Epistemology m a t h 4 w i s d o m - g m a i l +370 607 27 665 My work is in the Public Domain for all to share freely. 读物 书 影片 维基百科 Introduction E9F5FC Questions FFFFC0 Software 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 {$x+ye_1\Leftrightarrow \begin{pmatrix} x & -y \\ y & x \end{pmatrix}$} {$\mathbb{H}$} {$a + be_1e_2 + ce_1 + de_2\Leftrightarrow \begin{pmatrix} a & -b & -c & -d \\ b & a & d & c \\ c & -d & a & -b \\ d & -c & b & a \end{pmatrix}$} {$\mathbb{H}\oplus\mathbb{H}$} {$a + be_1e_2 + ce_2e_3 + de_1e_3 + ee_1 + fe_2 + ge_3 + he_1e_2e_3 =$} {$a + bk + c(-j\omega) + d(i\omega) + ei + fj + g(k\omega) + h(-\omega)$} {$\Leftrightarrow \begin{pmatrix} a & b & c & d & e & f & g & h \\ -b & a & d & -c & f & -e & -h & g \\ -c & -d & a & -b & -g & h & e & -f \\ -d & c & b & a & -h & -g & f & e \\ -e & -f & g & -h & a & -b & d & c \\ -f & e & h & g & b & a & c & -d \\ -g & -h & -e & -f & -d & -c & a & b \\ h & -g & f & -e & -c & d & -b & a \end{pmatrix}$}
This page was last changed on April 30, 2024, at 11:34 PM