find the root space decomposition for a semi-simple Lie algebra from a Cartan subalgebra

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

Calling Sequences

RootSpaceDecomposition()

Parameters

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

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 2 g is a common eigenvector for all these transformations, then . The -tuples are called the roots of with respect to the Cartan sub-algebra and the root space decomposition of g with respect to h.

•	The roots and root space decomposition enjoy the following basic properties.

1.	The eigenspaces or root spaces are each 1-dimensional.

2.	Ifis a root, then so is

3.	If and , then if is a root; otherwise

4.	If and , then The vectors define a 3-dimensional Lie algebra isomorphic to .

5.	If and and then where is the Killing form.

6.	The Killing form is non-degenerate on h.

7.	The number of linearly independent roots is

•	The command RootSpaceDecomposition returns a table describing the root space decomposition of g with respect to h. The indices of the table are the roots and the table entries are vectors in g defining the root spaces

•	The command Query/"RootSpaceDecomposition" will check that a given table defines a root space decomposition.

•	The commands SimpleLieAlgebraData and SimpleLieAlgebraProperties can be used to quickly obtain the root space decomposition for any simple classical matrix algebra.

Examples

Example 1.

In this example we initialize the simple Lie algebra (of trace-free matrices), calculate a Cartan subalgebra and a root space decomposition. We then illustrate the above properties of the root space decomposition.

First, we use the program SimpleLieAlgebraData to generate the Lie algebra data for s.

(2.1)

(2.2)

The program CartanSubalgebra calculates a Cartan subalgebra for .

sl3 >

(2.3)

Now compute the root space decomposition. We see that each root space is 1-dimensional (Property 1).

sl3 >

(2.4)

The roots are the indices for this table, given as column vectors. It is easy to see that the negative of any root is a root (Property 2).

sl3 >

RT := [Vector(2, {(1) = -1, (2) = -2}), Vector(2, {(1) = 1, (2) = 2}), Vector(2, {(1) = -1, (2) = 1}), Vector(2, {(1) = 2, (2) = 1}), Vector(2, {(1) = -2, (2) = -1}), Vector(2, {(1) = 1, (2) = -1})]

(2.5)

Here are the eigenvectors or root spaces.

sl3 >

(2.6)

The 2nd and 6th roots add to give the 4th root. This means that the Lie bracket of the 3rd and 4th vectors in (2.6) should be a multiple of the 1st vector (Property 3).

sl3 >

(2.7)

The 1st and 2nd roots are negatives of each other so the Lie bracket of the 1st and 2nd vectors in (2.6)should belong to the Cartan subalgebra (Property 4).

sl3 >

(2.8)

The vectors form a 3-dimensional Lie algebra (Property 4).

sl3 >

(2.9)

The Killing form restricted to the root spaces of the 2nd, 4rd and 6th roots is diagonal (Property 5).

sl3 >

Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})

(2.10)

Example 2.

We repeat the analysis of Example 1 using the Lie algebra This is a 21-dimensional Lie algebra of 7×7 matrices which preserve the quadratic form [[[0,I[3],0],[I[3],0,0],[0,0,1]]]. First, we use the program SimpleLieAlgebraData to generate the Lie algebra data for s.

(2.11)

(2.12)

The program CartanSubalgebra calculates a Cartan subalgebra for .

so43 >

(2.13)

Now compute the root space decomposition. We see that each root space is 1-dimensional (Property 1).

so43 >

(2.14)

(2.15)

The roots, given as column vectors, are obtained using LieAlgebraRoots. It is easy to see that the negative of any root is a root (Property 2).

so43 >

(2.16)

Here are the eigenvectors or root spaces.

so43 >

(2.17)

The 2nd and 12th roots add to give the 13th root. This means that the Lie bracket of the 1st and 7th vectors should be a multiple of the 10th vector (Property 3).

so43 >

Vector(3, {(1) = 1, (2) = 0, (3) = 1}), Vector(3, {(1) = 1, (2) = 0, (3) = 1})

(2.18)

so43 >

(2.19)

The Lie bracket of any root and its negative belongs to the Cartan subalgebra (Property 4).

so43 >

(2.20)

The Killing form restricted to the positive root space is diagonal (Property 5).

so43 >

PR := [Vector(3, {(1) = 0, (2) = 1, (3) = -1}), Vector(3, {(1) = 1, (2) = 0, (3) = 0}), Vector(3, {(1) = 1, (2) = 0, (3) = -1}), Vector(3, {(1) = 1, (2) = 1, (3) = 0}), Vector(3, {(1) = 0, (2) = 1, (3) = 1}), Vector(3, {(1) = 1, (2) = -1, (3) = 0}), Vector(3, {(1) = 0, (2) = 0, (3) = 1}), Vector(3, {(1) = 1, (2) = 0, (3) = 1}), Vector(3, {(1) = 0, (2) = 1, (3) = 0})]