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 RSA
Description
Examples
Let g be a semi-simple Lie algebra and h a Cartan subalgebra. Let h1, h2, ... ,hm be a basis for 𝔥. The linear transformations adhi are simultaneously diagonalizable over C -- if x∈ g is a common eigenvector for all these transformations, then adhix = hi , x = αi x . The m-tuples α = α1, α2, ... , αm are called the roots of 𝔤 with respect to the Cartan subalgebra 𝔥 and 𝔤=𝔥 ⊕⨁α ∈ Δ Rα the root space decomposition of g with respect to h.
Now suppose that 𝔤 is non-compact. To obtain the restricted root space decomposition, let g = k ⊕p be a Cartan decomposition of g. Let a={h1, .. hq} be a maximal Abelian subalgebra of p. Then the matrices adxfor x ∈𝔞 all commute and have real common eigenspaces. The resulting eigenspace decomposition 𝔤 = 𝔤0 ⊕⨁α ∈ Δ r Sα
, where 𝔤0 = Z𝔞 is the centralizer of a in g, called the restricted root space decomposition. The restricted roots Δr are q-tuples of real numbers. The common eigenspaces Sα need not be 1-dimensional. Eigenspaces associated to different roots are orthogonal with respect to the Killing form. If Θ is a Cartan involution which preserves 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 𝔞. The indices of the table are the roots α and the table entries are vectors in g defining the root spaces Sα.
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.
withDifferentialGeometry:withLieAlgebras:
Example 1.
We find the restricted root space decomposition for the Lie algebra so4,2. This is the 15-dimensional Lie algebra of matrices which are skew-symmetric with respect to the quadratic form 0I20I20000I2. We use the command SimpleLieAlgebraData to initialize so4,2.
LD≔SimpleLieAlgebraDataso(4, 2),so42:
DGsetupLD
Lie algebra: so42
To find a suitable candidate for the subspace a, we first calculate a Cartan subalgebra.
CSA≔CartanSubalgebra
CSA:=e1,e4,e15
Now we shall use the Signature command to find a subalgebra on which the Killing form is positive-definite.
B≔KillingForm
B:=8θ1θ1+8θ2θ3+8θ3θ2+8θ4θ4−8θ5θ6−8θ6θ5−8θ7θ11−8θ8θ12−8θ9θ13−8θ10θ14−8θ11θ7−8θ12θ8−8θ13θ9−8θ14θ10−8θ15θ15
Tensor:-QuadraticFormSignatureB,CSA
e1,e4,e15,
We can use the subspace a= e1, e4 to find the restricted root space decomposition for so4,2.
A≔e1,e4
A:=e1,e4
First calling sequence.
RRSD1≔RestrictedRootSpaceDecompositionA
RRSD1:=table0,1=e9,e10,1,−1=e2,−1,0=e11,e12,0,−1=e13,e14,−1,−1=e6,1,1=e5,−1,1=e3,1,0=e7,e8
Second calling sequence.
For the second calling sequence we first need the root space decomposition with respect to the Cartan subalgebra CSA.
RSD≔RootSpaceDecompositionCSA
RSD:=table0,−1,I=e13−Ie14,1,1,0=e5,1,0,−I=e7+Ie8,1,0,I=e7−Ie8,−1,1,0=e3,0,−1,−I=e13+Ie14,−1,−1,0=e6,0,1,I=e9−Ie10,−1,0,−I=e11+Ie12,1,−1,0=e2,−1,0,I=e11−Ie12,0,1,−I=e9+Ie10
RRSD≔RestrictedRootSpaceDecompositionRSD,CSA,A
RRSD:=table0,1=e9,e10,1,−1=e2,−1,0=e11,e12,0,−1=e13,e14,−1,−1=e6,1,1=e5,−1,1=e3,1,0=e7,e8
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 RSD are 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 RSD are 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 [x, y, z] →x, y. The restricted root space for x0,y0 is just the direct sum of the root spaces for the roots of the form x0, y0 , z. Finally, and this is the whole point, the restricted root spaces have a real basis.
RS≔RSD0,1,I,RSD0,1,−I
RS:=e9−Ie10,e9+Ie10
Tools:-CanonicalBasisRS
e9,e10
Example 2
We find a restricted root space decomposition for so*(8).
LD≔SimpleLieAlgebraDataso*(8),sos8:
Lie algebra: sos8
We calculate a Cartan subalgebra and a subspace on which the Killing form is positive-definite.
CSA:=e1,e6,e7,e12
B≔KillingForm:
e7,e12,e1,e6,
RestrictedRootSpaceDecompositione23,e28
table0,−2=e12+e20+e22,2,0=e7−e13−e17,1,−1=e2+e5−e25+e26,e3−e4+e24+e27,e8−e11−e16−e18,e9+e10+e15−e19,0,2=e12−e20−e22,−1,−1=e2−e5−e25−e26,e3+e4+e24−e27,e8+e11−e16+e18,e9−e10+e15+e19,1,1=e2−e5+e25+e26,e3+e4−e24+e27,e8+e11+e16−e18,e9−e10−e15−e19,−1,1=e2+e5+e25−e26,e3−e4−e24−e27,e8−e11+e16+e18,e9+e10−e15+e19,−2,0=e7+e13+e17
Note here that the restricted root spaces for so4,2 have dimensions 1or 4.
See Also
DifferentialGeometry
CartanSubalgebra
Killing
LieAlgebras
RootSpaceDecomposition
QuadraticeFormSignature
SimpleLieAlgebraData
SimpleLieAlgebraProperties
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