DifferentialGeometry/LieAlgebras/CompactRoots

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LieAlgebras[CompactRoots] - find the compact roots in a root system for a non-compact semi-simple real Lie algebra

Calling Sequences

CompactRoots()

Parameters

Delta - a list of column vectors, defining the root system, positive roots or simple roots of a non-compact semi-simple Lie algebra

A - a list of vectors in a Lie algebra, defining a subalgebra of the Cartan subalgebra on which the Killing form is negative-definite

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

Description

•	Let g be a semi-simple real Lie algebra. Then g is called compact if the Killing form of g is negative-definite, otherwise g is called non-compact.

•	Every non-compact semi-simple real Lie algebra g admits a Cartan decomposition g = t 4p . Here t is a subalgebra, p a subspace, [t, p] 4 p and [p, p] 4 t, that is, t and p define a symmetric pair. Moreover, the Killing form is negative-definite on t and positive-definite on p.

•	Let h be a Cartan subalgebra for g and let be the associated root system. Set a = h X p. Then the set of compact roots is defined to be

This means that if we choose a basis for a and extend to a basis for h, then the components of a compact root in the directions are 0. If determines the root space for then for With respect to the standard Cartan algebras for the non-compact, simple matrix algebras we consider here, the compact roots are precisely those which are purely imaginary complex numbers.

•	In the Satake diagram for a non-compact semi-simple real Lie algebra, the compact roots are given a different color from the other roots.

Examples

Example 1.

We find the compact roots for First we use the command SimpleLieAlgebraData to initialize the Lie algebra

(2.1)

For this example we use the command SimpleLieAlgebraProperties to generate the various properties of that we need.

su52 >

Here is the Cartan subalgebra.

su52 >

(2.2)

Here is the Cartan subalgebra decomposition

su52 >

(2.3)

We check that the restriction of the Killing form to the diagonal matrices with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices with real entries is positive-definite.

su52 >

K1 := Matrix(4, 4, {(1, 1) = -56, (1, 2) = -28, (1, 3) = 0, (1, 4) = 14, (2, 1) = -28, (2, 2) = -56, (2, 3) = 0, (2, 4) = 14, (3, 1) = 0, (3, 2) = 0, (3, 3) = -28, (3, 4) = -14, (4, 1) = 14, (4, 2) = 14, (4, 3) = -14, (4, 4) = -28})

(2.4)

su52 >

(2.5)

su52 >

K2 := Matrix(2, 2, {(1, 1) = 28, (1, 2) = 0, (2, 1) = 0, (2, 2) = 28})

(2.6)

The second list of vectors in (2.3) is therefore our subalgebra as described above.

Next we find the positive roots.

su52 >

(2.7)

The compact roots are:

su52 >

[Vector(6, {(1) = 0, (2) = 0, (3) = I, (4) = I, (5) = I, (6) = -I}), Vector(6, {(1) = 0, (2) = 0, (3) = -I, (4) = -I, (5) = I, (6) = 2*I}), Vector(6, {(1) = 0, (2) = 0, (3) = 0, (4) = 0, (5) = 2*I, (6) = I})]

(2.8)

[Vector(6, {(1) = 2*I, (2) = 2*I, (3) = -2*I, (4) = I, (5) = 0, (6) = 0}), Vector(6, {(1) = 0, (2) = 0, (3) = I, (4) = -2*I, (5) = 0, (6) = 0}), Vector(6, {(1) = 2*I, (2) = 2*I, (3) = -I, (4) = -I, (5) = 0, (6) = 0})]

(2.9)

Note that these roots all have purely imaginary components.

Example 2.

We find the compact roots for First we use the command SimpleLieAlgebraData to initialize the Lie algebra

(2.10)

We use the command SimpleLieAlgebraProperties to generate the various properties of that we need.

sp44 >

Here is the Cartan subalgebra.

sp44 >

(2.11)

Here is the Cartan subalgebra decomposition

sp44 >

(2.12)

The restriction of the Killing form to the diagonal matrices with imaginary entries is negative-definite. The restriction of the Killing form to the diagonal matrices with real entries is positive-definite.

sp44 >

K1 := Matrix(2, 2, {(1, 1) = -40, (1, 2) = 0, (2, 1) = 0, (2, 2) = -40})

(2.13)

sp44 >

(2.14)

sp44 >

K2 := Matrix(2, 2, {(1, 1) = 40, (1, 2) = 0, (2, 1) = 0, (2, 2) = 40})

(2.15)

The second list of vectors in (2.3) is therefore our subalgebra as described above.

Next we find the positive roots.

sp44 >

(2.16)

The compact roots are:

sp44 >

[Vector(4, {(1) = 0, (2) = 0, (3) = 2*I, (4) = 0}), Vector(4, {(1) = 0, (2) = 0, (3) = 0, (4) = 2*I})]

(2.17)

	See Also
	DifferentialGeometry, CartanDecomposition, Cartan Involution, CartanSubalgebra, DynkinDiagram, PositiveRoots, RootSpaceDecomposition, RestrictedRootSpaceDecomposition, SatakeDiagram, SimpleLieAlgebraProperties, SimpleRoots