associate to each positive root of a simple Lie algebra a vector in the Cartan subalgebra.

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LieAlgebras[RootToCartanSubalgebraElementH] - associate to each positive root of a simple Lie algebra a vector in the Cartan subalgebra.

Calling Sequences

RootToCartanSubalgebraElementH()

Parameters

alpha - a vector, defining a positive (or negative) root of a simple Lie algebra

RSD - a table, defining the root space decomposition of a simple Lie algebra

Description

•	Let g be a simple Lie algebra, h a Cartan subalgebra, and the root space decomposition of g with respect to h. For each root , there are vectors and such that and These conditions uniquely determine The procedure RootToCartanSubalgebraElementH() calculates the vector

•	Note that the vectors define the 3-dimensional Lie algebra .

•	The assignment is used to calculate the Cartan matrix for the Lie algebra .

Examples

Example 1.

We consider the Lie algebra This is the 24-dimensional real Lie algebra of 6×6 complex matrices which are trace-free and skew-Hermitian with respect to the quadratic form Q = (Matrix(2, 2, {(1, 1) = 0, (1, 2) = I[3], (2, 1) = I[3], (2, 2) = 0})) . We use the command SimpleLieAlgebraData to initialize this Lie algebra.

(2.1)

We use the command SimpleLieAlgebraProperties to obtain the Cartan subalgebra, the root space decomposition, and the simple roots.

su33 >

The result is a table. Here is the Cartan subalgebra for

su33 >

(2.2)

Here is the root space decomposition for

su33 >

(2.3)

Here are the positive roots.

su33 >

(2.4)

Let us find where is the first root (2.4)

su33 >

alpha := Vector(5, {(1) = 1, (2) = -1, (3) = 0, (4) = I, (5) = -I})

(2.5)

su33 >

(2.6)

We check that is in the Cartan subalgebra.

su33 >

(2.7)

Here are the root spaces for and

su33 >

(2.8)

su33 >

(2.9)

We check that defines a Lie subalgebra.

su33 >

(2.10)

If we scale the vectors X and Y then the structure equations take the standard form for .

su33 >

(2.11)

Example 2.

We illustrute how to use RootToCartanSubalgebraElementH to calculate the Cartan matrix for We first calculate the for the simple roots .

su33 >

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

(2.12)

su33 >

[_DG([["vector", su33, []], [[[1], -(1/2)*I], [[2], (1/2)*I], [[3], 1/2], [[4], -1/2]]]), _DG([["vector", su33, []], [[[2], -(1/2)*I], [[4], 1/2], [[5], -1/2]]]), _DG([["vector", su33, []], [[[5], 1]]]), _DG([["vector", su33, []], [[[2], (1/2)*I], [[4], 1/2], [[5], -1/2]]]), _DG([["vector", su33, []], [[[1], (1/2)*I], [[2], -(1/2)*I], [[3], 1/2], [[4], -1/2]]])]

(2.13)

Then we calculate the Killing form , restricted to subspace [

su33 >

B := Matrix(5, 5, {(1, 1) = 24, (1, 2) = -12, (1, 3) = 0, (1, 4) = 0, (1, 5) = 0, (2, 1) = -12, (2, 2) = 24, (2, 3) = -12, (2, 4) = 0, (2, 5) = 0, (3, 1) = 0, (3, 2) = -12, (3, 3) = 24, (3, 4) = -12, (3, 5) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -12, (4, 4) = 24, (4, 5) = -12, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = -12, (5, 5) = 24})

(2.14)

The Cartan matrix is given by normalizing the entries of

su33 >

C := Matrix(5, 5, proc (i, j) options operator, arrow; 2*B[i, j]/B[i, i] end proc)

C := Matrix(5, 5, {(1, 1) = 2, (1, 2) = -1, (1, 3) = 0, (1, 4) = 0, (1, 5) = 0, (2, 1) = -1, (2, 2) = 2, (2, 3) = -1, (2, 4) = 0, (2, 5) = 0, (3, 1) = 0, (3, 2) = -1, (3, 3) = 2, (3, 4) = -1, (3, 5) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1, (4, 4) = 2, (4, 5) = -1, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = -1, (5, 5) = 2})

(2.15)

The Lie algebra is a rank 5 simple Lie algebra of type "A". The matrix in (2.15) is therefore correct.

su33 >

Matrix(5, 5, {(1, 1) = 2, (1, 2) = -1, (1, 3) = 0, (1, 4) = 0, (1, 5) = 0, (2, 1) = -1, (2, 2) = 2, (2, 3) = -1, (2, 4) = 0, (2, 5) = 0, (3, 1) = 0, (3, 2) = -1, (3, 3) = 2, (3, 4) = -1, (3, 5) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1, (4, 4) = 2, (4, 5) = -1, (5, 1) = 0, (5, 2) = 0, (5, 3) = 0, (5, 4) = -1, (5, 5) = 2})