DifferentialGeometry/LieAlgebras/Query/RootSpaceDecomposition - Maple Help
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Query[RootSpaceDecomposition] - check that a table of roots and root spaces gives a root space decomposition for a semi-simple Lie algebra with respect to a given Cartan subalgebra

Calling Sequences

Query()

Parameters

CSA     - a list of vectors in a Lie algebra, defining the Cartan subalgebra of a semi-simple Lie algebra

RSD     - a table, defining a root space decomposition of a Lie algebra

options - the keyword argument output = "root"

Description

 • Let g be a semi-simple Lie algebra and h a Cartan subalgebra.  Let be a basis forThe linear transformations are simultaneously diagonalizable over C -- if  g is a common eigenvector for all these transformations, then . The $m$-tuples  ∈ ${\mathrm{ℂ}}^{m}$ are called the roots Δ of  with respect to the Cartan subalgebra . The eigenspace decomposition is called the root space decomposition of g with respect to h.
 • For each root and corresponding root space $x$, this query checks that .  It also checks that the span of the Cartan subalgebra and the root spaces is the full Lie algebra $𝔤$ .
 • With output = "root", this query will return the root  if the equations fail.

Examples

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

Example 1.

Check the root space decomposition for a 10-dimensional Lie algebra.

Here is the Lie algebra data structure.

 > $\mathrm{LD}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{alg},\left[10\right]\right],\left[\left[\left[1,2,3\right],1\right],\left[\left[1,3,2\right],-2\right],\left[\left[1,3,4\right],2\right],\left[\left[1,4,3\right],-1\right],\left[\left[1,5,6\right],1\right],\left[\left[1,6,5\right],-2\right],\left[\left[1,6,7\right],2\right],\left[\left[1,7,6\right],-1\right],\left[\left[1,8,9\right],1\right],\left[\left[1,9,8\right],-2\right],\left[\left[1,9,10\right],2\right],\left[\left[1,10,9\right],-1\right],\left[\left[2,3,1\right],1\right],\left[\left[2,5,8\right],2\right],\left[\left[2,6,9\right],1\right],\left[\left[2,8,5\right],-2\right],\left[\left[2,9,6\right],-1\right],\left[\left[3,4,1\right],1\right],\left[\left[3,5,9\right],1\right],\left[\left[3,6,8\right],2\right],\left[\left[3,6,10\right],2\right],\left[\left[3,7,9\right],1\right],\left[\left[3,8,6\right],-1\right],\left[\left[3,9,5\right],-2\right],\left[\left[3,9,7\right],-2\right],\left[\left[3,10,6\right],-1\right],\left[\left[4,6,9\right],1\right],\left[\left[4,7,10\right],2\right],\left[\left[4,9,6\right],-1\right],\left[\left[4,10,7\right],-2\right],\left[\left[5,6,1\right],1\right],\left[\left[5,8,2\right],2\right],\left[\left[5,9,3\right],1\right],\left[\left[6,7,1\right],1\right],\left[\left[6,8,3\right],1\right],\left[\left[6,9,2\right],2\right],\left[\left[6,9,4\right],2\right],\left[\left[6,10,3\right],1\right],\left[\left[7,9,3\right],1\right],\left[\left[7,10,4\right],2\right],\left[\left[8,9,1\right],1\right],\left[\left[9,10,1\right],1\right]\right]\right]\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e4}}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e7}}{-}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e10}}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e10}}{+}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e7}}{-}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e1}}\right]$ (2.1)

Initialize the Lie algebra.

 alg > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: alg}}$ (2.2)

Define a subalgebra and check that it is a Cartan subalgebra.

 alg > $\mathrm{CSA}≔\mathrm{evalDG}\left(\left[\mathrm{e1},\mathrm{e8}+\mathrm{e10}\right]\right)$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e8}}{+}{\mathrm{e10}}\right]$ (2.3)
 > $\mathrm{Query}\left(\mathrm{CSA},"CartanSubalgebra"\right)$
 ${\mathrm{true}}$ (2.4)

Define a table of roots and root spaces and check that it gives a root space decomposition.

 alg > $\mathrm{RSD}≔\mathrm{map}\left(\mathrm{evalDG},\mathrm{table}\left(\left[\left[2I,0\right]=\mathrm{e8}-I\mathrm{e9}-\mathrm{e10},\left[2I,2I\right]=\mathrm{e2}-I\mathrm{e3}-\mathrm{e4}-I\mathrm{e5}-\mathrm{e6}+I\mathrm{e7},\left[-2I,2I\right]=\mathrm{e2}+I\mathrm{e3}-\mathrm{e4}-I\mathrm{e5}+\mathrm{e6}+I\mathrm{e7},\left[0,2I\right]=\mathrm{e2}+\mathrm{e4}-I\mathrm{e5}-I\mathrm{e7},\left[-2I,0\right]=\mathrm{e8}+I\mathrm{e9}-\mathrm{e10},\left[2I,-2I\right]=\mathrm{e2}-I\mathrm{e3}-\mathrm{e4}+I\mathrm{e5}+\mathrm{e6}-I\mathrm{e7},\left[0,-2I\right]=\mathrm{e2}+\mathrm{e4}+I\mathrm{e5}+I\mathrm{e7},\left[-2I,-2I\right]=\mathrm{e2}+I\mathrm{e3}-\mathrm{e4}+I\mathrm{e5}-\mathrm{e6}-I\mathrm{e7}\right]\right)\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{0}{,}{2}{}{I}\right]{=}{\mathrm{e2}}{+}{\mathrm{e4}}{-}{I}{}{\mathrm{e5}}{-}{I}{}{\mathrm{e7}}{,}\left[{-}{2}{}{I}{,}{2}{}{I}\right]{=}{\mathrm{e2}}{+}{I}{}{\mathrm{e3}}{-}{\mathrm{e4}}{-}{I}{}{\mathrm{e5}}{+}{\mathrm{e6}}{+}{I}{}{\mathrm{e7}}{,}\left[{2}{}{I}{,}{2}{}{I}\right]{=}{\mathrm{e2}}{-}{I}{}{\mathrm{e3}}{-}{\mathrm{e4}}{-}{I}{}{\mathrm{e5}}{-}{\mathrm{e6}}{+}{I}{}{\mathrm{e7}}{,}\left[{2}{}{I}{,}{0}\right]{=}{\mathrm{e8}}{-}{I}{}{\mathrm{e9}}{-}{\mathrm{e10}}{,}\left[{-}{2}{}{I}{,}{-}{2}{}{I}\right]{=}{\mathrm{e2}}{+}{I}{}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{I}{}{\mathrm{e5}}{-}{\mathrm{e6}}{-}{I}{}{\mathrm{e7}}{,}\left[{2}{}{I}{,}{-}{2}{}{I}\right]{=}{\mathrm{e2}}{-}{I}{}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{I}{}{\mathrm{e5}}{+}{\mathrm{e6}}{-}{I}{}{\mathrm{e7}}{,}\left[{-}{2}{}{I}{,}{0}\right]{=}{\mathrm{e8}}{+}{I}{}{\mathrm{e9}}{-}{\mathrm{e10}}{,}\left[{0}{,}{-}{2}{}{I}\right]{=}{\mathrm{e2}}{+}{\mathrm{e4}}{+}{I}{}{\mathrm{e5}}{+}{I}{}{\mathrm{e7}}\right]\right)$ (2.5)
 alg > $\mathrm{Query}\left(\mathrm{CSA},\mathrm{RSD},"RootSpaceDecomposition"\right)$
 ${\mathrm{true}}$ (2.6)