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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(RSD, Y)

     PositiveRoots(Δ, Y)

     PositiveRoots(RtType, m, Δ1)

     PositiveRoots(CMatrix, Δ1)

Parameters

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

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

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

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

     m       - a non-negative integer

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

     Δ1      - (optional) a list of vectors of dimension m, defining a set of roots for a simple Lie algebra

 

Description

Examples

Description

• 

Let Δ ⊆ℂm be a list of roots for either an abstract root system or for a simple Lie algebra. In particular, Δ must have an even number of elements and if X ∈Δ, then −X ∈Δ. Write Δ = Δ  +⋃ Δ−where, if X ∈Δ+then −X ∈Δ− and if X ∈Δ−then −X ∈Δ+. The set Δ+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.

[i] Pick a vector Y ∈ℂm such that X⋅Y is real and non-zero for all X ∈ Δ. Then X ∈Δ is called positive if X⋅Y > 0. 

[ii] Pick a basis B for ℂm such that the components of every vector in Δ are real with respect to this basis. A vector X ∈Δ 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 Y is a single vector then method [i] is used. If Y is a list of vectors then method [ii] 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

> 

with⁡DifferentialGeometry:with⁡LieAlgebras:

 

Example 1.

We calculate the positive roots for the Lie algebra so4,4. This is the 18-dimensional Lie algebra of 6×6 matrices which are skew-symmetric with respect to the quadratic form Q = 0I4I40. We use the command SimpleLieAlgebraData to obtain the structure equations for this Lie algebra. The labels 'E' and 'omega' must be unassigned names.

> 

LD≔SimpleLieAlgebraData⁡so(4, 4),so44,labelformat=gl,labels=E,ω:

> 

DGsetup⁡LD

Lie algebra: so44

(2.1)

 

Find a Cartan subalgebra and the corresponding root space decomposition.

so44 > 

CSA_so44≔CartanSubalgebra⁡so44

CSA_so44:=E11,E22,E33,E44

(2.2)
so44 > 

RSD_so44≔RootSpaceDecomposition⁡CSA_so44

RSD_so44:=table1,0,0,−1=E14,0,1,0,−1=E24,0,1,−1,0=E23,0,−1,−1,0=E63,−1,0,1,0=E31,1,1,0,0=E16,0,−1,1,0=E32,0,−1,0,1=E42,0,0,−1,−1=E74,−1,0,0,−1=E54,0,−1,0,−1=E64,−1,0,0,1=E41,−1,−1,0,0=E52,0,1,1,0=E27,−1,0,−1,0=E53,−1,1,0,0=E21,0,0,−1,1=E43,1,0,1,0=E17,1,0,0,1=E18,1,0,−1,0=E13,0,1,0,1=E28,1,−1,0,0=E12,0,0,1,−1=E34,0,0,1,1=E38

(2.3)

 

We calculate the positive roots for so4, 4 using method [i].

so44 > 

PR_so44a≔PositiveRoots⁡RSD_so44,1,2,3,4

 

We calculate the positive roots for so3,3using method [ii].  We see that there are different possibilities for the choice of positive roots.

so44 > 

PR_so33b≔PositiveRoots⁡RSD_so44,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1

 

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

so44 > 

PositiveRoots⁡RSD_so44,4,3,2,1

 

Example 2.

To illustrate the second calling sequence we use the command LieAlgebraRoots to extract the roots from the table RSD_so44 giving the root space decomposition for so4,4.

so44 > 

Delta_so44≔LieAlgebraRoots⁡RSD_so44

so44 > 

PositiveRoots⁡Delta_so44,4,3,2,1

 

Example 3.

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

so44 > 

PR_so44a

 

Here are the simple roots for the first set of positive roots for su4, 4.  (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≔0,−1,1,0,1,1,0,0,0,0,−1,1,−1,1,0,0

 

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

so44 > 

Ca1≔GetComponents⁡PR_so44a,SR_so44a

Ca1:=1,0,0,1,0,1,0,0,1,0,0,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,1,0,0,1,1,1,0,1,1,1,1,2,1,1,1

(2.4)

 

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≔CartanMatrix⁡SR_so44a,RSD_so44

 

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

so44 > 

Ca2≔PositiveRoots⁡CM_so44a

 

We can check that, apart from their order and format, the coefficients in (2.4) and  are the same.

so44 > 

is⁡convert⁡Ca1,set=convert⁡map⁡convert,Ca2,list,set

true

(2.5)

 

Example 4.

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

so44 > 

infolevelPositiveRoots≔2

infolevelDifferentialGeometry:-LieAlgebras:-PositiveRoots:=2

(2.6)
so44 > 

PositiveRoots⁡CM_so44a

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]]

 

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 > 

SR_so44a,CM_so44a

so44 > 

CartanMatrixToStandardForm⁡CM_so44a

 

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

so44 > 

newCm,newSR,RootType≔CartanMatrixToStandardForm⁡CM_so44a,SR_so44a

 

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

so44 > 

newC≔GetComponents⁡PR_so44a,newSR

newC:=0,1,0,1,1,0,0,0,0,1,0,0,0,1,1,0,0,1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,1,0,0,1,1,1,0,1,1,1,1,1,2,1,1

(2.7)

 

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

so61 > 

newC1≔PositiveRoots⁡D,4

 

Again, the component lists (2.7) and  coincide, apart from their order and format.

so44 > 

is⁡convert⁡newC,set=convert⁡map⁡convert,newC1,list,set

true

(2.8)

 

Example 5.

We calculate a set of positive roots for so6,1. 

so61 > 

RemoveFrame⁡so44

0

(2.9)
> 

LD≔SimpleLieAlgebraData⁡so(6, 1),so61,labelformat=gl,labels=E,ω:

> 

DGsetup⁡LD

Lie algebra: so61

(2.10)

 

Find a Cartan subalgebra and the corresponding root space decomposition.

so61 > 

CSA_so61≔CartanSubalgebra⁡so61

CSA_so61:=E11,E34,E56

(2.11)
so61 > 

RSD_so61≔RootSpaceDecomposition⁡CSA_so61

RSD_so61:=table0,−I,0=E37+I⁢E47,1,0,0=E17,1,−I,0=E13+I⁢E14,0,I,I=E35−I⁢E36−I⁢E45−E46,0,−I,−I=E35+I⁢E36+I⁢E45−E46,−1,−I,0=E23+I⁢E24,−1,0,I=E25−I⁢E26,0,0,−I=E57+I⁢E67,−1,0,−I=E25+I⁢E26,−1,0,0=E27,0,I,0=E37−I⁢E47,0,I,−I=E35+I⁢E36−I⁢E45+E46,1,I,0=E13−I⁢E14,−1,I,0=E23−I⁢E24,0,−I,I=E35−I⁢E36+I⁢E45+E46,0,0,I=E57−I⁢E67,1,0,I=E15−I⁢E16,1,0,−I=E15+I⁢E16

(2.12)

 

The indices of this table give the roots for so6,1. Here are the roots, now as vectors.

so61 > 

Delta_so61≔LieAlgebraRoots⁡RSD_so61

 

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≔1,0,0,0,I,0,0,0,I

so61 > 

PR_so61≔PositiveRoots⁡Delta_so61,B

 

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≔SimpleRoots⁡PR_so61

so61 > 

CM_so61≔CartanMatrix⁡SR_so61,RSD_so61

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

so61 > 

Pr_so61b≔PositiveRoots⁡CM_so61,SR_so61

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

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

Apart from their order, the two sets of vectors  and  coincide.

 

Example 6.

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

> 

CM_F4≔CartanMatrix⁡F,4

> 

PR_F4≔PositiveRoots⁡CM_F4

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]]

> 

nops⁡PR_F4

24

(2.13)

 

See Also

DifferentialGeometry

Details for PositiveRoots

DGzip

GetComponents

LieAlgebras

RootSpaceDecomposition

SimpleLieAlgebraData

SimpleRoots