LieAlgebraRoots - Maple Help

LieAlgebras[LieAlgebraRoots] - find a root or the roots for a semi-simple Lie algebra from a root space and the Cartan subalgebra; or from a root space decomposition

Calling Sequences

LieAlgebraRoots()

LieAlgebraRoots()

Parameters

X     - a vector in a Lie algebra, defining a root space

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

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

Description

 • Let g be a Lie algebra and h a Cartan subalgebra. Let be a basis for $\mathrm{𝔥}$. A root for g with respect to this basis is a non-zero $m$-tuple of complex numbers such that  $x$  (*)  for some .
 • The set of  which satisfy (*) is called the root space of g defined by and denoted by  A basic theorem in the structure theory of semi-simple Lie algebras asserts that the root spaces  are 1-dimensional.
 • The first calling sequence calculates the root ${\mathrm{α}}_{}$ for the given root space $X$. If is not a root space, then an empty vector is returned.
 • The second calling sequence simply returns the indices, as column vectors, for the table defining the root space decomposition.

Examples

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

Example 1.

Use the command SimpleLieAlgebraData to initialize the simple Lie algebra This is a 15-dimensional Lie algebra of skew-Hermitian matrices.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("su\left(4\right)",\mathrm{su4},\mathrm{labelformat}="gl",\mathrm{labels}=\left['E','\mathrm{\theta }'\right]\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{2}{}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e12}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e12}}\right]{=}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{-}{2}{}{\mathrm{e1}}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{-}{2}{}{\mathrm{e2}}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e15}}\right]{=}{-}{2}{}{\mathrm{e1}}{-}{2}{}{\mathrm{e2}}{-}{2}{}{\mathrm{e3}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e4}}\right]{,}\left[{\mathrm{Ei11}}{,}{\mathrm{Ei22}}{,}{\mathrm{Ei33}}{,}{\mathrm{E12}}{,}{\mathrm{E23}}{,}{\mathrm{E34}}{,}{\mathrm{E13}}{,}{\mathrm{E24}}{,}{\mathrm{E14}}{,}{\mathrm{Ei12}}{,}{\mathrm{Ei23}}{,}{\mathrm{Ei34}}{,}{\mathrm{Ei13}}{,}{\mathrm{Ei24}}{,}{\mathrm{Ei14}}\right]{,}\left[{\mathrm{thetai11}}{,}{\mathrm{thetai22}}{,}{\mathrm{thetai33}}{,}{\mathrm{θ12}}{,}{\mathrm{θ23}}{,}{\mathrm{θ34}}{,}{\mathrm{θ13}}{,}{\mathrm{θ24}}{,}{\mathrm{θ14}}{,}{\mathrm{thetai12}}{,}{\mathrm{thetai23}}{,}{\mathrm{thetai34}}{,}{\mathrm{thetai13}}{,}{\mathrm{thetai24}}{,}{\mathrm{thetai14}}\right]$ (2.1)
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: su4}}$ (2.2)

The explicit matrices defining $\mathrm{su}\left(4\right)$are given by the StandardRepresentation command.

 su4 > $\mathrm{StandardRepresentation}\left(\mathrm{su4}\right)$
 $\left[\left[\begin{array}{cccc}{I}& {0}& {0}& {0}\\ {0}& {-}{I}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {I}& {0}& {0}\\ {0}& {0}& {-}{I}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {I}& {0}\\ {0}& {0}& {0}& {-}{I}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {-}{1}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}\\ {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {1}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrr}{0}& {0}& {0}& {-}{1}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {I}& {0}& {0}\\ {I}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {I}& {0}\\ {0}& {I}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {I}\\ {0}& {0}& {I}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {I}& {0}\\ {0}& {0}& {0}& {0}\\ {I}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}\\ {0}& {I}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{cccc}{0}& {0}& {0}& {I}\\ {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}\\ {I}& {0}& {0}& {0}\end{array}\right]\right]$ (2.3)

The diagonal matrices determine a Cartan subalgebra.

 su4 > $\mathrm{CSA}≔\left[\mathrm{Ei11},\mathrm{Ei22},\mathrm{Ei33}\right]$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{Ei11}}{,}{\mathrm{Ei22}}{,}{\mathrm{Ei33}}\right]$ (2.4)

We use the Query command to check that (2.4) is a Cartan subalgebra.

 su4 > $\mathrm{Query}\left(\mathrm{CSA},"CartanSubalgebra"\right)$
 ${\mathrm{true}}$ (2.5)

Find the root for the root space

 > $\mathrm{LieAlgebraRoots}\left(\mathrm{E14}-I\mathrm{Ei14},\mathrm{CSA}\right)$
 $\left[\begin{array}{c}{-}{I}\\ {0}\\ {-}{I}\end{array}\right]$ (2.6)

Note that the command RootSpace  performs the inverse operation to  - given a root, the command returns the corresponding root space.

 su4 > $\mathrm{RootSpace}\left(⟨-I,0,-I⟩,\mathrm{CSA}\right)$
 ${\mathrm{E14}}{-}{I}{}{\mathrm{Ei14}}$ (2.7)

Example 2.

If the complete root space decomposition is given as a table, then the command returns the indices of that table as column vectors.

 su4 > $\mathrm{RSD}≔\mathrm{RootSpaceDecomposition}\left(\mathrm{CSA}\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{-}{I}{,}{I}{,}{I}\right]{=}{\mathrm{E24}}{+}{I}{}{\mathrm{Ei24}}{,}\left[{0}{,}{I}{,}{-}{2}{}{I}\right]{=}{\mathrm{E34}}{-}{I}{}{\mathrm{Ei34}}{,}\left[{0}{,}{-}{I}{,}{2}{}{I}\right]{=}{\mathrm{E34}}{+}{I}{}{\mathrm{Ei34}}{,}\left[{I}{,}{-}{2}{}{I}{,}{I}\right]{=}{\mathrm{E23}}{-}{I}{}{\mathrm{Ei23}}{,}\left[{-}{I}{,}{2}{}{I}{,}{-}{I}\right]{=}{\mathrm{E23}}{+}{I}{}{\mathrm{Ei23}}{,}\left[{I}{,}{0}{,}{I}\right]{=}{\mathrm{E14}}{+}{I}{}{\mathrm{Ei14}}{,}\left[{-}{I}{,}{0}{,}{-}{I}\right]{=}{\mathrm{E14}}{-}{I}{}{\mathrm{Ei14}}{,}\left[{I}{,}{-}{I}{,}{-}{I}\right]{=}{\mathrm{E24}}{-}{I}{}{\mathrm{Ei24}}{,}\left[{-}{2}{}{I}{,}{I}{,}{0}\right]{=}{\mathrm{E12}}{-}{I}{}{\mathrm{Ei12}}{,}\left[{I}{,}{I}{,}{-}{I}\right]{=}{\mathrm{E13}}{+}{I}{}{\mathrm{Ei13}}{,}\left[{-}{I}{,}{-}{I}{,}{I}\right]{=}{\mathrm{E13}}{-}{I}{}{\mathrm{Ei13}}{,}\left[{2}{}{I}{,}{-}{I}{,}{0}\right]{=}{\mathrm{E12}}{+}{I}{}{\mathrm{Ei12}}\right]\right)$ (2.8)
 su4 > $\mathrm{LieAlgebraRoots}\left(\mathrm{RSD}\right)$
 $\left[\left[\begin{array}{c}{-}{I}\\ {I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {I}\\ {-}{2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{0}\\ {-}{I}\\ {2}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {-}{2}{}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {2}{}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {0}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {0}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {-}{I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{-}{2}{}{I}\\ {I}\\ {0}\end{array}\right]{,}\left[\begin{array}{c}{I}\\ {I}\\ {-}{I}\end{array}\right]{,}\left[\begin{array}{c}{-}{I}\\ {-}{I}\\ {I}\end{array}\right]{,}\left[\begin{array}{c}{2}{}{I}\\ {-}{I}\\ {0}\end{array}\right]\right]$ (2.9)