JordanMatrices - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

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

 • Let be the algebra of real numbers, the complex numbers, the quaternions, the octonions, or one of their split versions. A Jordan matrix is a square matrix with entries in which is Hermitian with respect to the conjugation in the algebra, that is,  More generally, if is the diagonal matrix ${I}_{\mathrm{pq}}=\left[\begin{array}{rr}{I}_{p}& 0\\ 0& -{I}_{q}\end{array}\right]$ and then $J$ is called a Jordan matrix with respect to . 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  square matrices with entries in . With the keyword argument signature = [p, q] a basis for the Jordan matrices with respect to ${I}_{\mathrm{pq}}$ is determined.
 • The Jordan product of 2 Jordan matrices and is the symmetric product . The set of Jordan matrices with Jordan product is an algebra which is denoted by $\mathrm{𝕁}\left(n\mathit{,}\mathrm{𝔸}\right)$or .
 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{LieAlgebras}\right):$$\mathrm{with}\left(\mathrm{Tools}\right):$

Example 1.

In this example we construct a basis for the Jordan algebra of matrices over the quaternions. The first step is to use the command AlgebraLibraryData to retrieve the structure equations for the quaternions.

 > $\mathrm{AD}≔\mathrm{AlgebraLibraryData}\left("Quaternions",Q\right)$
 ${\mathrm{AD}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}\right]$ (2.1)

Initialize this algebra.

 > $\mathrm{DGsetup}\left(\mathrm{AD},'\left[e,i,j,k\right]','\left[\mathrm{\omega }\right]'\right)$
 ${\mathrm{algebra name: Q}}$ (2.2)

Generate a basis for the Jordan algebra of matrices over the quaternions.

 > $M≔\mathrm{JordanMatrices}\left(3,Q\right)$

We form the general element of $\mathrm{𝕁}\left(\mathit{3}\mathit{,}\mathrm{ℚ}\right)$ and check it is Hermitian.

 Q > $C≔\left[\mathrm{seq}\left(c‖n,n=1..15\right)\right]$
 ${C}{:=}\left[{\mathrm{c1}}{,}{\mathrm{c2}}{,}{\mathrm{c3}}{,}{\mathrm{c4}}{,}{\mathrm{c5}}{,}{\mathrm{c6}}{,}{\mathrm{c7}}{,}{\mathrm{c8}}{,}{\mathrm{c9}}{,}{\mathrm{c10}}{,}{\mathrm{c11}}{,}{\mathrm{c12}}{,}{\mathrm{c13}}{,}{\mathrm{c14}}{,}{\mathrm{c15}}\right]$ (2.3)
 Q > $J≔\mathrm{evalDG}\left(\mathrm{DGzip}\left(C,M,"plus"\right)\right)$

Here is the conjugate transpose of J.

 Q > $\mathrm{Jdagger}≔{\mathrm{DGconjugate}\left(J\right)}^{\mathrm{%T}}$

We see that J is Hermitian.

 Q > $J\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixMinus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Jdagger}$

Now define two elements of $\mathrm{𝕁}\left(\mathit{3}\mathit{,}\mathrm{ℚ}\right)$ and calculate their Jordan product.

 Q > $A≔\mathrm{evalDG}\left(M\left[8\right]+M\left[12\right]\right)$
 Q > $B≔\mathrm{evalDG}\left(M\left[7\right]+M\left[15\right]\right)$
 Q > $\mathrm{JordanProduct}\left(A,B\right)$

Example 2.

In this example we construct a basis for the Jordan matrices over the split octonions with respect to the inner product . First we retrieve the structure equations for the split octonions and initialize.

 Q > $\mathrm{AD}≔\mathrm{AlgebraLibraryData}\left("Octonions",\mathrm{Os},\mathrm{type}="Split"\right)$
 ${\mathrm{AD}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e1}}{.}{\mathrm{e5}}{=}{\mathrm{e5}}{,}{\mathrm{e1}}{.}{\mathrm{e6}}{=}{\mathrm{e6}}{,}{\mathrm{e1}}{.}{\mathrm{e7}}{=}{\mathrm{e7}}{,}{\mathrm{e1}}{.}{\mathrm{e8}}{=}{\mathrm{e8}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e2}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e2}}{.}{\mathrm{e6}}{=}{\mathrm{e5}}{,}{\mathrm{e2}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e2}}{.}{\mathrm{e8}}{=}{\mathrm{e7}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e3}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e3}}{.}{\mathrm{e6}}{=}{\mathrm{e8}}{,}{\mathrm{e3}}{.}{\mathrm{e7}}{=}{\mathrm{e5}}{,}{\mathrm{e3}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e4}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e4}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e4}}{.}{\mathrm{e7}}{=}{\mathrm{e6}}{,}{\mathrm{e4}}{.}{\mathrm{e8}}{=}{\mathrm{e5}}{,}{\mathrm{e5}}{.}{\mathrm{e1}}{=}{\mathrm{e5}}{,}{\mathrm{e5}}{.}{\mathrm{e2}}{=}{\mathrm{e6}}{,}{\mathrm{e5}}{.}{\mathrm{e3}}{=}{\mathrm{e7}}{,}{\mathrm{e5}}{.}{\mathrm{e4}}{=}{\mathrm{e8}}{,}{{\mathrm{e5}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e5}}{.}{\mathrm{e6}}{=}{\mathrm{e2}}{,}{\mathrm{e5}}{.}{\mathrm{e7}}{=}{\mathrm{e3}}{,}{\mathrm{e5}}{.}{\mathrm{e8}}{=}{\mathrm{e4}}{,}{\mathrm{e6}}{.}{\mathrm{e1}}{=}{\mathrm{e6}}{,}{\mathrm{e6}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e6}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e8}}{,}{\mathrm{e6}}{.}{\mathrm{e4}}{=}{\mathrm{e7}}{,}{\mathrm{e6}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e6}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e6}}{.}{\mathrm{e7}}{=}{\mathrm{e4}}{,}{\mathrm{e6}}{.}{\mathrm{e8}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e7}}{.}{\mathrm{e1}}{=}{\mathrm{e7}}{,}{\mathrm{e7}}{.}{\mathrm{e2}}{=}{\mathrm{e8}}{,}{\mathrm{e7}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e7}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e6}}{,}{\mathrm{e7}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e7}}{.}{\mathrm{e6}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e7}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e7}}{.}{\mathrm{e8}}{=}{\mathrm{e2}}{,}{\mathrm{e8}}{.}{\mathrm{e1}}{=}{\mathrm{e8}}{,}{\mathrm{e8}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e7}}{,}{\mathrm{e8}}{.}{\mathrm{e3}}{=}{\mathrm{e6}}{,}{\mathrm{e8}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e5}}{,}{\mathrm{e8}}{.}{\mathrm{e5}}{=}{-}{\mathrm{e4}}{,}{\mathrm{e8}}{.}{\mathrm{e6}}{=}{\mathrm{e3}}{,}{\mathrm{e8}}{.}{\mathrm{e7}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e8}}}^{{2}}{=}{\mathrm{e1}}\right]$ (2.4)
 Q > $\mathrm{DGsetup}\left(\mathrm{AD}\right)$
 ${\mathrm{algebra name: Os}}$ (2.5)

Here are the Jordan matrices we seek.

 Os > $M≔\mathrm{JordanMatrices}\left(2,\mathrm{Os},\mathrm{signature}=\left[1,1\right]\right)$

We form the general element of and check that it is Hermitian.

 Q > $C≔\left[\mathrm{seq}\left(c‖n,n=1..10\right)\right]$
 ${C}{:=}\left[{\mathrm{c1}}{,}{\mathrm{c2}}{,}{\mathrm{c3}}{,}{\mathrm{c4}}{,}{\mathrm{c5}}{,}{\mathrm{c6}}{,}{\mathrm{c7}}{,}{\mathrm{c8}}{,}{\mathrm{c9}}{,}{\mathrm{c10}}\right]$ (2.6)
 Q > $J≔\mathrm{evalDG}\left(\mathrm{DGzip}\left(C,M,"plus"\right)\right)$

Here is the conjugate transpose of J.

 Q > $\mathrm{Jdagger}≔{\mathrm{DGconjugate}\left(J\right)}^{\mathrm{%T}}$
 Os > $\mathrm{I22}≔\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,-1\right]\right]\right)$
 Os > $\mathrm{evalDG}\left(\mathrm{I22}·J\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&MatrixMinus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{evalDG}\left(\mathrm{Jdagger}·\mathrm{I22}\right)$

Now define two elements of $\mathrm{𝕁}\left(\mathit{2}\mathit{,}\mathrm{ℚs}\right)$ and calculate their Jordan product.

 Q > $A≔\mathrm{evalDG}\left(M\left[8\right]+M\left[10\right]\right)$
 Q > $B≔\mathrm{evalDG}\left(M\left[1\right]+M\left[4\right]\right)$
 Q > $\mathrm{JordanProduct}\left(A,B\right)$