retrieve the structure equations for various classical algebras (quaternions, octonions, Clifford algebras, and low dimensional Jordan algebras) - Maple Programming Help

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LieAlgebras[AlgebraLibraryData] - retrieve the structure equations for various classical algebras (quaternions, octonions, Clifford algebras, and low dimensional Jordan algebras)

Calling Sequences

   AlgebraLibraryData(AlgType, AlgName, options)

    

Parameters

     AlgType    - a string, "Real", "Complex", "Quaternions", "Octonions", "Clifford(n)", "Jordan(n, Real)", "Jordan(n, Complex)", "Jordan(n, Quaternions)", "Jordan(n, Octonions)" where n is a positive integer

     AlgName    - a name or a string, the frame name for the algebra being created

     options    - the keyword arguments type = "Standard" or type ="Split", version = 1 or version =2, quadraticform = Q where Q is a non-singular symmetric matrix.

  

 

Description

Examples

Description

• 

 The command AlgebraLibraryData retrieves the structure equations for any of the following real algebras: the real numbers ℝ, the complex numbers ℂ, the quaternions ℍ, the octonions 𝕆, the Clifford algebras ℂln, Q on ℝn with respect to the quadratic form Q, and the Jordan algebras 𝕁n,,𝕁n,, 𝕁n,𝕆 for small values of n.

• 

The keyword argument type ="Split" may be applied to the algebras ℂ, ℍ,𝕆 to obtain their split forms. The argument type ="Split" can be applied to 𝕁n,,𝕁n,, 𝕁n,𝕆 to obtain the Jordan algebras defined over the split complex numbers, the split quaternions, or the split octonions.

• 

There are two generally accepted versions of the structure equations for the octonions. These are described in Example 2.

• 

The keyword argument quadraticform = Q can be used create the general Clifford algebras, defined with respect to a quadratic form. See Example 3.

• 

For the following small values of n, the structure equations have been stored in Maple and are available without computation: 𝕁n,ℝ for n = 2, 3, 4,5; 𝕁n,ℂ for n = 2, 3,4,5;  𝕁n,ℍ for n = 2, 3, 4; 𝕁n,𝕆 for n = 2, 3. More generally, Jordan algebras can be created using the command JordanMatrices, JordanProduct, and AlgebraData.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

We define the quaternions and the split quaternions and compare their multiplication tables:

AD1a ≔ AlgebraLibraryDataQuaternions,H:

DGsetupAD1a,'e,i,j,k','ω'

algebra name: H

(2.1)

 

Here are the split quaternions.

H > 

AD1b ≔ AlgebraLibraryDataQuaternions,Hs,type=Split:

H > 

DGsetupAD1b,'e,i,j,k',ω

algebra name: Hs

(2.2)

 

We see that the off-diagonal products in the multiplication tables are the same. For the quaternions j2 = k2 = e while for the split quaternions .j2 = k2 = e 

Hs > 

MultiplicationTableH,AlgebraTable,MultiplicationTableHs,AlgebraTable

| eijk---- ---- ---- ---- ---- e| eijki| iekjj| jkeik| kjie,| eijk---- ---- ---- ---- ---- e| eijki| iekjj| jkeik| kjie

(2.3)

 

Example 2.

Various conventions can be found in the literature for the multiplication table for the octonions, differing by a labeling of the basis elements. The command AlgebraLibraryData provides 2 different conventions. For the first, the multiplication rules are defined by the formula

 eiej = δije0 + γijkek ,

where the γijk are the components of a 3-form determined by γ123 = γ145 =γ176 = γ246=γ257= γ347=γ365 = 1. These multiplication rules are summarized using the Fano plane mnemonic:

 

 

 

The triple of integers lying on a straight line or circle coincide with the non-zero coefficients of γ.

AD2a ≔ AlgebraLibraryDataOctonions,O1:

DGsetupAD2a,'e0,e1,e2,e3,e4,e5,e6,e7',ω

algebra name: O1

(2.4)
O1 > 

MultiplicationTableO1,AlgebraTable

| e0e1e2e3e4e5e6e7---- ---- ---- ---- ---- ---- ---- ---- ---- e0| e0e1e2e3e4e5e6e7e1| e1e0e3e2e5e4e7e6e2| e2e3e0e1e6e7e4e5e3| e3e2e1e0e7e6e5e4e4| e4e5e6e7e0e1e2e3e5| e5e4e7e6e1e0e3e2e6| e6e7e4e5e2e3e0e1e7| e7e6e5e4e3e2e1e0

(2.5)

 

For the second version, the non-zero components of the 3-form γ are γ124 = γ137 =γ457 = γ267=γ235= γ346=γ 156= 1. 

O1 > 

AD2b ≔ AlgebraLibraryDataOctonions,O2,version=2:

O1 > 

DGsetupAD2b,'e0,e1,e2,e3,e4,e5,e6,e7',ω

algebra name: O2

(2.6)
O2 > 

MultiplicationTableO2,AlgebraTable

| e0e1e2e3e4e5e6e7---- ---- ---- ---- ---- ---- ---- ---- ---- e0| e0e1e2e3e4e5e6e7e1| e1e0e4e7e2e6e5e3e2| e2e4e0e5e1e3e7e6e3| e3e7e5e0e6e2e4e1e4| e4e2e1e6e0e7e3e5e5| e5e6e3e2e7e0e1e4e6| e6e5e7e4e3e1e0e2e7| e7e3e6e1e5e4e2e0

(2.7)

Both versions have split counterparts.

 

Example 3.

Let V be a vector space with basis e1, e2 , ..., en and let Q be a non-degenerate quadratic form on V. The Clifford algebra ℂln, Q is the algebra generated by products of the vectors ei , subject to the multiplication rules

 

ei ej + ejei=2 Qei,eje0.

 

A vector space basis for the Clifford algebra is the identity e0 and the ordered products ei1ei2 ..&period;  eir, where 1 i1 < i2 < ..&period; <ir  n. The dimension of &complexes;ln&comma; Q is 2n&period; The default choice for the quadratic form Q is given by the identity matrix In.

 

We first display the multiplication tables for &complexes;l3 &period;

O2 > 

AD3a ≔ AlgebraLibraryDataClifford(3)&comma;Cl3&colon;

O2 > 

DGsetupAD3a&comma;&apos;e0&comma;e1&comma;e2&comma;e3&comma;e12&comma;e13&comma;e23&comma;e123&apos;&comma;&apos;&omega;&apos;

algebra name: Cl3

(2.8)
O2 > 

MultiplicationTableCl3&comma;AlgebraTable

| e0e1e2e3e12e13e23e123---- ---- ---- ---- ---- ---- ---- ---- ---- e0| e0e1e2e3e12e13e23e123e1| e1e0e12e13e2e3e123e23e2| e2e12e0e23e1e123e3e13e3| e3e13e23e0e123e1e2e12e12| e12e2e1e123e0e23e13e3e13| e13e3e123e1e23e0e12e2e23| e23e123e3e2e13e12e0e1e123| e123e23e13e12e3e2e1e0

(2.9)

 

We note that the Clifford algebras are always associative.

 

Here is the multiplication table for &complexes;l3&comma; I12.

Cl3 > 

I12 ≔ Matrix1&comma;0&comma;0&comma;0&comma;1&comma;0&comma;0&comma;0&comma;1

I12:=100010001

(2.10)
O2 > 

AD3b ≔ AlgebraLibraryDataClifford(3)&comma;Cl3Q&comma;quadraticform&equals;I12&colon;

O2 > 

DGsetupAD3b&comma;&apos;e0&comma;e1&comma;e2&comma;e3&comma;e12&comma;e13&comma;e23&comma;e123&apos;&comma;&apos;&omega;&apos;

algebra name: Cl3Q

(2.11)
O2 > 

MultiplicationTableCl3Q&comma;AlgebraTable

| e0e1e2e3e12e13e23e123---- ---- ---- ---- ---- ---- ---- ---- ---- e0| e0e1e2e3e12e13e23e123e1| e1e0e12e13e2e3e123e23e2| e2e12e0e23e1e123e3e13e3| e3e13e23e0e123e1e2e12e12| e12e2e1e123e0e23e13e3e13| e13e3e123e1e23e0e12e2e23| e23e123e3e2e13e12e0e1e123| e123e23e13e12e3e2e1e0

(2.12)

 

Finally, we remark that the quaternions &quaternions; and the Clifford algebra &complexes;l2 are isomorphic.

O2 > 

AD3b ≔ AlgebraLibraryDataClifford(2)&comma;Cl2&colon;

O2 > 

DGsetupAD3b&comma;&apos;e0&comma;e1&comma;e2&comma;e3&apos;&comma;&apos;&omega;&apos;

algebra name: Cl2

(2.13)
O2 > 

MultiplicationTableH&comma;AlgebraTable&comma;MultiplicationTableCl2&comma;AlgebraTable

| eijk---- ---- ---- ---- ---- e| eijki| iekjj| jkeik| kjie&comma;| e0e1e2e3---- ---- ---- ---- ---- e0| e0e1e2e3e1| e1e0e3e2e2| e2e3e0e1e3| e3e2e1e0

(2.14)

 

Example 4.

Here are the structure equations for the Jordan algebra &Jopf;2&comma; &quaternions;. This is the algebra of 2 × 2 Hermitian matrices with quaternionic entries and the product ab&equals; 12ab&plus;ba&period;

Cl3 > 

AD4 ≔ AlgebraLibraryDataJordan(2, Quaternions)&comma;J2H&colon;

Cl2 > 

DGsetupAD4

algebra name: J2H

(2.15)
J2 > 

MultiplicationTableJ2H&comma;AlgebraTable

| e1e2e3e4e5e6---- ---- ---- ---- ---- ---- ---- e1| e10e112e312e412e512e6e2| 0e1e212e312e412e512e6e3| 12e312e3e1&plus;e20e10e10e1e4| 12e412e40e1e1&plus;e20e10e1e5| 12e512e50e10e1e1&plus;e20e1e6| 12e612e60e10e10e1e1&plus;e2

(2.16)

See Also

DifferentialGeometry

AlgebraData

Algebra Inverse

AlgebraNorm

DGsetup

MultiplicationTable

 


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