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LieAlgebras[AlgebraData] - find the structure equations for a real algebra defined by a list of matrices and a multiplication procedure

Calling Sequences

     AlgebraData(A, mu, algname)

Parameters

     A        - a list of square matrices, with entries which are real numbers, complex numbers or vectors in an algebra.

     mu       - a 2 argument procedure defining a multiplication rule for the matrices A.

     algname  - an unassigned name or string

  

Description

• 

 Let A = A1, A2, ..., AN be a list of square matrices with entries which are real numbers ℝ, complex numbers ℂ or vectors in an algebra 𝔸 . In most applications, the algebra 𝔸 is one that can be created by the AlgebraLibraryData command such as the quaternions, octonions, or a Clifford algebra. The matrices A must be linearly independent over ℝ. The multiplication procedure must return a matrix μAi,Aj which is a real linear combination of the matrices in A, μ Ai,Aj = Cijk Ak.. The algebra defined in this manner need not be commutative, skew-commutative or associative.

• 

The command AlgebraData returns the algebra data structure specified by the structure constants Cijk which can be subsequently initialized with DGsetup.

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

For the first example, we simply take A to be the standard basis for the vector space of 2 × 2 matrices and let μ be the usual matrix product.

 

A:=Matrix1,0,0,0,Matrix0,1,0,0,Matrix0,0,1,0,Matrix0,0,0,1

A:=1000,0100,0010,0001

(2.1)

μ1:=a,b→a.b

μ1:=a,b→a.b

(2.2)

 

The AlgebraData commands produces the usual multiplication table for 2 × 2 matrices (Here e1 denotes the first matrix in the list A, e2 the second, and so on).

AD1:=AlgebraDataA,μ1,alg1

AD1:=e12=e1,e1.e2=e2,e2.e3=e1,e2.e4=e2,e3.e1=e3,e3.e2=e4,e4.e3=e3,e42=e4

(2.3)

DGsetupAD1

algebra name: alg1

(2.4)

 

This algebra is non-commutative but associative.

alg1 > 

Queryalg1,Commutative

false

(2.5)
alg1 > 

Queryalg1,Associative

true

(2.6)

 

Example 2.

Again let A be the standard basis for the vector space of 2 × 2  matrices and but now let μ be the Jordan product ( a b = 12ab+ ba.

 

alg1 > 

A:=Matrix1,0,0,0,Matrix0,1,0,0,Matrix0,0,1,0,Matrix0,0,0,1

A:=1000,0100,0010,0001

(2.7)
alg1 > 

μ3:=JordanProduct

μ3:=DifferentialGeometry:-LieAlgebras:-JordanProduct

(2.8)

 

The structure equations for this Jordan algebra are:

AD2:=AlgebraDataA,μ3,alg2

AD2:=e12=e1,e1.e2=12e2,e1.e3=12e3,e2.e1=12e2,e2.e3=12e1+12e4,e2.e4=12e2,e3.e1=12e3,e3.e2=12e1+12e4,e3.e4=12e3,e4.e2=12e2,e4.e3=12e3,e42=e4

(2.9)

DGsetupAD2

algebra name: alg2

(2.10)

 

This time the algebra is commutative but not associative.

alg2 > 

Queryalg2,Commutative

true

(2.11)
alg2 > 

Queryalg2,Associative

false

(2.12)

 

Example 3.

Now let μ be the matrix commutator.

 

alg1 > 

A:=Matrix1,0,0,0,Matrix0,1,0,0,Matrix0,0,1,0,Matrix0,0,0,1

A:=1000,0100,0010,0001

(2.13)
alg1 > 

μ3:=a,b→a.bb.a

μ3:=a,b→a.bb.a

(2.14)

 

Now the structure equations

AD3:=AlgebraDataA,μ3,alg3

AD3:=e1.e2=e2,e1.e3=e3,e2.e1=e2,e2.e3=e1e4,e2.e4=e2,e3.e1=e3,e3.e2=e1+e4,e3.e4=e3,e4.e2=e2,e4.e3=e3

(2.15)

 

coincide with the structure equations for the Lie algebra of 2 ×2 matrices.

alg2 > 

LD:=LieAlgebraDataA,Liealg3

LD:=e1,e2=e2,e1,e3=e3,e2,e3=e1e4,e2,e4=e2,e3,e4=e3

(2.16)

 

Note that in (2.15) both products e1e3 and e3e1 must be specified but in (2.16) only the product e1, e3 is calculated and stored.

 

Example 4.

In this example we shall calculate the structure equations for the Jordan algebra of 3 × 3 matrices over the quaternions ℚ. First we create the quaternions with AlgebraLibraryData.

alg2 > 

AD4a:=AlgebraLibraryDataQuaternions,Qn

AD4a:=e12=e1,e1.e2=e2,e1.e3=e3,e1.e4=e4,e2.e1=e2,e22=e1,e2.e3=e4,e2.e4=e3,e3.e1=e3,e3.e2=e4,e32=e1,e3.e4=e2,e4.e1=e4,e4.e2=e3,e4.e3=e2,e42=e1

(2.17)
alg2 > 

DGsetupAD4a

algebra name: Qn

(2.18)

 

We use JordanMatrices to generate a basis for the space of 3 ×3 Hermitian matrices with entries in ℚn.

Qn > 

J:=JordanMatrices3,Qn

J:=e10e10e10e10e10e10e10e10e1,0e10e10e10e1e10e10e10e10e1,0e10e10e10e10e10e10e10e1e1,0e1e10e1e10e10e10e10e10e1,0e10e1e10e10e10e1e10e10e1,0e10e10e10e10e1e10e1e10e1,0e1e20e1e20e10e10e10e10e1,0e1e30e1e30e10e10e10e10e1,0e1e40e1e40e10e10e10e10e1,0e10e1e20e10e10e1e20e10e1,0e10e1e30e10e10e1e30e10e1,0e10e1e40e10e10e1e40e10e1,0e10e10e10e10e1e20e1e20e1,0e10e10e10e10e1e30e1e30e1,0e10e10e10e10e1e40e1e40e1

(2.19)

 

We find the structure equations for this 15-dimensional algebra, initialize the algebra, and display the structure equations in the form of a multiplication table.

M > 

AD4:=AlgebraDataJ,JordanProduct,J3Qn:

Qn > 

DGsetupAD4

algebra name: J3Qn

(2.20)
J3Qn > 

interfacertablesize=17

10

(2.21)
Qn > 

MultiplicationTableAlgebraTable

| e1e2e3e4e5e6e7e8e9e10e11e12e13e14e15---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- ---- e1| e10e10e112e412e50e112e712e812e912e1012e1112e120e10e10e1e2| 0e1e20e112e40e112e612e712e812e90e10e10e112e1312e1412e15e3| 0e10e1e30e112e512e60e10e10e112e1012e1112e1212e1312e1412e15e4| 12e412e40e1e1+e212e612e50e10e10e112e1312e1412e1512e1012e1112e12e5| 12e50e112e512e6e1+e312e412e1312e1412e150e10e10e112e712e812e9e6| 0e112e612e612e512e4e2+e312e1012e1112e1212e712e812e90e10e10e1e7| 12e712e70e10e112e1312e10e1+e20e10e112e612e1512e1412e512e1212e11e8| 12e812e80e10e112e1412e110e1e1+e20e112e1512e612e1312e1212e512e10e9| 12e912e90e10e112e1512e120e10e1e1+e212e1412e1312e612e1112e1012e5e10| 12e100e112e1012e130e112e712e612e1512e14e1+e30e10e112e412e912e8e11| 12e110e112e1112e140e112e812e1512e612e130e1e1+e30e112e912e412e7e12| 12e120e112e1212e150e112e912e1412e1312e60e10e1e1+e312e812e712e4e13| 0e112e1312e1312e1012e70e112e512e1212e1112e412e912e8e2+e30e10e1e14| 0e112e1412e1412e1112e80e112e1212e512e1012e912e412e70e1e2+e30e1e15| 0e112e1512e1512e1212e90e112e1112e1012e512e812e712e40e10e1e2+e3

(2.22)

 

 

 

See Also

DifferentialGeometry,  LieAlgebras, AlgebraLibraryData, AlgebraInverse, AlgebraNorm , JordanMatrices, JordanProduct, LieAlgebraData, MultiplicationTable


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