AlgebraLibraryData - Maple Help

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

 • The command AlgebraLibraryData retrieves the structure equations for any of the following real algebras: the real numbers $\mathrm{ℝ}$, the complex numbers the quaternions the octonions the Clifford algebras on ${\mathrm{ℝ}}^{n}$ with respect to the quadratic form $Q$, and the Jordan algebras  for small values of $n$.
 • The keyword argument type ="Split" may be applied to the algebras  to obtain their split forms. The argument type ="Split" can be applied to  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 the structure equations have been stored in Maple and are available without computation: for for for for More generally, Jordan algebras can be created using the command JordanMatrices, JordanProduct, and AlgebraData.

Examples

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

Example 1.

We define the quaternions and the split quaternions and compare their multiplication tables:

 > $\mathrm{AD1a}≔\mathrm{AlgebraLibraryData}\left("Quaternions",H\right):$
 > $\mathrm{DGsetup}\left(\mathrm{AD1a},'\left[e,i,j,k\right]','\left[\mathrm{\omega }\right]'\right)$
 ${\mathrm{algebra name: H}}$ (2.1)

Here are the split quaternions.

 H > $\mathrm{AD1b}≔\mathrm{AlgebraLibraryData}\left("Quaternions",\mathrm{Hs},\mathrm{type}="Split"\right):$
 H > $\mathrm{DGsetup}\left(\mathrm{AD1b},'\left[e,i,j,k\right]',\left[\mathrm{\omega }\right]\right)$
 ${\mathrm{algebra name: Hs}}$ (2.2)

We see that the off-diagonal products in the multiplication tables are the same. For the quaternions while for the split quaternions .

 Hs > $\mathrm{MultiplicationTable}\left(H,"AlgebraTable"\right),\mathrm{MultiplicationTable}\left(\mathrm{Hs},"AlgebraTable"\right)$
 $\left[\begin{array}{cccccc}{}& {|}& {e}& {i}& {j}& {k}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {e}& {|}& {e}& {i}& {j}& {k}\\ {i}& {|}& {i}& {-}{e}& {k}& {-}{j}\\ {j}& {|}& {j}& {-}{k}& {-}{e}& {i}\\ {k}& {|}& {k}& {j}& {-}{i}& {-}{e}\end{array}\right]{,}\left[\begin{array}{cccccc}{}& {|}& {e}& {i}& {j}& {k}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {e}& {|}& {e}& {i}& {j}& {k}\\ {i}& {|}& {i}& {-}{e}& {k}& {-}{j}\\ {j}& {|}& {j}& {-}{k}& {e}& {-}{i}\\ {k}& {|}& {k}& {j}& {i}& {e}\end{array}\right]$ (2.3)

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

where the are the components of a 3-form determined by 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 $\mathrm{γ}$.

 > $\mathrm{AD2a}≔\mathrm{AlgebraLibraryData}\left("Octonions",\mathrm{O1}\right):$
 > $\mathrm{DGsetup}\left(\mathrm{AD2a},'\left[\mathrm{e0},\mathrm{e1},\mathrm{e2},\mathrm{e3},\mathrm{e4},\mathrm{e5},\mathrm{e6},\mathrm{e7}\right]',\left[\mathrm{\omega }\right]\right)$
 ${\mathrm{algebra name: O1}}$ (2.4)
 O1 > $\mathrm{MultiplicationTable}\left(\mathrm{O1},"AlgebraTable"\right)$
 $\left[\begin{array}{cccccccccc}{}& {|}& {\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}& {\mathrm{e6}}& {\mathrm{e7}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e0}}& {|}& {\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}& {\mathrm{e6}}& {\mathrm{e7}}\\ {\mathrm{e1}}& {|}& {\mathrm{e1}}& {-}{\mathrm{e0}}& {\mathrm{e3}}& {-}{\mathrm{e2}}& {\mathrm{e5}}& {-}{\mathrm{e4}}& {-}{\mathrm{e7}}& {\mathrm{e6}}\\ {\mathrm{e2}}& {|}& {\mathrm{e2}}& {-}{\mathrm{e3}}& {-}{\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e6}}& {\mathrm{e7}}& {-}{\mathrm{e4}}& {-}{\mathrm{e5}}\\ {\mathrm{e3}}& {|}& {\mathrm{e3}}& {\mathrm{e2}}& {-}{\mathrm{e1}}& {-}{\mathrm{e0}}& {\mathrm{e7}}& {-}{\mathrm{e6}}& {\mathrm{e5}}& {-}{\mathrm{e4}}\\ {\mathrm{e4}}& {|}& {\mathrm{e4}}& {-}{\mathrm{e5}}& {-}{\mathrm{e6}}& {-}{\mathrm{e7}}& {-}{\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}\\ {\mathrm{e5}}& {|}& {\mathrm{e5}}& {\mathrm{e4}}& {-}{\mathrm{e7}}& {\mathrm{e6}}& {-}{\mathrm{e1}}& {-}{\mathrm{e0}}& {-}{\mathrm{e3}}& {\mathrm{e2}}\\ {\mathrm{e6}}& {|}& {\mathrm{e6}}& {\mathrm{e7}}& {\mathrm{e4}}& {-}{\mathrm{e5}}& {-}{\mathrm{e2}}& {\mathrm{e3}}& {-}{\mathrm{e0}}& {-}{\mathrm{e1}}\\ {\mathrm{e7}}& {|}& {\mathrm{e7}}& {-}{\mathrm{e6}}& {\mathrm{e5}}& {\mathrm{e4}}& {-}{\mathrm{e3}}& {-}{\mathrm{e2}}& {\mathrm{e1}}& {-}{\mathrm{e0}}\end{array}\right]$ (2.5)

For the second version, the non-zero components of the 3-form $\mathrm{γ}$ are

 O1 > $\mathrm{AD2b}≔\mathrm{AlgebraLibraryData}\left("Octonions",\mathrm{O2},\mathrm{version}=2\right):$
 O1 > $\mathrm{DGsetup}\left(\mathrm{AD2b},'\left[\mathrm{e0},\mathrm{e1},\mathrm{e2},\mathrm{e3},\mathrm{e4},\mathrm{e5},\mathrm{e6},\mathrm{e7}\right]',\left[\mathrm{\omega }\right]\right)$
 ${\mathrm{algebra name: O2}}$ (2.6)
 O2 > $\mathrm{MultiplicationTable}\left(\mathrm{O2},"AlgebraTable"\right)$
 $\left[\begin{array}{cccccccccc}{}& {|}& {\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}& {\mathrm{e6}}& {\mathrm{e7}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e0}}& {|}& {\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}& {\mathrm{e6}}& {\mathrm{e7}}\\ {\mathrm{e1}}& {|}& {\mathrm{e1}}& {-}{\mathrm{e0}}& {\mathrm{e4}}& {\mathrm{e7}}& {-}{\mathrm{e2}}& {\mathrm{e6}}& {-}{\mathrm{e5}}& {-}{\mathrm{e3}}\\ {\mathrm{e2}}& {|}& {\mathrm{e2}}& {-}{\mathrm{e4}}& {-}{\mathrm{e0}}& {\mathrm{e5}}& {\mathrm{e1}}& {-}{\mathrm{e3}}& {\mathrm{e7}}& {-}{\mathrm{e6}}\\ {\mathrm{e3}}& {|}& {\mathrm{e3}}& {-}{\mathrm{e7}}& {-}{\mathrm{e5}}& {-}{\mathrm{e0}}& {\mathrm{e6}}& {\mathrm{e2}}& {-}{\mathrm{e4}}& {\mathrm{e1}}\\ {\mathrm{e4}}& {|}& {\mathrm{e4}}& {\mathrm{e2}}& {-}{\mathrm{e1}}& {-}{\mathrm{e6}}& {-}{\mathrm{e0}}& {\mathrm{e7}}& {\mathrm{e3}}& {-}{\mathrm{e5}}\\ {\mathrm{e5}}& {|}& {\mathrm{e5}}& {-}{\mathrm{e6}}& {\mathrm{e3}}& {-}{\mathrm{e2}}& {-}{\mathrm{e7}}& {-}{\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e4}}\\ {\mathrm{e6}}& {|}& {\mathrm{e6}}& {\mathrm{e5}}& {-}{\mathrm{e7}}& {\mathrm{e4}}& {-}{\mathrm{e3}}& {-}{\mathrm{e1}}& {-}{\mathrm{e0}}& {\mathrm{e2}}\\ {\mathrm{e7}}& {|}& {\mathrm{e7}}& {\mathrm{e3}}& {\mathrm{e6}}& {-}{\mathrm{e1}}& {\mathrm{e5}}& {-}{\mathrm{e4}}& {-}{\mathrm{e2}}& {-}{\mathrm{e0}}\end{array}\right]$ (2.7)

Both versions have split counterparts.

Example 3.

Let be a vector space with basis and let be a non-degenerate quadratic form on The Clifford algebra is the algebra generated by products of the vectors subject to the multiplication rules

.

A vector space basis for the Clifford algebra is the identity and the ordered products , where . The dimension of is The default choice for the quadratic form $Q$ is given by the identity matrix ${I}_{n}$.

We first display the multiplication tables for

 O2 > $\mathrm{AD3a}≔\mathrm{AlgebraLibraryData}\left("Clifford\left(3\right)",\mathrm{Cl3}\right):$
 O2 > $\mathrm{DGsetup}\left(\mathrm{AD3a},'\left[\mathrm{e0},\mathrm{e1},\mathrm{e2},\mathrm{e3},\mathrm{e12},\mathrm{e13},\mathrm{e23},\mathrm{e123}\right]','\left[\mathrm{\omega }\right]'\right)$
 ${\mathrm{algebra name: Cl3}}$ (2.8)
 O2 > $\mathrm{MultiplicationTable}\left(\mathrm{Cl3},"AlgebraTable"\right)$
 $\left[\begin{array}{cccccccccc}{}& {|}& {\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e12}}& {\mathrm{e13}}& {\mathrm{e23}}& {\mathrm{e123}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e0}}& {|}& {\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e12}}& {\mathrm{e13}}& {\mathrm{e23}}& {\mathrm{e123}}\\ {\mathrm{e1}}& {|}& {\mathrm{e1}}& {-}{\mathrm{e0}}& {\mathrm{e12}}& {\mathrm{e13}}& {-}{\mathrm{e2}}& {-}{\mathrm{e3}}& {\mathrm{e123}}& {-}{\mathrm{e23}}\\ {\mathrm{e2}}& {|}& {\mathrm{e2}}& {-}{\mathrm{e12}}& {-}{\mathrm{e0}}& {\mathrm{e23}}& {\mathrm{e1}}& {-}{\mathrm{e123}}& {-}{\mathrm{e3}}& {\mathrm{e13}}\\ {\mathrm{e3}}& {|}& {\mathrm{e3}}& {-}{\mathrm{e13}}& {-}{\mathrm{e23}}& {-}{\mathrm{e0}}& {\mathrm{e123}}& {\mathrm{e1}}& {\mathrm{e2}}& {-}{\mathrm{e12}}\\ {\mathrm{e12}}& {|}& {\mathrm{e12}}& {\mathrm{e2}}& {-}{\mathrm{e1}}& {\mathrm{e123}}& {-}{\mathrm{e0}}& {\mathrm{e23}}& {-}{\mathrm{e13}}& {-}{\mathrm{e3}}\\ {\mathrm{e13}}& {|}& {\mathrm{e13}}& {\mathrm{e3}}& {-}{\mathrm{e123}}& {-}{\mathrm{e1}}& {-}{\mathrm{e23}}& {-}{\mathrm{e0}}& {\mathrm{e12}}& {\mathrm{e2}}\\ {\mathrm{e23}}& {|}& {\mathrm{e23}}& {\mathrm{e123}}& {\mathrm{e3}}& {-}{\mathrm{e2}}& {\mathrm{e13}}& {-}{\mathrm{e12}}& {-}{\mathrm{e0}}& {-}{\mathrm{e1}}\\ {\mathrm{e123}}& {|}& {\mathrm{e123}}& {-}{\mathrm{e23}}& {\mathrm{e13}}& {-}{\mathrm{e12}}& {-}{\mathrm{e3}}& {\mathrm{e2}}& {-}{\mathrm{e1}}& {\mathrm{e0}}\end{array}\right]$ (2.9)

We note that the Clifford algebras are always associative.

Here is the multiplication table for .

 Cl3 > $\mathrm{I12}≔\mathrm{Matrix}\left(\left[\left[1,0,0\right],\left[0,-1,0\right],\left[0,0,-1\right]\right]\right)$
 ${\mathrm{I12}}{:=}\left[\begin{array}{rrr}{1}& {0}& {0}\\ {0}& {-}{1}& {0}\\ {0}& {0}& {-}{1}\end{array}\right]$ (2.10)
 O2 > $\mathrm{AD3b}≔\mathrm{AlgebraLibraryData}\left("Clifford\left(3\right)",\mathrm{Cl3Q},\mathrm{quadraticform}=\mathrm{I12}\right):$
 O2 > $\mathrm{DGsetup}\left(\mathrm{AD3b},'\left[\mathrm{e0},\mathrm{e1},\mathrm{e2},\mathrm{e3},\mathrm{e12},\mathrm{e13},\mathrm{e23},\mathrm{e123}\right]','\left[\mathrm{\omega }\right]'\right)$
 ${\mathrm{algebra name: Cl3Q}}$ (2.11)
 O2 > $\mathrm{MultiplicationTable}\left(\mathrm{Cl3Q},"AlgebraTable"\right)$
 $\left[\begin{array}{cccccccccc}{}& {|}& {\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e12}}& {\mathrm{e13}}& {\mathrm{e23}}& {\mathrm{e123}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e0}}& {|}& {\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e12}}& {\mathrm{e13}}& {\mathrm{e23}}& {\mathrm{e123}}\\ {\mathrm{e1}}& {|}& {\mathrm{e1}}& {-}{\mathrm{e0}}& {\mathrm{e12}}& {\mathrm{e13}}& {-}{\mathrm{e2}}& {-}{\mathrm{e3}}& {\mathrm{e123}}& {-}{\mathrm{e23}}\\ {\mathrm{e2}}& {|}& {\mathrm{e2}}& {-}{\mathrm{e12}}& {\mathrm{e0}}& {\mathrm{e23}}& {-}{\mathrm{e1}}& {-}{\mathrm{e123}}& {\mathrm{e3}}& {-}{\mathrm{e13}}\\ {\mathrm{e3}}& {|}& {\mathrm{e3}}& {-}{\mathrm{e13}}& {-}{\mathrm{e23}}& {\mathrm{e0}}& {\mathrm{e123}}& {-}{\mathrm{e1}}& {-}{\mathrm{e2}}& {\mathrm{e12}}\\ {\mathrm{e12}}& {|}& {\mathrm{e12}}& {\mathrm{e2}}& {\mathrm{e1}}& {\mathrm{e123}}& {\mathrm{e0}}& {\mathrm{e23}}& {\mathrm{e13}}& {\mathrm{e3}}\\ {\mathrm{e13}}& {|}& {\mathrm{e13}}& {\mathrm{e3}}& {-}{\mathrm{e123}}& {\mathrm{e1}}& {-}{\mathrm{e23}}& {\mathrm{e0}}& {-}{\mathrm{e12}}& {-}{\mathrm{e2}}\\ {\mathrm{e23}}& {|}& {\mathrm{e23}}& {\mathrm{e123}}& {-}{\mathrm{e3}}& {\mathrm{e2}}& {-}{\mathrm{e13}}& {\mathrm{e12}}& {-}{\mathrm{e0}}& {-}{\mathrm{e1}}\\ {\mathrm{e123}}& {|}& {\mathrm{e123}}& {-}{\mathrm{e23}}& {-}{\mathrm{e13}}& {\mathrm{e12}}& {\mathrm{e3}}& {-}{\mathrm{e2}}& {-}{\mathrm{e1}}& {\mathrm{e0}}\end{array}\right]$ (2.12)

Finally, we remark that the quaternions $\mathrm{ℍ}$ and the Clifford algebra are isomorphic.

 O2 > $\mathrm{AD3b}≔\mathrm{AlgebraLibraryData}\left("Clifford\left(2\right)",\mathrm{Cl2}\right):$
 O2 > $\mathrm{DGsetup}\left(\mathrm{AD3b},'\left[\mathrm{e0},\mathrm{e1},\mathrm{e2},\mathrm{e3}\right]','\left[\mathrm{\omega }\right]'\right)$
 ${\mathrm{algebra name: Cl2}}$ (2.13)
 O2 > $\mathrm{MultiplicationTable}\left(H,"AlgebraTable"\right),\mathrm{MultiplicationTable}\left(\mathrm{Cl2},"AlgebraTable"\right)$
 $\left[\begin{array}{cccccc}{}& {|}& {e}& {i}& {j}& {k}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {e}& {|}& {e}& {i}& {j}& {k}\\ {i}& {|}& {i}& {-}{e}& {k}& {-}{j}\\ {j}& {|}& {j}& {-}{k}& {-}{e}& {i}\\ {k}& {|}& {k}& {j}& {-}{i}& {-}{e}\end{array}\right]{,}\left[\begin{array}{cccccc}{}& {|}& {\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e0}}& {|}& {\mathrm{e0}}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}\\ {\mathrm{e1}}& {|}& {\mathrm{e1}}& {-}{\mathrm{e0}}& {\mathrm{e3}}& {-}{\mathrm{e2}}\\ {\mathrm{e2}}& {|}& {\mathrm{e2}}& {-}{\mathrm{e3}}& {-}{\mathrm{e0}}& {\mathrm{e1}}\\ {\mathrm{e3}}& {|}& {\mathrm{e3}}& {\mathrm{e2}}& {-}{\mathrm{e1}}& {-}{\mathrm{e0}}\end{array}\right]$ (2.14)

Example 4.

Here are the structure equations for the Jordan algebra . This is the algebra of 2 × 2 Hermitian matrices with quaternionic entries and the product

 Cl3 > $\mathrm{AD4}≔\mathrm{AlgebraLibraryData}\left("Jordan\left(2, Quaternions\right)",\mathrm{J2H}\right):$
 Cl2 > $\mathrm{DGsetup}\left(\mathrm{AD4}\right)$
 ${\mathrm{algebra name: J2H}}$ (2.15)
 J2 > $\mathrm{MultiplicationTable}\left(\mathrm{J2H},"AlgebraTable"\right)$
 $\left[\begin{array}{cccccccc}{}& {|}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}& {\mathrm{e6}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {|}& {\mathrm{e1}}& {0}{}{\mathrm{e1}}& \frac{{1}}{{2}}{}{\mathrm{e3}}& \frac{{1}}{{2}}{}{\mathrm{e4}}& \frac{{1}}{{2}}{}{\mathrm{e5}}& \frac{{1}}{{2}}{}{\mathrm{e6}}\\ {\mathrm{e2}}& {|}& {0}{}{\mathrm{e1}}& {\mathrm{e2}}& \frac{{1}}{{2}}{}{\mathrm{e3}}& \frac{{1}}{{2}}{}{\mathrm{e4}}& \frac{{1}}{{2}}{}{\mathrm{e5}}& \frac{{1}}{{2}}{}{\mathrm{e6}}\\ {\mathrm{e3}}& {|}& \frac{{1}}{{2}}{}{\mathrm{e3}}& \frac{{1}}{{2}}{}{\mathrm{e3}}& {\mathrm{e1}}{+}{\mathrm{e2}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}\\ {\mathrm{e4}}& {|}& \frac{{1}}{{2}}{}{\mathrm{e4}}& \frac{{1}}{{2}}{}{\mathrm{e4}}& {0}{}{\mathrm{e1}}& {\mathrm{e1}}{+}{\mathrm{e2}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}\\ {\mathrm{e5}}& {|}& \frac{{1}}{{2}}{}{\mathrm{e5}}& \frac{{1}}{{2}}{}{\mathrm{e5}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}& {\mathrm{e1}}{+}{\mathrm{e2}}& {0}{}{\mathrm{e1}}\\ {\mathrm{e6}}& {|}& \frac{{1}}{{2}}{}{\mathrm{e6}}& \frac{{1}}{{2}}{}{\mathrm{e6}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}& {0}{}{\mathrm{e1}}& {\mathrm{e1}}{+}{\mathrm{e2}}\end{array}\right]$ (2.16)