\documentclass[landscape,fleqn]{seminar}
\usepackage{czech}
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\usepackage{pause}
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\begin{document}
\centerslidesfalse
\slideframe{none}
\setlength{\slidewidth}{24.7cm}
\setlength{\slideheight}{16cm}
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\parindent=0pt
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\begin{slide}
\begin{center}
{\Huge\bfseries\sffamily\color{head} \MakeUppercase{Cliffordova algebra}}
\end{center}
\vspace*{2cm}
\emph{1843 ---} William Rowan Hamilton\pause

\emph{1844 ---} Hermann G\"unther Grassmann\pause

\emph{1845 ---} Arthur Cayley\pause

\emph{1878 ---} William Kingdon Clifford\pause

\emph{1925 ---} Hermann Klaus Hugo Weyl\pause

\emph{1926 ---} Elie Joseph Cartan\pause

\emph{1954 ---} Claude Chevalley
\end{slide}
\begin{slide}
\section{Komplexn� ��sla, kvaterniony, oktoniony}
\begin{description}
\item[\emph{$A$} ---] algebra s jedni�kou $1$ nad polem $\RR$\pause
\item[\emph{sdru�en�}---] operace $a \zob \bar{a}$ tj. $\bar{\bar{a}}=a$,
$\overline{ab} = \bar{b} \bar{a}$\pause
\item[\emph{zdvojen�}---] $A\oplus A = \{ (a,b) \sd a,b\in A\}$\pause
\begin{description}
\item{n�soben� ---} $(a,b)(c,d) = (ac - \bar{d}b, b\bar{c} + da)$, jedni�ka
$(1,0)$\pause
\item{identifikace ---} ozna�me $e=(0,1)$, pak $(a,b) = a+be$\pause
\item{sdru�en� ---} $\overline{(a,b)} =(\bar{a},-b)$, plat� $e
a=\bar{a}e$, $a(be)=(ba)e$\pause
\end{description}
\item{\emph{konstrukce t�les} ---}\pause
$
\quad 
\xymatrix@!=1.1cm{
\RR \ar[r]^{\text{zdvojen�}}\pause& \CC \ar[r]^{\text{zdvojen�}}\pause& \HH
\ar[r]^{\text{zdvojen�}}\pause& \OO
}
$\pause
\item{\emph{kvaterniony} ---} nekomutativn�, $h=a+be$, $a=\alpha+\beta i$,
$b=\gamma+\delta i$\pause
\begin{description}
\item{vnit�n� sou�in ---} $(g,h)=\tfrac{1}{2}(\bar{g}h+\bar{h}g)$\pause
\item{inverzn� prvek ---} $h^{-1} = \bar{h}/(h,h)$\pause
\end{description}
\item{\emph{oktoniony} ---} nekomutativn�, neasociativn� zdvojen�
$\HH$
\end{description}
\end{slide}
\begin{slide}
\section{V�ce o oktonionech}
B�zi oktonion� nad $\RR$ ozna��me $1, e_1, \ldots, e_7$.\pause
\opengraphsfile{octo}
\begin{center}
\begin{mfpic}[45]{-1}{1}{0}{2}
\pen{1pt}
\tlabel[cc](1.2,0){$e_5$}
\tlabel[cc](-1.2,0){$e_6$}
\tlabel[cc](0,1.932){$e_7$}
\tlabel[cc](-0.7,0.866){$e_1$}
\tlabel[cc](0.7,0.866){$e_2$}
\tlabel[cc](0,-0.2){$e_3$}
\tlabel[cc](0.2,0.577){$e_4$}
\arrow\lines{(0,0),(-0.5,0)}
\arrow\lines{(-1,0),(-0.25,1.299)}
\arrow\lines{(0,1.732),(0.75,0.433)}
\arrow\lines{(0.5,0.866),(-0.5,0.288)}
\arrow\lines{(-0.5,0.866),(0.5,0.288)}
\arrow\lines{(0,0),(0,1.155)}
\arrow\lines{(0,0.866),(0.25,0.866)}
\arrow\lines{(0.5,0.866),(0.167,0.289)}
\arrow\lines{(0,0),(-0.333,0.577)}
\lines{(1,0),(-1,0)}
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\lines{(-0.5,0.866),(0.5,0.866)}
\lines{(0.5,0.866),(0,0)}
\lines{(0,0),(-0.5,0.866)}
\end{mfpic}\pause
\end{center}
\closegraphsfile
�ist� imagin�rn� kvaterniony resp. oktoniony lze vyu��t k definici
vektorov�ho sou�inu v $\RR^3$ resp. $\RR^7$.\pause
$$
a \times b = \frac{1}{2}(ab-ba), \quad\pause a\times b \perp a,\quad\pause a\times
b\perp b, \quad \pause |a\times b|^2 = |a|^2|b|^2-(a,b)^2
$$
\end{slide}
\begin{slide}
\section{Z�kladn� pojmy}
\emph{Definice:} Nech� $V$ je vektorov� prostor nad $\KK$, $g\colon V\zob
\KK$ kvadratick� forma. Zkonstruujme $T(V)$ tenzorovou algebru $V$ a $I$ ide�l
generovan� $v\otimes v+g(v)$ pro $v\in V$. \emph{Cliffordova algebra $C(V,g)$} je definov�na
jako $T(V)/I$.\pause

\emph{Definice$'$:} Asociativn� algebra s jedni�kou nad polem $\KK$
s~gener�tory $1$, $V$ a spl�uj�c� $v^2=-g(v)$ pro $v\in V$.\pause

Kvadratickou formu $g$ nad poli $\RR$ a $\CC$ lze uv�st do
diagon�ln�ho tvaru.
$$
\RR\colon g \sim
\left(
\begin{smallmatrix}
\pm 1 & \\
      & \ddots & \\
      &        & \pm 1 \\
      &        &       &  0 \\
      &        &       &      & \ddots \\
      &        &       &      &        & 0
\end{smallmatrix}\right), \qquad\pause
\CC\colon g \sim \left(\begin{smallmatrix}
1  \\
 & \ddots \\
 &        & 1 \\
 &        &   & 0 \\
 &        &   &   & \ddots \\
 &        &   &   &        & 0
\end{smallmatrix}\right)\pause
$$

Z ka�d�ho zobrazen� $f\colon V\zob B$, kde $B$ je asociativn� algebra
s jedni�kou a $f(v)f(v)=-g(v)$, 
lze jednozna�n� zkonstruovat $f\colon C(V,g) \zob B$.
\end{slide}
\begin{slide}
\section{Cliffordova algebra je superalgebra}
\emph{$\ZZ_2$-gradace:} Specieln� vezm�me inverzi ${}^* \colon v\zob -v$. 
Z�ejm� plat� ${{}^*}^2=\ide$ a
$$C(V,g)=C^0(V,g)\oplus C^1(V,g).$$
Tak� plat� $(AB)^*=A^*B^*$ tak�e 
$
C^i(V,g)C^j(V,g) \subset C^{i+j}(V,g).\pause
$

\emph{Tenzorov� sou�in} algeber $A\otimes B$.
$$
(a\otimes b)(a'\otimes b') = (aa')\otimes (bb').\pause
$$

\emph{Jin� tenzorov� sou�in} pro $\ZZ_2$ graduovan� algebry
$A\hat{\otimes} B$
$$
(a\hat{\otimes} b)(a'\hat{\otimes} b')
=(-1)^{|b||a'|}(aa')\hat{\otimes} (bb').\pause
$$
\emph{V�ta:} Nech� $V=V_1\oplus V_2$ je ortogon�ln� rozklad. Pak plat�
$$
C(V,g) \iso C(V_1,g_1) \hat{\otimes} C(V_2,g_2).
$$ 
\end{slide}
\begin{slide}
\section{P��klady re�ln�ch Cliffordov�ch algeber}
$e_1,\ldots, e_n$ je b�ze vektorov�ho prostoru $V$\pause
\begin{description}
\item[\emph{$C(\RR,1,0)$}] $\iso \CC$: $1\mapsto 1$ a $e_1\mapsto i$\pause
\item[\emph{$C(\RR,0,1)$}] $\iso \RR\oplus\RR$: $1\mapsto (1,1)$ a
$e_1\mapsto (1,-1)$\pause
\item[\emph{$C(\RR^2,2,0)$}] $\iso \HH$:  $1\mapsto 1$, $e_1\mapsto i$
a $e_2\mapsto j$\pause
\item[\emph{$C(\RR^2,0,2)$}] $\iso \Mat_2(\RR)$:
$1\mapsto\left(\begin{smallmatrix}
1 & 0 \\
0 & 1
\end{smallmatrix}\right)$,
$e_1\mapsto\left(\begin{smallmatrix}
1 & 0 \\
0 & -1
\end{smallmatrix}\right)$ a
$e_2\mapsto\left(\begin{smallmatrix}
0 & 1 \\
1 & 0
\end{smallmatrix}\right)$\pause
\item[\emph{$C(\RR^2,1,1)$}] $\iso \Mat_2(\RR)$:
$1\mapsto\left(\begin{smallmatrix}
1 & 0 \\
0 & 1
\end{smallmatrix}\right)$,
$e_1\mapsto\left(\begin{smallmatrix}
1 & 0 \\
0 & -1
\end{smallmatrix}\right)$ a
$e_2\mapsto\left(\begin{smallmatrix}
0 & 1 \\
-1 & 0
\end{smallmatrix}\right)$\pause
\end{description}

\vspace*{-0.1cm}

Na prvn� pohled nap�. vid�me $\CC\otimes\HH \iso \Mat_2(\CC)$.\pause

\vspace*{-0.1cm}

Dal�� Cliffordovy algebry m��eme dostat indukc�. Nap�. plat�
$C(\RR,k+2,0)\iso C(\RR,0,k)\otimes C(\RR,2,0)$~a~$C(\RR,0,k+2)\iso C(\RR,k,0)\otimes C(\RR,0,2)$ 
\newline
konkr�tn� jsou
izomorfizmy v obou p��padech d�ny jako
$e_i\mapsto 1\otimes e_i$, $i=1,2$ a $e_i\mapsto e_{i-2}\otimes
e_1e_2$, $i\ge 3$.\pause

\vspace*{-0.1cm}

Obecn� plat�
$C(\RR,k+8,\ell)=C(\RR,k,\ell+8)=C(\RR,k,\ell)\otimes\Mat_{16}(\RR)$.
\end{slide}
\begin{slide}
\vspace*{-0.5cm}
\hspace*{-1cm}
{\tiny
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%\rowcolor{jemnemodra}
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& \multicolumn{1}{>{\columncolor{jemnecerv}}c}{$\begin{array}{c}\Mat_{16}(\HH)\\ \oplus \\
\Mat_{16}(\HH)\end{array}$}\pauselevel{=16}\pause 
& \multicolumn{1}{>{\columncolor{jemnecerv}}c}{$\Mat_{32}(\HH)$}\pauselevel{=16}\pause 
& \multicolumn{1}{>{\columncolor{jemnecerv}}c}{$\Mat_{64}(\CC)$}\pauselevel{=16}\pause
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\Mat_{128}(\RR)\end{array}$}\pauselevel{=16}\pause 
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%
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%
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%
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\Mat_{32}(\HH)\end{array}$}\pauselevel{=16}\pause \\
%
\multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\Mat_2(\HH)$}\pauselevel{=5}\pause 
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& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\begin{array}{c}\Mat_8(\RR)\\
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& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\Mat_{16}(\CC)$}\pauselevel{=10}\pause 
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\Mat_{16}(\HH)\end{array}$}\pauselevel{=12}\pause 
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%
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& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\Mat_8(\CC)$}\pauselevel{=8}\pause 
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%
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&
\multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\begin{array}{c}\Mat_4(\HH)\\\oplus\\
\Mat_4(\HH)\end{array}$}\pauselevel{=8}\pause 
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%
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\RR\end{array}$}\pauselevel{=2}\pause
& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\Mat_2(\RR)$}\pauselevel{=3}\pause
& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\Mat_2(\CC)$}\pauselevel{=4}\pause
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\oplus\\ \Mat_2(\HH)\end{array}$}\pauselevel{=6}\pause
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& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\Mat_8(\CC)$}\pauselevel{=8}\pause
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& \multicolumn{1}{>{\columncolor{jemnecerv}}c}{$\begin{array}{c}\Mat_{16}(\RR)\\ \oplus\\
\Mat_{16}(\RR)\end{array}$}\pauselevel{=16}\pause \\
%
\multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\RR$}\pauselevel{=1}\pause
& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\CC$}\pauselevel{=2}\pause
& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\HH$}\pauselevel{=3}\pause
& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\begin{array}{c}\HH\\ \oplus\\
\HH\end{array}$}\pauselevel{=4}\pause
& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\Mat_2(\HH)$}\pauselevel{=5}\pause
& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\Mat_4(\CC)$}\pauselevel{=6}\pause
& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\Mat_8(\RR)$}\pauselevel{=7}\pause
& \multicolumn{1}{>{\columncolor{jemnemodra}}c}{$\begin{array}{c}\Mat_8(\RR)\\
\oplus\\ \Mat_8(\RR)\end{array}$}\pauselevel{=8}\pause
&\multicolumn{1}{>{\columncolor{jemnecerv}}c}{$\Mat_{16}(\RR)$}\pauselevel{=16}\pause \\
\end{tabular}
}
\end{slide}
\begin{slide}
\section{Komplexn� Cliffordovy algebry}
Zde je situace podstatn� jednodu���.\pause
\begin{description}
\item[\emph{$C(\CC,1)$}] $\iso \CC\oplus\CC$\pause
\item[\emph{$C(\CC,2)$}] $\iso \Mat_2(C)$\pause
\end{description}
a pro vy��� dimenze plat� $C(\CC,k+2)\iso C(\CC,k)\otimes C(\CC,2)
\iso C(\CC,k)\otimes \Mat_2(\CC)$.\pause

Komplexn� Cliffordovy algebry $C(\CC,2k)$ z�sk�me komplexifikac�
re�ln� Cliffordovy algebry $C(\RR,k,k)$ a komplexn� Cliffordovy
algebry $C(\CC,2k+1)$ komplexifikac� $C(\RR,k,k+1)$.\pause

Komplexn� Cliffordovu algebru $C(\CC,4)$ z�sk�me komplexifikac�
libovoln� re�ln� Cliffordovy algebry $C(\RR,k,\ell)$ pro
$k+\ell=4$.
\end{slide}
\begin{slide}
\section{Reprezentace Cliffordov�ch algeber}
\emph{Definice:} 
Nech� $W^0\oplus W^1$ je $\ZZ_2$-graduovan� modul nad $\RR$ nebo
$\CC$. 
Zobrazen� $f\colon C(V,g) W \zob W$ nazveme 
\emph{Cliffordovou akc�}, jestli�e je sud�, tj.
$$
f(C^0(V,g),W^{0,1}) \subset W^{0,1}, \qquad
f(C^1(V,g),W^{0,1}) \subset W^{1,0}.\pause
$$
Ekvivalentn� m��eme mluvit o \emph{reprezentaci} $C(V,g)\zob\End(W)$.\pause

Zavedeme je�t� operaci \emph{reverze} v Cliffordov� algeb�e. Tenzorov�
algebra $T(V)$ m� involuci danou a homogenn�ch prvc�ch obr�cen�m
po�ad� $v_1\otimes\cdots\otimes v_k\zob v_k\otimes\cdots\otimes v_1$.
Proto�e zachov�v� ide�l $I$, p�enese se i na Cliffordovu algebru
$$
(\quad)^T\colon C(V,g)\zob C(V,g).
$$
\end{slide}
\begin{slide}
\section{Jedna zaj�mav� reprezentace}
Uva�ujme $W=\Lambda V$, pro $v\in V$ definujeme \emph{vn�j�� sou�in}
$$
v\ww \colon \Lambda^k V \ni v_1\ww\ldots\ww v_k \zob v\ww
v_1\ww\ldots\ww v_k \in \Lambda^{k+1} V\pause
$$
a \emph{vnit�n� sou�in}
$$
v\zuz \colon \Lambda^k V \ni v_1\ww\ldots\ww v_k \zob
\sum_{i=1}^n (-1)^{i-1}G(v,v_i) v_1\ww\ldots\ww \hat{v}_i\ww \ldots
\ww v_k \in\Lambda^{k-1} V\pause
$$

Pro $v\in V$ a $\rho\in\Lambda V$ je d�no zobrazen� $f$ jako
$$
f(v,\rho) = \frac{1}{\sqrt{2}} (v\ww\rho - v\zuz\rho), \quad
\text{p�i�em�}
(v\ww - v\zuz )^2\rho = -2g(v)\rho,\pause
$$
tak�e celkov� dostaneme
$f(v)^2=-g(v)$
a $f$ definuje Cliffordovu akci.
\end{slide}
\begin{slide}
\section{Cliffordovy grupy}
\begin{description}
\item[\emph{Cliffordova grupa $\Pin(V,g)$} ---] mno�ina v�ech
invertibiln�ch prvk� $A\in C(V,g)$ takov�, �e $AVA^{-1}\subset V$ spolu s
n�soben�m v $C(V,g)$.\pause
\item[\emph{speci�ln� Cliffordova grupa $\Spin(V,g)$} ---] podgrupa sud�ch prvk�
Cliffordovy grupy\pause
\end{description}

Ka�d� prvek $A$ Cliffordovy grupy indukuje line�rn� transformaci
$f(A)\colon V\ni v \zob AvA^{-1} \in V$,
kter� je ortogon�ln� transformac� $\OO(V,g)$. Nav�c je zobrazen�
$\Pin(V,g)\ni A\zob f(A)\in\OO(V,g)$ reprezentac� $\Pin(V,g)$, kter�
��k�me \emph{vektorov� reprezentace}.\pause

Pokud $v\in \Pin(V,g)\cap V$, potom $g(v)\ne 0$ a $f(v)\colon V \zob V$ 
je \emph{zrcadlen�} $V$ vzhledem k nadrovin� \uv{kolm�} na $v$.\pause

Je-li $\dim V=2k+1$, pak $f(\Pin(V,g))=f(\Spin(V,g))=\SO(V,g)$ a je-li
$\dim V=2k$, pak $f(\Pin(V,g))=\OO(V,g)$ a $f(\Spin(V,g))=\SO(V,g)$.
\end{slide}
\end{document}
