}}{,}{-}\frac{{1}}{{2}}{}{\mathrm{e2}}{-}\frac{{I}}{{2}}{}{\mathrm{e3}}{,}{-}\frac{{1}}{{2}}{}{\mathrm{e2}}{+}\frac{{I}}{{2}}{}{\mathrm{e3}}{,}{-}{\mathrm{e4}}{+}{I}{}{\mathrm{e5}}{,}{-}{\mathrm{e4}}{-}{I}{}{\mathrm{e5}}\right]{,}\left[{I}{}{\mathrm{e1}}{-}{\mathrm{e6}}{,}{I}{}{\mathrm{e1}}{+}{\mathrm{e6}}{,}{-}\frac{{1}}{{2}}{}{\mathrm{e2}}{-}\frac{{I}}{{2}}{}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{I}{}{\mathrm{e5}}{,}{-}\frac{{1}}{{2}}{}{\mathrm{e2}}{+}\frac{{I}}{{2}}{}{\mathrm{e3}}{-}{\mathrm{e4}}{-}{I}{}{\mathrm{e5}}{,}{-}\frac{{I}}{{2}}{}{\mathrm{e2}}{+}\frac{{1}}{{2}}{}{\mathrm{e3}}{+}{I}{}{\mathrm{e4}}{+}{\mathrm{e5}}{,}{-}\frac{{I}}{{2}}{}{\mathrm{e2}}{-}\frac{{1}}{{2}}{}{\mathrm{e3}}{+}{I}{}{\mathrm{e4}}{-}{\mathrm{e5}}\right]$ (2.6)

Here is the split real form of the Lie algebra so31.

 so31 > $\mathrm{LDS}≔\mathrm{LieAlgebraData}\left(\mathrm{BasisS},\mathrm{so31S}\right)$
 ${\mathrm{LDS}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e2}}\right]$ (2.7)
 so31 > $\mathrm{DGsetup}\left(\mathrm{LDS},\left['\mathrm{h1}','\mathrm{h2}','\mathrm{x1}','\mathrm{x2}','\mathrm{y1}','\mathrm{y2}'\right],\left['\mathrm{\theta }'\right]\right)$
 ${\mathrm{Lie algebra: so31S}}$ (2.8)

The multiplication table is:

 so31S > $\mathrm{MultiplicationTable}\left("LieTable"\right)$
 $\left[\begin{array}{cccccccc}{}& {|}& {\mathrm{h1}}& {\mathrm{h2}}& {\mathrm{x1}}& {\mathrm{x2}}& {\mathrm{y1}}& {\mathrm{y2}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{h1}}& {|}& {0}& {0}& {2}{}{\mathrm{x1}}& {0}& {-}{2}{}{\mathrm{y1}}& {0}\\ {\mathrm{h2}}& {|}& {0}& {0}& {0}& {2}{}{\mathrm{x2}}& {0}& {-}{2}{}{\mathrm{y2}}\\ {\mathrm{x1}}& {|}& {-}{2}{}{\mathrm{x1}}& {0}& {0}& {0}& {-}{\mathrm{h1}}& {0}\\ {\mathrm{x2}}& {|}& {0}& {-}{2}{}{\mathrm{x2}}& {0}& {0}& {0}& {-}{\mathrm{h2}}\\ {\mathrm{y1}}& {|}& {2}{}{\mathrm{y1}}& {0}& {\mathrm{h1}}& {0}& {0}& {0}\\ {\mathrm{y2}}& {|}& {0}& {2}{}{\mathrm{y2}}& {0}& {\mathrm{h2}}& {0}& {0}\end{array}\right]$ (2.9)

from which one sees immediately that  is a Cartan subalgebra for which the root space decomposition is real. Therefore this is a real form.

 so31S > $\mathrm{RootSpaceDecomposition}\left(\left[\mathrm{h1},\mathrm{h2}\right]\right)$
 ${\mathrm{table}}\left(\left[\left[{0}{,}{-}{2}\right]{=}{\mathrm{y2}}{,}\left[{2}{,}{0}\right]{=}{\mathrm{x1}}{,}\left[{-}{2}{,}{0}\right]{=}{\mathrm{y1}}{,}\left[{0}{,}{2}\right]{=}{\mathrm{x2}}\right]\right)$ (2.10)

Here is the compact form of the Lie algebra so31.

 so31 > $\mathrm{LDS}≔\mathrm{LieAlgebraData}\left(\mathrm{BasisC},\mathrm{so31C}\right)$
 ${\mathrm{LDS}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e2}}\right]$ (2.11)
 so31 > $\mathrm{DGsetup}\left(\mathrm{LDS},\left[E\right],\left[\mathrm{\omega }\right]\right)$
 ${\mathrm{Lie algebra: so31C}}$ (2.12)

The Killing form is negative-definite and so this is indeed the compact form.

 so31C > $\mathrm{Killing}\left(\right)$
 $\left[\begin{array}{rrrrrr}{-}{8}& {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{8}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{8}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{8}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{8}& {0}\\ {0}& {0}& {0}& {0}& {0}& {-}{8}\end{array}\right]$ (2.13)

By construction, the first 2 vectors define a Cartan subalgebra. The root vectors are pure imaginary.

 so31C > $\mathrm{RootSpaceDecomposition}\left(\left[\mathrm{E1},\mathrm{E2}\right]\right)$
 ${\mathrm{table}}\left(\left[\left[{0}{,}{2}{}{I}\right]{=}{\mathrm{E4}}{-}{I}{}{\mathrm{E6}}{,}\left[{-}{2}{}{I}{,}{0}\right]{=}{\mathrm{E3}}{+}{I}{}{\mathrm{E5}}{,}\left[{0}{,}{-}{2}{}{I}\right]{=}{\mathrm{E4}}{+}{I}{}{\mathrm{E6}}{,}\left[{2}{}{I}{,}{0}\right]{=}{\mathrm{E3}}{-}{I}{}{\mathrm{E5}}\right]\right)$ (2.14)