RootSpace - Maple Help

LieAlgebras[RootSpace] - find a root space for a semi-simple Lie algebra from a Cartan subalgebra or a root space decomposition

Calling Sequences

RootSpace(RV, CSA)

RootSpace(RV, RSD)

Parameters

RV    - a column vector

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 theorem of semi-simple Lie algebras asserts that the root spaces are 1-dimensional.
 • The first call sequence calculates the root space for a given root. If is not a root, then the zero vector (in $\mathrm{𝔤}$) is returned.
 • The second calling sequence simply returns the table entry in the table of root spaces corresponding to the root ${\mathrm{α}}_{}$.

Examples

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

Example 1.

Use the command SimpleLieAlgebraData to obtain the Lie algebra data for the simple Lie algebra This is the 15-dimensional Lie algebra of trace-free, skew-Hermitian matrices.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("su\left(4\right)",\mathrm{su4},\mathrm{labelformat}="gl",\mathrm{labels}=\left['U','\mathrm{\eta }'\right]\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{2}{}{\mathrm{e15}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e15}}\right]{=}{-}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e9}}{,}\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}}{-}{2}{}{\mathrm{e2}}{,}\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}}{-}{2}{}{\mathrm{e3}}{,}\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{e3}}{,}\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}}{,}\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}}{,}\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{Ui11}}{,}{\mathrm{Ui22}}{,}{\mathrm{Ui33}}{,}{\mathrm{U12}}{,}{\mathrm{U23}}{,}{\mathrm{U34}}{,}{\mathrm{U13}}{,}{\mathrm{U24}}{,}{\mathrm{U14}}{,}{\mathrm{Ui12}}{,}{\mathrm{Ui23}}{,}{\mathrm{Ui34}}{,}{\mathrm{Ui13}}{,}{\mathrm{Ui24}}{,}{\mathrm{Ui14}}\right]{,}\left[{\mathrm{etai11}}{,}{\mathrm{etai22}}{,}{\mathrm{etai33}}{,}{\mathrm{η12}}{,}{\mathrm{η23}}{,}{\mathrm{η34}}{,}{\mathrm{η13}}{,}{\mathrm{η24}}{,}{\mathrm{η14}}{,}{\mathrm{etai12}}{,}{\mathrm{etai23}}{,}{\mathrm{etai34}}{,}{\mathrm{etai13}}{,}{\mathrm{etai24}}{,}{\mathrm{etai14}}\right]$ (2.1)

Initialize the Lie algebra $\mathrm{su}\left(4\right).$

 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: su4}}$ (2.2)

The command StandardRepresentation will produce the actual matrices defining $\mathrm{su}\left(4\right)$. (This command only applies to Lie algebras constructed by the SimpleLieAlgebraData  procedure.)

 su4 > $\mathrm{StandardRepresentation}\left(\mathrm{su4}\right)$

The Lie algebra elements corresponding to the complex diagonal matrices define a Cartan subalgebra.

 su4 > $\mathrm{CSA}≔\left[\mathrm{Ui11},\mathrm{Ui22},\mathrm{Ui33}\right]$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{Ui11}}{,}{\mathrm{Ui22}}{,}{\mathrm{Ui33}}\right]$ (2.3)

We check this is indeed a Cartan subalgebra using the Query command

 su4 > $\mathrm{Query}\left(\mathrm{CSA},"CartanSubalgebra"\right)$
 ${\mathrm{true}}$ (2.4)

Here is the root space corresponding to the root <I, I, -I>.

 su4 > $X≔\mathrm{RootSpace}\left(⟨I,I,2I⟩,\mathrm{CSA}\right)$
 ${X}{:=}{\mathrm{U34}}{-}{I}{}{\mathrm{Ui34}}$ (2.5)

We check that the X is an eigenvector for the elements of the Cartan subalgebra.

 su4 > $B≔\left[\mathrm{seq}\left(\mathrm{LieBracket}\left(h,X\right),h=\mathrm{CSA}\right)\right]$
 ${B}{:=}\left[{I}{}{\mathrm{U34}}{+}{\mathrm{Ui34}}{,}{I}{}{\mathrm{U34}}{+}{\mathrm{Ui34}}{,}{2}{}{I}{}{\mathrm{U34}}{+}{2}{}{\mathrm{Ui34}}\right]$ (2.6)
 su4 > $\mathrm{GetComponents}\left(B,\left[X\right]\right)$
 $\left[\left[{I}\right]{,}\left[{I}\right]{,}\left[{2}{}{I}\right]\right]$ (2.7)

The column vector <I, I, I> is not a root

 su4 > $\mathrm{RootSpace}\left(⟨I,I,I⟩,\mathrm{CSA}\right)$
 ${0}{}{\mathrm{Ui11}}$ (2.8)

Example 2.

Here is the full root space decomposition for the Lie algebra $\mathrm{su}\left(4\right)$from Example 1.

 su4 > $\mathrm{RSD}≔\mathrm{RootSpaceDecomposition}\left(\mathrm{CSA}\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{2}{}{I}{,}{I}{,}{I}\right]{=}{\mathrm{U14}}{-}{I}{}{\mathrm{Ui14}}{,}\left[{-}{I}{,}{I}{,}{0}\right]{=}{\mathrm{U12}}{+}{I}{}{\mathrm{Ui12}}{,}\left[{I}{,}{2}{}{I}{,}{I}\right]{=}{\mathrm{U24}}{-}{I}{}{\mathrm{Ui24}}{,}\left[{I}{,}{0}{,}{-}{I}\right]{=}{\mathrm{U13}}{-}{I}{}{\mathrm{Ui13}}{,}\left[{0}{,}{I}{,}{-}{I}\right]{=}{\mathrm{U23}}{-}{I}{}{\mathrm{Ui23}}{,}\left[{-}{I}{,}{-}{I}{,}{-}{2}{}{I}\right]{=}{\mathrm{U34}}{+}{I}{}{\mathrm{Ui34}}{,}\left[{0}{,}{-}{I}{,}{I}\right]{=}{\mathrm{U23}}{+}{I}{}{\mathrm{Ui23}}{,}\left[{-}{I}{,}{-}{2}{}{I}{,}{-}{I}\right]{=}{\mathrm{U24}}{+}{I}{}{\mathrm{Ui24}}{,}\left[{-}{2}{}{I}{,}{-}{I}{,}{-}{I}\right]{=}{\mathrm{U14}}{+}{I}{}{\mathrm{Ui14}}{,}\left[{-}{I}{,}{0}{,}{I}\right]{=}{\mathrm{U13}}{+}{I}{}{\mathrm{Ui13}}{,}\left[{I}{,}{I}{,}{2}{}{I}\right]{=}{\mathrm{U34}}{-}{I}{}{\mathrm{Ui34}}{,}\left[{I}{,}{-}{I}{,}{0}\right]{=}{\mathrm{U12}}{-}{I}{}{\mathrm{Ui12}}\right]\right)$ (2.9)

The second calling sequence for RootSpace simply converts the given root vector to a list and extracts the corresponding root space from the root space decomposition table.

 su4 > $\mathrm{RootSpace}\left(⟨I,I,2I⟩,\mathrm{RSD}\right)$
 ${\mathrm{U34}}{-}{I}{}{\mathrm{Ui34}}$ (2.10)