 RestrictedRootSpaceDecomposition - Maple Help

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LieAlgebras[RestrictedRootSpaceDecomposition] - find the real root space decomposition of a non-compact semi-simple Lie algebra with respect to an Abelian subalgebra

Calling Sequences

RestrictedRootSpaceDecomposition(A)

RestrictedRootSpaceDecomposition(RSD$,$CSA, A)

Parameters

A     - a list of vectors, defining an Abelian subalgebra of a non-compact, semi-simple Lie algebra

RSD   - a table, defining a root space decomposition

CSA   - a list of vectors, defining the Cartan subalgebra used to calculate the root space decomposition $\mathrm{RSA}$ 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 $x$∈ g is a common eigenvector for all these transformations, then . The $m$-tuples  are called the roots of with respect to the Cartan subalgebra and  the root space decomposition of g with respect to h.
 • Now suppose that is non-compact. To obtain the restricted root space decomposition, let g = k ⊕p be a Cartan decomposition of g. Let a={ be a maximal Abelian subalgebra of p. Then the matrices $\mathrm{ad}\left(x\right)$for  all commute and have real common eigenspaces. The resulting eigenspace decomposition
 • , where  is the centralizer of a in g, called the restricted root space decomposition. The restricted roots are q-tuples of real numbers. The common eigenspacesneed not be 1-dimensional. Eigenspaces associated to different roots are orthogonal with respect to the Killing form. If is a Cartan involution which preserves $\mathrm{𝔥}$, then a can be chosen as a subalgebra of h.
 • The command RestrictedRootSpaceDecomposition returns a table describing the root space decomposition of g with respect to $\mathrm{𝔞}$. The indices of the table are the roots ${\mathrm{\alpha }}_{}$ and the table entries are vectors in g defining the root spaces ${S}_{{\mathrm{α}}_{}}.$
 • For the second calling sequence, the restricted roots are determined by restricting the roots in the root space decomposition (as functionals on h) to the subalgebra a. Examples

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

Example 1.

We find the restricted root space decomposition for the Lie algebra $\mathrm{so}\left(4,2\right)$. This is the 15-dimensional Lie algebra of matrices which are skew-symmetric with respect to the quadratic form $\left[\begin{array}{rrr}0& {I}_{2}& 0\\ {I}_{2}& 0& 0\\ 0& 0& {I}_{2}\end{array}\right]$. We use the command SimpleLieAlgebraData to initialize $\mathrm{so}\left(4,2\right)$.

 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("so\left(4, 2\right)",\mathrm{so42}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: so42}}$ (2.1)

To find a suitable candidate for the subspace a, we first calculate a Cartan subalgebra.

 so42 > $\mathrm{CSA}≔\mathrm{CartanSubalgebra}\left(\right)$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e4}}{,}{\mathrm{e15}}\right]$ (2.2)

Now we shall use the Signature command to find a subalgebra on which the Killing form is positive-definite.

 > $B≔\mathrm{KillingForm}\left(\right)$
 ${B}{:=}{8}{}{\mathrm{θ1}}{}{\mathrm{θ1}}{+}{8}{}{\mathrm{θ2}}{}{\mathrm{θ3}}{+}{8}{}{\mathrm{θ3}}{}{\mathrm{θ2}}{+}{8}{}{\mathrm{θ4}}{}{\mathrm{θ4}}{-}{8}{}{\mathrm{θ5}}{}{\mathrm{θ6}}{-}{8}{}{\mathrm{θ6}}{}{\mathrm{θ5}}{-}{8}{}{\mathrm{θ7}}{}{\mathrm{θ11}}{-}{8}{}{\mathrm{θ8}}{}{\mathrm{θ12}}{-}{8}{}{\mathrm{θ9}}{}{\mathrm{θ13}}{-}{8}{}{\mathrm{θ10}}{}{\mathrm{θ14}}{-}{8}{}{\mathrm{θ11}}{}{\mathrm{θ7}}{-}{8}{}{\mathrm{θ12}}{}{\mathrm{θ8}}{-}{8}{}{\mathrm{θ13}}{}{\mathrm{θ9}}{-}{8}{}{\mathrm{θ14}}{}{\mathrm{θ10}}{-}{8}{}{\mathrm{θ15}}{}{\mathrm{θ15}}$ (2.3)
 so42 > $\mathrm{Tensor}:-\mathrm{QuadraticFormSignature}\left(B,\mathrm{CSA}\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e15}}\right]{,}\left[{}\right]\right]$ (2.4)

We can use the subspace a to find the restricted root space decomposition for $\mathrm{so}\left(4,2\right)$.

 so42 > $A≔\left[\mathrm{e1},\mathrm{e4}\right]$
 ${A}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]$ (2.5)

First calling sequence.

 so42 > $\mathrm{RRSD1}≔\mathrm{RestrictedRootSpaceDecomposition}\left(A\right)$
 ${\mathrm{RRSD1}}{:=}{\mathrm{table}}\left(\left[\left[{0}{,}{1}\right]{=}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{,}\left[{1}{,}{-}{1}\right]{=}\left[{\mathrm{e2}}\right]{,}\left[{-}{1}{,}{0}\right]{=}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{,}\left[{0}{,}{-}{1}\right]{=}\left[{\mathrm{e13}}{,}{\mathrm{e14}}\right]{,}\left[{-}{1}{,}{-}{1}\right]{=}\left[{\mathrm{e6}}\right]{,}\left[{1}{,}{1}\right]{=}\left[{\mathrm{e5}}\right]{,}\left[{-}{1}{,}{1}\right]{=}\left[{\mathrm{e3}}\right]{,}\left[{1}{,}{0}\right]{=}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]\right]\right)$ (2.6)

Second calling sequence.

For the second calling sequence we first need the root space decomposition with respect to the Cartan subalgebra CSA.

 so42 > $\mathrm{RSD}≔\mathrm{RootSpaceDecomposition}\left(\mathrm{CSA}\right)$
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{0}{,}{-}{1}{,}{I}\right]{=}{\mathrm{e13}}{-}{I}{}{\mathrm{e14}}{,}\left[{1}{,}{1}{,}{0}\right]{=}{\mathrm{e5}}{,}\left[{1}{,}{0}{,}{-}{I}\right]{=}{\mathrm{e7}}{+}{I}{}{\mathrm{e8}}{,}\left[{1}{,}{0}{,}{I}\right]{=}{\mathrm{e7}}{-}{I}{}{\mathrm{e8}}{,}\left[{-}{1}{,}{1}{,}{0}\right]{=}{\mathrm{e3}}{,}\left[{0}{,}{-}{1}{,}{-}{I}\right]{=}{\mathrm{e13}}{+}{I}{}{\mathrm{e14}}{,}\left[{-}{1}{,}{-}{1}{,}{0}\right]{=}{\mathrm{e6}}{,}\left[{0}{,}{1}{,}{I}\right]{=}{\mathrm{e9}}{-}{I}{}{\mathrm{e10}}{,}\left[{-}{1}{,}{0}{,}{-}{I}\right]{=}{\mathrm{e11}}{+}{I}{}{\mathrm{e12}}{,}\left[{1}{,}{-}{1}{,}{0}\right]{=}{\mathrm{e2}}{,}\left[{-}{1}{,}{0}{,}{I}\right]{=}{\mathrm{e11}}{-}{I}{}{\mathrm{e12}}{,}\left[{0}{,}{1}{,}{-}{I}\right]{=}{\mathrm{e9}}{+}{I}{}{\mathrm{e10}}\right]\right)$ (2.7)
 so42 > $\mathrm{RRSD}≔\mathrm{RestrictedRootSpaceDecomposition}\left(\mathrm{RSD},\mathrm{CSA},A\right)$
 ${\mathrm{RRSD}}{:=}{\mathrm{table}}\left(\left[\left[{0}{,}{1}\right]{=}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{,}\left[{1}{,}{-}{1}\right]{=}\left[{\mathrm{e2}}\right]{,}\left[{-}{1}{,}{0}\right]{=}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{,}\left[{0}{,}{-}{1}\right]{=}\left[{\mathrm{e13}}{,}{\mathrm{e14}}\right]{,}\left[{-}{1}{,}{-}{1}\right]{=}\left[{\mathrm{e6}}\right]{,}\left[{1}{,}{1}\right]{=}\left[{\mathrm{e5}}\right]{,}\left[{-}{1}{,}{1}\right]{=}\left[{\mathrm{e3}}\right]{,}\left[{1}{,}{0}\right]{=}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]\right]\right)$ (2.8)

It is instructive to compare the root space decomposition RSD (equation (2.6) and the restricted root space decomposition RRSD(equation (2.8)). First, note that the roots for RSDare vectors in 3-dimensions (since the Cartan subalgebra is 3-dimensional) while the roots for RRSD are vectors in 2-dimensions (since the subspace a is 2-dimensional). Second, we see that the first 2 components of the roots for RSDare all real and the 3rd component is pure imaginary. This reflects the fact that the basis we have used for the Cartan subalgebra is adapted to the Cartan decomposition. Third, we see that the restricted roots are just the projections [The restricted root space for is just the direct sum of the root spaces for the roots of the form Finally, and this is the whole point, the restricted root spaces have a real basis.

 so42 > $\mathrm{RS}≔\left[\mathrm{RSD}\left[\left[0,1,I\right]\right],\mathrm{RSD}\left[\left[0,1,-I\right]\right]\right]$
 ${\mathrm{RS}}{:=}\left[{\mathrm{e9}}{-}{I}{}{\mathrm{e10}}{,}{\mathrm{e9}}{+}{I}{}{\mathrm{e10}}\right]$ (2.9)
 so42 > $\mathrm{Tools}:-\mathrm{CanonicalBasis}\left(\mathrm{RS}\right)$
 $\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]$ (2.10)

Example 2

We find a restricted root space decomposition for so*(8).

 so42 > $\mathrm{LD}≔\mathrm{SimpleLieAlgebraData}\left("so*\left(8\right)",\mathrm{sos8}\right):$
 so42 > $\mathrm{DGsetup}\left(\mathrm{LD}\right)$
 ${\mathrm{Lie algebra: sos8}}$ (2.11)

We calculate a Cartan subalgebra and a subspace on which the Killing form is positive-definite.

 su33 > $\mathrm{CSA}≔\mathrm{CartanSubalgebra}\left(\right)$
 ${\mathrm{CSA}}{:=}\left[{\mathrm{e1}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}{,}{\mathrm{e12}}\right]$ (2.12)
 su33 > $B≔\mathrm{KillingForm}\left(\right):$
 su33 > $\mathrm{Tensor}:-\mathrm{QuadraticFormSignature}\left(B,\mathrm{CSA}\right)$
 $\left[\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{,}\left[{}\right]\right]$ (2.13)
 so42 > $\mathrm{RestrictedRootSpaceDecomposition}\left(\left[\mathrm{e23},\mathrm{e28}\right]\right)$
 ${\mathrm{table}}\left(\left[\left[{0}{,}{-}{2}\right]{=}\left[{\mathrm{e12}}{+}{\mathrm{e20}}{+}{\mathrm{e22}}\right]{,}\left[{2}{,}{0}\right]{=}\left[{\mathrm{e7}}{-}{\mathrm{e13}}{-}{\mathrm{e17}}\right]{,}\left[{1}{,}{-}{1}\right]{=}\left[{\mathrm{e2}}{+}{\mathrm{e5}}{-}{\mathrm{e25}}{+}{\mathrm{e26}}{,}{\mathrm{e3}}{-}{\mathrm{e4}}{+}{\mathrm{e24}}{+}{\mathrm{e27}}{,}{\mathrm{e8}}{-}{\mathrm{e11}}{-}{\mathrm{e16}}{-}{\mathrm{e18}}{,}{\mathrm{e9}}{+}{\mathrm{e10}}{+}{\mathrm{e15}}{-}{\mathrm{e19}}\right]{,}\left[{0}{,}{2}\right]{=}\left[{\mathrm{e12}}{-}{\mathrm{e20}}{-}{\mathrm{e22}}\right]{,}\left[{-}{1}{,}{-}{1}\right]{=}\left[{\mathrm{e2}}{-}{\mathrm{e5}}{-}{\mathrm{e25}}{-}{\mathrm{e26}}{,}{\mathrm{e3}}{+}{\mathrm{e4}}{+}{\mathrm{e24}}{-}{\mathrm{e27}}{,}{\mathrm{e8}}{+}{\mathrm{e11}}{-}{\mathrm{e16}}{+}{\mathrm{e18}}{,}{\mathrm{e9}}{-}{\mathrm{e10}}{+}{\mathrm{e15}}{+}{\mathrm{e19}}\right]{,}\left[{1}{,}{1}\right]{=}\left[{\mathrm{e2}}{-}{\mathrm{e5}}{+}{\mathrm{e25}}{+}{\mathrm{e26}}{,}{\mathrm{e3}}{+}{\mathrm{e4}}{-}{\mathrm{e24}}{+}{\mathrm{e27}}{,}{\mathrm{e8}}{+}{\mathrm{e11}}{+}{\mathrm{e16}}{-}{\mathrm{e18}}{,}{\mathrm{e9}}{-}{\mathrm{e10}}{-}{\mathrm{e15}}{-}{\mathrm{e19}}\right]{,}\left[{-}{1}{,}{1}\right]{=}\left[{\mathrm{e2}}{+}{\mathrm{e5}}{+}{\mathrm{e25}}{-}{\mathrm{e26}}{,}{\mathrm{e3}}{-}{\mathrm{e4}}{-}{\mathrm{e24}}{-}{\mathrm{e27}}{,}{\mathrm{e8}}{-}{\mathrm{e11}}{+}{\mathrm{e16}}{+}{\mathrm{e18}}{,}{\mathrm{e9}}{+}{\mathrm{e10}}{-}{\mathrm{e15}}{+}{\mathrm{e19}}\right]{,}\left[{-}{2}{,}{0}\right]{=}\left[{\mathrm{e7}}{+}{\mathrm{e13}}{+}{\mathrm{e17}}\right]\right]\right)$ (2.14)

Note here that the restricted root spaces for have dimensions 1or 4.