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
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,ℂ,𝕁n,ℍ, 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,ℂ,𝕁n,ℍ, 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.
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
Here are the split quaternions.
AD1b≔AlgebraLibraryDataQuaternions,Hs,type=Split:
DGsetupAD1b,e,i,j,k,ω
algebra name: Hs
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
MultiplicationTableH,AlgebraTable,MultiplicationTableHs,AlgebraTable
| eijk---- ---- ---- ---- ---- e| eijki| i−ek−jj| j−k−eik| kj−i−e,| eijk---- ---- ---- ---- ---- e| eijki| i−ek−jj| j−ke−ik| kjie
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
MultiplicationTableO1,AlgebraTable
| e0e1e2e3e4e5e6e7---- ---- ---- ---- ---- ---- ---- ---- ---- e0| e0e1e2e3e4e5e6e7e1| e1−e0e3−e2e5−e4−e7e6e2| e2−e3−e0e1e6e7−e4−e5e3| e3e2−e1−e0e7−e6e5−e4e4| e4−e5−e6−e7−e0e1e2e3e5| e5e4−e7e6−e1−e0−e3e2e6| e6e7e4−e5−e2e3−e0−e1e7| e7−e6e5e4−e3−e2e1−e0
For the second version, the non-zero components of the 3-form γ are γ124 = γ137 =γ457 = γ267=γ235= γ346=γ 156= 1.
AD2b≔AlgebraLibraryDataOctonions,O2,version=2:
DGsetupAD2b,e0,e1,e2,e3,e4,e5,e6,e7,ω
algebra name: O2
MultiplicationTableO2,AlgebraTable
| e0e1e2e3e4e5e6e7---- ---- ---- ---- ---- ---- ---- ---- ---- e0| e0e1e2e3e4e5e6e7e1| e1−e0e4e7−e2e6−e5−e3e2| e2−e4−e0e5e1−e3e7−e6e3| e3−e7−e5−e0e6e2−e4e1e4| e4e2−e1−e6−e0e7e3−e5e5| e5−e6e3−e2−e7−e0e1e4e6| e6e5−e7e4−e3−e1−e0e2e7| e7e3e6−e1e5−e4−e2−e0
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 + ej⋅ei=−2 Qei,eje0.
A vector space basis for the Clifford algebra is the identity e0 and the ordered products ei1⋅ei2⋅ ... ⋅ eir, where 1 ≤i1 < i2 < ... <ir ≤ n. The dimension of ℂln, Q is 2n. The default choice for the quadratic form Q is given by the identity matrix In.
We first display the multiplication tables for ℂl3 .
AD3a≔AlgebraLibraryDataClifford(3),Cl3:
DGsetupAD3a,e0,e1,e2,e3,e12,e13,e23,e123,ω
algebra name: Cl3
MultiplicationTableCl3,AlgebraTable
| e0e1e2e3e12e13e23e123---- ---- ---- ---- ---- ---- ---- ---- ---- e0| e0e1e2e3e12e13e23e123e1| e1−e0e12e13−e2−e3e123−e23e2| e2−e12−e0e23e1−e123−e3e13e3| e3−e13−e23−e0e123e1e2−e12e12| e12e2−e1e123−e0e23−e13−e3e13| e13e3−e123−e1−e23−e0e12e2e23| e23e123e3−e2e13−e12−e0−e1e123| e123−e23e13−e12−e3e2−e1e0
We note that the Clifford algebras are always associative.
Here is the multiplication table for ℂl3, I12.
I12≔Matrix1,0,0,0,−1,0,0,0,−1
I12:=1000−1000−1
AD3b≔AlgebraLibraryDataClifford(3),Cl3Q,quadraticform=I12:
DGsetupAD3b,e0,e1,e2,e3,e12,e13,e23,e123,ω
algebra name: Cl3Q
MultiplicationTableCl3Q,AlgebraTable
| e0e1e2e3e12e13e23e123---- ---- ---- ---- ---- ---- ---- ---- ---- e0| e0e1e2e3e12e13e23e123e1| e1−e0e12e13−e2−e3e123−e23e2| e2−e12e0e23−e1−e123e3−e13e3| e3−e13−e23e0e123−e1−e2e12e12| e12e2e1e123e0e23e13e3e13| e13e3−e123e1−e23e0−e12−e2e23| e23e123−e3e2−e13e12−e0−e1e123| e123−e23−e13e12e3−e2−e1e0
Finally, we remark that the quaternions ℍ and the Clifford algebra ℂl2 are isomorphic.
AD3b≔AlgebraLibraryDataClifford(2),Cl2:
DGsetupAD3b,e0,e1,e2,e3,ω
algebra name: Cl2
MultiplicationTableH,AlgebraTable,MultiplicationTableCl2,AlgebraTable
| eijk---- ---- ---- ---- ---- e| eijki| i−ek−jj| j−k−eik| kj−i−e,| e0e1e2e3---- ---- ---- ---- ---- e0| e0e1e2e3e1| e1−e0e3−e2e2| e2−e3−e0e1e3| e3e2−e1−e0
Example 4.
Here are the structure equations for the Jordan algebra 𝕁2, ℍ. This is the algebra of 2 × 2 Hermitian matrices with quaternionic entries and the product a⋅b= 12ab+ba.
AD4≔AlgebraLibraryDataJordan(2, Quaternions),J2H:
DGsetupAD4
algebra name: J2H
MultiplicationTableJ2H,AlgebraTable
| e1e2e3e4e5e6---- ---- ---- ---- ---- ---- ---- e1| e10e112e312e412e512e6e2| 0e1e212e312e412e512e6e3| 12e312e3e1+e20e10e10e1e4| 12e412e40e1e1+e20e10e1e5| 12e512e50e10e1e1+e20e1e6| 12e612e60e10e10e1e1+e2
See Also
DifferentialGeometry
AlgebraData
Algebra Inverse
AlgebraNorm
DGsetup
MultiplicationTable
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