LieAlgebras[JordanMatrices] - find the basis for a Jordan algebra of matrices
LieAlgebras[JordanProduct] - find the Jordan product of two Jordan matrices
Calling Sequences
JordanMatrices(n, alg, option )
JordanProduct(A, B)
Parameters
n - an integer
alg - a name or string of an initialized algebra, the string "R" or the string "C"
option - the keyword argument signature = [p, q], where p and q are integers and p + q = n
A, B - square matrices
Description
Examples
Let 𝔸 be the algebra of real numbers, the complex numbers, the quaternions, the octonions, or one of their split versions. A Jordan matrix J is a square matrix with entries in 𝔸 which is Hermitian with respect to the conjugation in the algebra, that is, J = J†. More generally, if Ipq is the n × n diagonal matrix Ipq=Ip00−Iq and IpqJ = J†Ipq , then J is called a Jordan matrix with respect to Ipq . The set of such matrices is always a real vector space.
The command JordanMatrix(n, alg) returns a list of matrices which form a basis for the real vector space of n × n square matrices with entries in 𝔸 . With the keyword argument signature = [p, q] a basis for the Jordan matrices with respect to Ipq is determined.
The Jordan product of 2 Jordan matrices A and B is the symmetric product A ∘B =1/2A⋅B +B⋅A. The set of Jordan matrices with Jordan product is an algebra which is denoted by 𝕁n,𝔸 or 𝕁n, 𝔸, Ipq.
The structure equations for a general Jordan algebra can be calculated with the command AlgebraData. The structure equations for a few low dimensional Jordan algebras are also available through the command AlgebraLibraryData.
withDifferentialGeometry:withLieAlgebras:withTools:
Example 1.
In this example we construct a basis for the Jordan algebra of 3 ×3 matrices over the quaternions. The first step is to use the command AlgebraLibraryData to retrieve the structure equations for the quaternions.
AD≔AlgebraLibraryDataQuaternions,Q
AD:=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
Initialize this algebra.
DGsetupAD,e,i,j,k,ω
algebra name: Q
Generate a basis for the Jordan algebra of 3 ×3 matrices over the quaternions.
M≔JordanMatrices3,Q
M:=e0e0e0e0e0e0e0e0e,0e0e0e0ee0e0e0e0e,0e0e0e0e0e0e0e0ee,0ee0ee0e0e0e0e0e,0e0ee0e0e0ee0e0e,0e0e0e0e0ee0ee0e,0ei0e−i0e0e0e0e0e,0ej0e−j0e0e0e0e0e,0ek0e−k0e0e0e0e0e,0e0ei0e0e0e−i0e0e,0e0ej0e0e0e−j0e0e,0e0ek0e0e0e−k0e0e,0e0e0e0e0ei0e−i0e,0e0e0e0e0ej0e−j0e,0e0e0e0e0ek0e−k0e
We form the general element of 𝕁3,ℚ and check it is Hermitian.
C≔seqc‖n,n=1..15
C:=c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11,c12,c13,c14,c15
J≔evalDGDGzipC,M,plus
J:=c1ec4e+c7i+c8j+c9kc5e+c10i+c11j+c12kc4e−c7i−c8j−c9kc2ec6e+c13i+c14j+c15kc5e−c10i−c11j−c12kc6e−c13i−c14j−c15kc3e
Here is the conjugate transpose of J.
Jdagger≔DGconjugateJ%T
Jdagger:=c1ec4e+c7i+c8j+c9kc5e+c10i+c11j+c12kc4e−c7i−c8j−c9kc2ec6e+c13i+c14j+c15kc5e−c10i−c11j−c12kc6e−c13i−c14j−c15kc3e
We see that J is Hermitian.
J&MatrixMinusJdagger
0e0e0e0e0e0e0e0e0e
Now define two elements of 𝕁3,ℚ and calculate their Jordan product.
A≔evalDGM8+M12
A:=0ejk−j0e0e−k0e0e
B≔evalDGM7+M15
B:=0ei0e−i0ek0e−k0e
JordanProductA,B
0e12e12i12e0e12j−12i−12j0e
Example 2.
In this example we construct a basis for the 2 ×2 Jordan matrices over the split octonions with respect to the inner product I22 = 100−1. First we retrieve the structure equations for the split octonions and initialize.
AD≔AlgebraLibraryDataOctonions,Os,type=Split
AD:=e12=e1,e1.e2=e2,e1.e3=e3,e1.e4=e4,e1.e5=e5,e1.e6=e6,e1.e7=e7,e1.e8=e8,e2.e1=e2,e22=−e1,e2.e3=e4,e2.e4=−e3,e2.e5=−e6,e2.e6=e5,e2.e7=−e8,e2.e8=e7,e3.e1=e3,e3.e2=−e4,e32=−e1,e3.e4=e2,e3.e5=−e7,e3.e6=e8,e3.e7=e5,e3.e8=−e6,e4.e1=e4,e4.e2=e3,e4.e3=−e2,e42=−e1,e4.e5=−e8,e4.e6=−e7,e4.e7=e6,e4.e8=e5,e5.e1=e5,e5.e2=e6,e5.e3=e7,e5.e4=e8,e52=e1,e5.e6=e2,e5.e7=e3,e5.e8=e4,e6.e1=e6,e6.e2=−e5,e6.e3=−e8,e6.e4=e7,e6.e5=−e2,e62=e1,e6.e7=e4,e6.e8=−e3,e7.e1=e7,e7.e2=e8,e7.e3=−e5,e7.e4=−e6,e7.e5=−e3,e7.e6=−e4,e72=e1,e7.e8=e2,e8.e1=e8,e8.e2=−e7,e8.e3=e6,e8.e4=−e5,e8.e5=−e4,e8.e6=e3,e8.e7=−e2,e82=e1
DGsetupAD
algebra name: Os
Here are the Jordan matrices we seek.
M≔JordanMatrices2,Os,signature=1,1
M:=e10e10e10e1,0e1e1−e10e1,0e1e2e20e1,0e1e3e30e1,0e1e4e40e1,0e1e5e50e1,0e1e6e60e1,0e1e7e70e1,0e1e8e80e1,0e10e10e1e1
We form the general element of 𝕁2,𝕆s and check that it is Hermitian.
C≔seqc‖n,n=1..10
C:=c1,c2,c3,c4,c5,c6,c7,c8,c9,c10
J:=c1e1c2e1+c3e2+c4e3+c5e4+c6e5+c7e6+c8e7+c9e8−c2e1+c3e2+c4e3+c5e4+c6e5+c7e6+c8e7+c9e8c10e1
Jdagger:=c1e1−c2e1−c3e2−c4e3−c5e4−c6e5−c7e6−c8e7−c9e8c2e1−c3e2−c4e3−c5e4−c6e5−c7e6−c8e7−c9e8c10e1
I22≔Matrix1,0,0,−1
I22:=100−1
evalDGI22·J&MatrixMinusevalDGJdagger·I22
0e10e10e10e1
Now define two elements of 𝕁2,ℚs and calculate their Jordan product.
A≔evalDGM8+M10
A:=0e1e7e7e1
B≔evalDGM1+M4
B:=e1e3e30e1
0e112e3+12e712e3+12e70e1
See Also
DifferentialGeometry
LieAlgebras
AlgebraData
AlgebraLibraryData
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