find the positive roots from a set of roots or a root space decomposition, list the positive roots for a given root type

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LieAlgebras[PositiveRoots] - find the positive roots from a set of roots or a root space decomposition, list the positive roots for a given root type

Calling Sequences

PositiveRoots()

PositiveRoots(

PositiveRoots(, Delta1)

Parameters

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

Y - a vector of dimension or a list of vectors of dimension where is the rank of the Lie algebra

Delta - a list of vectors of dimension , defining a set of roots for a simple Lie algebra or an abstract root system

RT - a string, a root type "A", "B", "C", "D"

m - a non-negative integer

CM - a square matrix, the Cartan matrix of a simple Lie algebra

Delta1 - (optional) a list of vectors of dimension , defining a set of roots for a simple Lie algebra

Description

•	Let be a list of roots for either an abstract root system or for a simple Lie algebra. In particular, must have an even number of elements and if then . Write where, if then and if then The set is called the set of positive roots. The choice of positive roots for a given root system is not unique.

•

There are two convenient ways to pick a set of positive roots. [1] Pick a vector such that X$Y is real and non-zero for all . Then is called positive if X$Y >0. [2] Pick a basis B for such that the components of every vector in are real with respect to this basis. A vector is called positive if its first non-zero component with respect to B is positive.

•	The first and second calling sequences calculate a set of positive roots for roots of a given root space decomposition or for a given list of roots. If is a single vector then method [1] is used. If is a list of vectors then method [2] is used.

•	For a given root "A", "B", "C", "D", the positive roots can always be given by specific linear combinations of simple roots. These linear combinations are returned by the third calling sequence.

•	The positive roots can also be constructed from the Cartan matrix. This method is implemented with the 4th calling sequence. When the Cartan matrix is in standard form, the results of the 3rd and 4th calling sequences are the same.

•	For more information on the last two calling sequences, see Details for PositiveRoots.

Examples

Example 1.

We calculate the positive roots for the Lie algebra This is the 18-dimensional Lie algebra of matrices which are skew-symmetric with respect to the quadratic form Q = [[[0,I[4]],[I[4],0]]].

We use the command SimpleLieAlgebraData to obtain the structure equations for this Lie algebra.

(2.1)

Find a Cartan subalgebra and the corresponding root space decomposition.

so44 >

(2.2)

so44 >

(2.3)

We calculate the positive roots for using method [1].

so44 >

(2.4)

We calculate the positive roots for using method [2]. We see that there are different possiblities for the choice of positive roots.

so44 >

(2.5)

To get the same result from method [1] chose a different vector for the 2nd argument.

so44 >

(2.6)

Example 2.

To illustrate the second calling sequence we use the command LieAlgebraRoots to extract the roots from the table giving the root space decomposition for

so44 >

(2.7)

so44 >

(2.8)

Example 3.

To illustrate the underlying theory which supports the 4th calling sequence, we return to Example 1.

so44 >

(2.9)

Here are the simple roots for the first set of positive roots for (We can calculate simple roots using the command SimpleRoots but we wish to use this particular ordering of the roots for this example and the next.)

so44 >

$SR_so44a := [Vector(4, {(1) = 0, (2) = -1, (3) = 1, (4) = 0}), Vector(4, {(1) = 1, (2) = 1, (3) = 0, (4) = 0}), Vector(4, {(1) = 0, (2) = 0, (3) = -1, (4) = 1}), Vector(4, {(1) = -1, (2) = 1, (3) = 0, (4) = 0})]$

(2.10)

Here are the positive roots expressed as linear combinations of the simple roots.

so44 >

(2.11)

We can use the 4th calling sequence to reconstruct the positive roots from the simple roots and the Cartan matrix. Here is the Cartan matrix (note that it is not in standard form).

so44 >

$CM_so44a := Matrix(4, 4, {(1, 1) = 2, (1, 2) = -1, (1, 3) = -1, (1, 4) = -1, (2, 1) = -1, (2, 2) = 2, (2, 3) = 0, (2, 4) = 0, (3, 1) = -1, (3, 2) = 0, (3, 3) = 2, (3, 4) = 0, (4, 1) = -1, (4, 2) = 0, (4, 3) = 0, (4, 4) = 2})$

(2.12)

Here are the components of the positive roots, as calculated from the Cartan matrix.

so44 >

(2.13)

We can check that, apart from their order and format, the coefficients in (2.11) and (2.13) are the same.

so44 >

(2.14)

Example 4.

The inductive construction of the positive roots by root height can be traced with infolevel.

so44 >

(2.15)

so44 >

The roots at level 1 are:
[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]

The roots at level 2 are:
[[1, 1, 0, 0], [1, 0, 1, 0], [1, 0, 0, 1]]
The roots at level 3 are:
[[1, 1, 1, 0], [1, 1, 0, 1], [1, 0, 1, 1]]

The roots at level 4 are:
[[1, 1, 1, 1]]
The roots at level 5 are:
[[2, 1, 1, 1]]

(2.16)

Example 5.

We continue with the results from Example 2. This time we re-order the simple roots so that the Cartan matrix is in standard form. Here are the roots and Cartan matrix from before.

so44 >

(2.17)

so44 >

(2.18)

Now we change the ordering of the simple roots to put the Cartan matrix into standard form.

so44 >

(2.19)

Here are the components of the positive roots in terms of this new ordering of the simple roots.

so44 >

(2.20)

Here are the components of the positive roots for the root type D, using the 3rd calling sequence.

so61 >

(2.21)

Again, the component lists (2.20) and (2.21) coincide, apart from their order and format.

so44 >

(2.22)

Example 5.

We calculate a set of positive roots for

so61 >

(2.23)

(2.24)

Find a Cartan subalgebra and the corresponding root space decomposition.

so61 >

(2.25)

so61 >

(2.26)

The indices of this table give the roots for Here are the roots, now as vectors.

so61 >

(2.27)

Since the last two components of the roots are complex we must use a basis with complex components to calculate the positive roots.

so61 >

B := [Vector(3, {(1) = 1, (2) = 0, (3) = 0}), Vector(3, {(1) = 0, (2) = I, (3) = 0}), Vector(3, {(1) = 0, (2) = 0, (3) = I})]

(2.28)

so61 >

$PR_so61 := [Vector(3, {(1) = 1, (2) = 0, (3) = 0}), Vector(3, {(1) = 0, (2) = I, (3) = I}), Vector(3, {(1) = 1, (2) = I, (3) = 0}), Vector(3, {(1) = 0, (2) = 0, (3) = I}), Vector(3, {(1) = 0, (2) = I, (3) = 0}), Vector(3, {(1) = 1, (2) = -I, (3) = 0}), Vector(3, {(1) = 0, (2) = I, (3) = -I}), Vector(3, {(1) = 1, (2) = 0, (3) = -I}), Vector(3, {(1) = 1, (2) = 0, (3) = I})]$

(2.29)

Again, let's find the simple roots and then check that the positive roots can be reconstructed from the simple roots via the Cartan matrix.

so61 >

$SR_so61 := [Vector(3, {(1) = 0, (2) = 0, (3) = I}), Vector(3, {(1) = 1, (2) = -I, (3) = 0}), Vector(3, {(1) = 0, (2) = I, (3) = -I})]$

(2.30)

so61 >

$CM_so61 := Matrix(3, 3, {(1, 1) = 2, (1, 2) = 0, (1, 3) = -1, (2, 1) = 0, (2, 2) = 2, (2, 3) = -1, (3, 1) = -2, (3, 2) = -1, (3, 3) = 2})$

(2.31)

We use the PositiveRoots command to generate the coefficients of the positive roots.

so61 >

The roots at level 1 are:
[[1, 0, 0], [0, 1, 0], [0, 0, 1]]

The roots at level 2 are:
  [[1, 0, 1], [0, 1, 1]]
The roots at level 3 are:
  [[2, 0, 1], [1, 1, 1]]
The roots at level 4 are:
  [[2, 1, 1]]
The roots at level 5 are:
  [[2, 1, 2]]

$Pr_so61b := [Vector(3, {(1) = 0, (2) = 0, (3) = I}), Vector(3, {(1) = 1, (2) = -I, (3) = 0}), Vector(3, {(1) = 0, (2) = I, (3) = -I}), Vector(3, {(1) = 0, (2) = I, (3) = 0}), Vector(3, {(1) = 1, (2) = 0, (3) = -I}), Vector(3, {(1) = 0, (2) = I, (3) = I}), Vector(3, {(1) = 1, (2) = 0, (3) = 0}), Vector(3, {(1) = 1, (2) = 0, (3) = I}), Vector(3, {(1) = 1, (2) = I, (3) = 0})]$

(2.32)

Apart from their order, the two sets of vectors (2.29) and (2.32) coincide.

Example 6.

The positive roots for the exceptional Lie algebras can be generated by the 4th calling sequence.

$CM_F4 := Matrix(4, 4, {(1, 1) = 2, (1, 2) = -1, (1, 3) = 0, (1, 4) = 0, (2, 1) = -1, (2, 2) = 2, (2, 3) = -2, (2, 4) = 0, (3, 1) = 0, (3, 2) = -1, (3, 3) = 2, (3, 4) = -1, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1, (4, 4) = 2})$

(2.33)

The roots at level 1 are:
[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]

The roots at level 2 are:
[[1, 1, 0, 0], [0, 1, 1, 0], [0, 0, 1, 1]]
The roots at level 3 are:
[[1, 1, 1, 0], [0, 1, 2, 0], [0, 1, 1, 1]]

The roots at level 4 are:
  [[1, 1, 2, 0], [1, 1, 1, 1], [0, 1, 2, 1]]
The roots at level 5 are:
  [[1, 2, 2, 0], [1, 1, 2, 1], [0, 1, 2, 2]]
The roots at level 6 are:
  [[1, 2, 2, 1], [1, 1, 2, 2]]

The roots at level 7 are:
  [[1, 2, 3, 1], [1, 2, 2, 2]]
The roots at level 8 are:
  [[1, 2, 3, 2]]
The roots at level 9 are:
  [[1, 2, 4, 2]]
The roots at level 10 are:
  [[1, 3, 4, 2]]

The roots at level 11 are:
[[2, 3, 4, 2]]