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Clifford Algebras, Z2-Graded Modules

Clifford Modules

A module {$M$} over an algebra {$A$} is simply the module {$M$} over the ring {$A$}.

A left {$A$}-module {$M$} consists of an abelian group {$(M, +)$} and an operation {$· : A × M → M$} such that for all {$a_1, a_2$} in {$A$} and {$x, y$} in {$M$}, we have

• {$a\cdot (x+y)=a\cdot x+a\cdot y$}
• {$(a_1+a_2)\cdot x=a_1\cdot x+a_2\cdot x$}
• {$(a_1a_2)\cdot x=a_1\cdot (a_2\cdot x)$}
• {$ 1\cdot x=x $}

Thus a module is an abelian group (like a vector space) upon which there is an operation (like matrix multiplication).

Consider the simple modules.

{$\mathbb{R}$}

The simple module is a one-dimensional space {$(v)\cong\mathbb{R}$}.

{$(x)(v)=xv$}

{$\mathbb{C}$}

The simple module is a two-dimensional space {$\mathbb{R}^2$} with operation given by

{$x+ye_1\Leftrightarrow \begin{pmatrix} x & -y \\ y & x \end{pmatrix}$}

{$\mathbb{H}$}

{$a + be_1e_2 + ce_1 + de_2\Leftrightarrow \begin{pmatrix} a & -b & -c & -d \\ b & a & d & c \\ c & -d & a & -b \\ d & -c & b & a \end{pmatrix}$}

{$\mathbb{H}\oplus\mathbb{H}$}

{$a + be_1e_2 + ce_2e_3 + de_1e_3 + ee_1 + fe_2 + ge_3 + he_1e_2e_3 = $}

{$a + bk + c(-j\omega) + d(i\omega) + ei + fj + g(k\omega) + h(-\omega)$}

{$\Leftrightarrow \begin{pmatrix} a & b & c & d & e & f & g & h \\ -b & a & d & -c & f & -e & -h & g \\ -c & -d & a & -b & -g & h & e & -f \\ -d & c & b & a & -h & -g & f & e \\ -e & -f & g & -h & a & -b & d & c \\ -f & e & h & g & b & a & c & -d \\ -g & -h & -e & -f & -d & -c & a & b \\ h & -g & f & -e & -c & d & -b & a \end{pmatrix}$}