DirectSum - Maple Help

LieAlgebras[DirectSum] - create the direct sum of a list of Lie algebras

Calling Sequences

DirectSum(Summands, AlgName)

Parameters

Summands    - a list of Lie algebra data structures or names of Lie algebras

AlgName     - a name or string, the name of the direct sum Lie algebra being created

Description

 • The direct sum of two Lie algebras ${\mathrm{𝔤}}_{1}$ and is the vector space direct sum  with Lie bracket

, where   and  .

 • DirectSum(Summands, AlgName) creates a Lie algebra data structure for the direct sum of the Lie algebras listed in the first argument. The name given to the direct sum algebra is AlgName. The structure equations for the direct sum are displayed.
 • A Lie algebra data structure contains the structure constants in a standard format used by the LieAlgebras package.  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 command DirectSum is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form DirectSum(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-DirectSum(...).

Examples

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

Example 1.

First we define 3 Lie algebra data structures and initialize their Lie algebras.  We display the multiplication tables.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[2\right]\right],\left[\left[\left[1,2,1\right],1\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{L1},\left[x\right],\left[a\right]\right):$
 Alg1 > $\mathrm{L2}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg2},\left[3\right]\right],\left[\left[\left[1,2,1\right],1\right],\left[\left[1,3,2\right],-2\right],\left[\left[2,3,3\right],1\right]\right]\right]\right):$
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2},\left[y\right],\left[b\right]\right):$
 Alg2 > $\mathrm{L3}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg3},\left[1\right]\right],\left[\right]\right]\right):$
 Alg2 > $\mathrm{DGsetup}\left(\mathrm{L3},\left[z\right],\left[c\right]\right):$
 Alg3 > $\mathrm{MultiplicationTable}\left(\mathrm{Alg1},"LieBracket"\right),\mathrm{MultiplicationTable}\left(\mathrm{Alg2},"LieBracket"\right),\mathrm{MultiplicationTable}\left(\mathrm{Alg3},"LieBracket"\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}\right]{,}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e3}}\right]{,}\left[{}\right]$ (2.1)

Create the direct sum of the Lie algebra data structures L1 and L2.

 Alg3 > $\mathrm{L4}≔\mathrm{DirectSum}\left(\left[\mathrm{L1},\mathrm{L2}\right],\mathrm{Alg4}\right)$
 ${\mathrm{L4}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}\right]$ (2.2)
 Alg3 > $\mathrm{DGsetup}\left(\mathrm{L4}\right):$
 Alg4 > $\mathrm{MultiplicationTable}\left(\mathrm{Alg4},"LieTable"\right)$

Create the direct sum of the Lie algebras Alg1, Alg2 and the Lie algebra data structure L3.

 Alg4 > $\mathrm{L5}≔\mathrm{DirectSum}\left(\left[\mathrm{Alg1},\mathrm{Alg2},\mathrm{L3}\right],\mathrm{Alg5}\right)$
 ${\mathrm{L5}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e5}}\right]$ (2.3)
 Alg4   > $\mathrm{DGsetup}\left(\mathrm{L5}\right):$
 Alg2 > $\mathrm{MultiplicationTable}\left(\mathrm{Alg5},"LieTable"\right)$