AlgebraData - Maple Help
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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

 • Letbe a list of square matrices with entries which are real numbers $\mathrm{ℝ}$, complex numbers $\mathrm{ℂ}$ or vectors in an algebra . In most applications, the algebra is one that can be created by the AlgebraLibraryData command such as the quaternions, octonions, or a Clifford algebra. The matrices must be linearly independent over $\mathrm{ℝ}$. The multiplication procedure must return a matrix which is a real linear combination of the matrices in . 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 which can be subsequently initialized with DGsetup.

Examples

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

Example 1.

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

 > $A≔\left[\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[1,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[0,1\right]\right]\right)\right]$
 > $\mathrm{μ1}≔\left(a,b\right)→\mathrm{.}\left(a,b\right)$
 ${\mathrm{μ1}}{:=}\left({a}{,}{b}\right){→}{a}{.}{b}$ (2.1)

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

 > $\mathrm{AD1}≔\mathrm{AlgebraData}\left(A,\mathrm{μ1},\mathrm{alg1}\right)$
 ${\mathrm{AD1}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{{\mathrm{e4}}}^{{2}}{=}{\mathrm{e4}}\right]$ (2.2)
 > $\mathrm{DGsetup}\left(\mathrm{AD1}\right)$
 ${\mathrm{algebra name: alg1}}$ (2.3)

This algebra is non-commutative but associative.

 alg1 > $\mathrm{Query}\left(\mathrm{alg1},"Commutative"\right)$
 ${\mathrm{false}}$ (2.4)
 alg1 > $\mathrm{Query}\left(\mathrm{alg1},"Associative"\right)$
 ${\mathrm{true}}$ (2.5)

Example 2.

Again let be the standard basis for the vector space of matrices and but now let be the Jordan product (

 alg1 > $A≔\left[\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[1,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[0,1\right]\right]\right)\right]$
 alg1 > $\mathrm{μ3}≔\mathrm{JordanProduct}$
 ${\mathrm{μ3}}{:=}{\mathrm{DifferentialGeometry:-LieAlgebras:-JordanProduct}}$ (2.6)

The structure equations for this Jordan algebra are:

 > $\mathrm{AD2}≔\mathrm{AlgebraData}\left(A,\mathrm{μ3},\mathrm{alg2}\right)$
 ${\mathrm{AD2}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}\frac{{1}}{{2}}{}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}\frac{{1}}{{2}}{}{\mathrm{e3}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}\frac{{1}}{{2}}{}{\mathrm{e2}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}\frac{{1}}{{2}}{}{\mathrm{e1}}{+}\frac{{1}}{{2}}{}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}\frac{{1}}{{2}}{}{\mathrm{e2}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}\frac{{1}}{{2}}{}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}\frac{{1}}{{2}}{}{\mathrm{e1}}{+}\frac{{1}}{{2}}{}{\mathrm{e4}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}\frac{{1}}{{2}}{}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}\frac{{1}}{{2}}{}{\mathrm{e2}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}\frac{{1}}{{2}}{}{\mathrm{e3}}{,}{{\mathrm{e4}}}^{{2}}{=}{\mathrm{e4}}\right]$ (2.7)
 > $\mathrm{DGsetup}\left(\mathrm{AD2}\right)$
 ${\mathrm{algebra name: alg2}}$ (2.8)

This time the algebra is commutative but not associative.

 alg2 > $\mathrm{Query}\left(\mathrm{alg2},"Commutative"\right)$
 ${\mathrm{true}}$ (2.9)
 alg2 > $\mathrm{Query}\left(\mathrm{alg2},"Associative"\right)$
 ${\mathrm{false}}$ (2.10)

Example 3.

Now let be the matrix commutator.

 alg1 > $A≔\left[\mathrm{Matrix}\left(\left[\left[1,0\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,1\right],\left[0,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[1,0\right]\right]\right),\mathrm{Matrix}\left(\left[\left[0,0\right],\left[0,1\right]\right]\right)\right]$
 alg1 > $\mathrm{μ3}≔\left(a,b\right)→\mathrm{.}\left(a,b\right)-\mathrm{.}\left(b,a\right)$
 ${\mathrm{μ3}}{:=}\left({a}{,}{b}\right){→}{a}{.}{b}{-}{b}{.}{a}$ (2.11)

Now the structure equations

 > $\mathrm{AD3}≔\mathrm{AlgebraData}\left(A,\mathrm{μ3},\mathrm{alg3}\right)$
 ${\mathrm{AD3}}{:=}\left[{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{-}{\mathrm{e2}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e1}}{-}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e1}}{+}{\mathrm{e4}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e2}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}\right]$ (2.12)

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

 alg2 > $\mathrm{LD}≔\mathrm{LieAlgebraData}\left(A,\mathrm{Liealg3}\right)$
 ${\mathrm{LD}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{-}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e3}}\right]$ (2.13)

Note that in (2.12) both products ${e}_{1}\cdot {e}_{3}$ and must be specified but in (2.13) only the product is calculated and stored.

Example 4.

In this example we shall calculate the structure equations for the Jordan algebra of matrices over the quaternions $\mathrm{ℚ}$. First we create the quaternions with AlgebraLibraryData.

 alg2 > $\mathrm{AD4a}≔\mathrm{AlgebraLibraryData}\left("Quaternions",\mathrm{Qn}\right)$
 ${\mathrm{AD4a}}{:=}\left[{{\mathrm{e1}}}^{{2}}{=}{\mathrm{e1}}{,}{\mathrm{e1}}{.}{\mathrm{e2}}{=}{\mathrm{e2}}{,}{\mathrm{e1}}{.}{\mathrm{e3}}{=}{\mathrm{e3}}{,}{\mathrm{e1}}{.}{\mathrm{e4}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e1}}{=}{\mathrm{e2}}{,}{{\mathrm{e2}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e2}}{.}{\mathrm{e3}}{=}{\mathrm{e4}}{,}{\mathrm{e2}}{.}{\mathrm{e4}}{=}{-}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e1}}{=}{\mathrm{e3}}{,}{\mathrm{e3}}{.}{\mathrm{e2}}{=}{-}{\mathrm{e4}}{,}{{\mathrm{e3}}}^{{2}}{=}{-}{\mathrm{e1}}{,}{\mathrm{e3}}{.}{\mathrm{e4}}{=}{\mathrm{e2}}{,}{\mathrm{e4}}{.}{\mathrm{e1}}{=}{\mathrm{e4}}{,}{\mathrm{e4}}{.}{\mathrm{e2}}{=}{\mathrm{e3}}{,}{\mathrm{e4}}{.}{\mathrm{e3}}{=}{-}{\mathrm{e2}}{,}{{\mathrm{e4}}}^{{2}}{=}{-}{\mathrm{e1}}\right]$ (2.14)
 alg2 > $\mathrm{DGsetup}\left(\mathrm{AD4a}\right)$
 ${\mathrm{algebra name: Qn}}$ (2.15)

We use JordanMatrices to generate a basis for the space of Hermitian matrices with entries in $\mathrm{ℚn}.$

 Qn > $J≔\mathrm{JordanMatrices}\left(3,\mathrm{Qn}\right)$

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 > $\mathrm{AD4}≔\mathrm{AlgebraData}\left(J,\mathrm{JordanProduct},\mathrm{J3Qn}\right):$
 Qn > $\mathrm{DGsetup}\left(\mathrm{AD4}\right)$
 ${\mathrm{algebra name: J3Qn}}$ (2.16)
 J3Qn > $\mathrm{interface}\left(\mathrm{rtablesize}=17\right)$
 ${10}$ (2.17)
 Qn > $\mathrm{MultiplicationTable}\left("AlgebraTable"\right)$

 See Also