 Complexify - Maple Help

LieAlgebras[Complexify] find the complexification of a Lie algebra

Calling Sequences

Complexify(AlgName1, AlgName2)

Parameters

AlgName1  - name or string, the name of a Lie algebra g

AlgName2  - name or string, the name for the complexification of g Description

 • The complexification of a real Lie algebra $\mathrm{𝔤}$ of dimension $n$ is a real Lie algebra of dimension $2n$.  If is a basis for $\mathrm{𝔤}$, then , where , is a basis for the complexification of $\mathrm{𝔤}$,
 • Complexify(AlgName1, AlgName2) calculates the complexification of the Lie algebra g defined by AlgName1.
 • A Lie algebra data structure is returned for the complexified Lie algebra with name AlgName2. The structure equations for the complexification are displayed. (A Lie algebra data structure contains the structure constants of a Lie algebra in a standard format used by the LieAlgebras package).
 • The command Complexify is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Complexify(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Complexify(...). Examples

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

Example 1.

First we initialize a Lie algebra and then display its multiplication table.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[3\right]\right],\left[\left[\left[1,3,1\right],1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,3,2\right],1\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

We complexify Alg1 and call the result Alg2.

 Alg1 > $\mathrm{L2}≔\mathrm{Complexify}\left(\mathrm{Alg1},\mathrm{Alg2}\right)$
 ${\mathrm{L2}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{+}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e4}}{+}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e4}}{-}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{-}{\mathrm{e2}}\right]$ (2.1)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2}\right):$

We note that the original Lie algebra [e1, e2, e3], as a subalgebra of its complexification, admits a symmetric complement.

 Alg2 > $\mathrm{Query}\left(\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right],\left[\mathrm{e4},\mathrm{e5},\mathrm{e6}\right],"SymmetricPair"\right)$
 ${\mathrm{true}}$ (2.2)
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