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dimensione + finita $n$ e per $\varphi$ un suo prodotto, hermitiano o scalare + dipendentemente dal contesto. + \end{note} + + \begin{definition} [azione di un gruppo] + Sia $G$ un gruppo e sia $X$ un insieme. Un'\textbf{azione} di $G$ + su $X$ (a sinistra) è un'applicazione $G \times V \to X$ tale + che $(g, x) \mapsto g.x$ e che: + + \begin{enumerate}[(i)] + \item $e.x = x$ $\forall x \in X$, + \item $g.(h.x) = (gh).x$ $\forall x \in X$, $\forall g$, $h \in G$. + \end{enumerate} + \end{definition} + + Si può dunque definire un'applicazione $f_g$, che, dato $g \in X$, + è tale che $f_g(x) = g.x$ $\forall x \in X$. Tale applicazione è + bigettiva, dacché $f_{g\inv}$ è una sua inversa, sia destra che sinistra. + La definizione equivale a dare un omomorfismo da $G$ a $S_X$ associando + a $g$ l'applicazione $f_g$, dove $S_X$ è il gruppo delle bigezioni + di $X$ con la composizione. \\ + + L'azione di $G$ si dice \textit{fedele} se $g \mapsto f_g$ è iniettivo + (ossia se $f_g(x) = x \forall x \in X \implies g=e$). + + \begin{enumerate}[(i)] + \item Per ogni insieme $X$, $G = S_X$ agisce su $X$ in modo tale + che $g.x = g(x)$ $\forall x \in X$, + + \item $\forall$ gruppo $G$, $G$ agisce su $X = G$ tramite + $g.g' = gg'$, + + \item Si può chiaramente definire un'azione destra in modo + analogo, con la notazione $(g, x) \mapsto x.g$. + \end{enumerate} + + Se $X$ subisce un'azione di $G$, si dice che $X$ è un $G$-insieme. + Si introduce la relazione di equivalenza $x \sim_G y \defiff \exists g \in G \mid g.x = y$. Le classi di equivalenza si chiamano \textbf{orbite} + di $G$ (i.e.~$O_X = \{ g.x \mid g \in G \}$). \\ + + \begin{example} + \begin{enumerate}[(i)] + \item Se $G = \GL(n, \KK)$, $G$ opera su $M(n, \KK)$ tramite + la similitudine. Le orbite sono le classi di similitudine + della matrici. + + \item Se $G = \GL(n, \KK)$, $G$ opera su $\Sym(n, \KK)$ + tramite la congruenza. Le orbite sono le classi di congruenza + delle matrici simmetriche. Analogamente si può fare per la + matrici hermitiane. + + \item Se $G = O_n$, esso opera su $\RR^n$ tramite la + moltiplicazione. Le orbite sono le sfere di raggio $\norm x$. + \end{enumerate} + \end{example} + + \begin{definition} + Lo \textbf{stabilizzatore} di un punto $x \in X$ è + $\Stab_G(X) = \{g \in G \mid g.x = x \}$, sottogruppo + di $G$. + \end{definition} + + \begin{example} + Sia $H \subseteq G$ e sia $X = G/H$. $X$ è un $G$-insieme + tramite l'azione $g'.(gH) = g'gH$. Vale in particolare + che $\Stab_G(eH) = H$. + \end{example} + + \begin{proposition} + Sia $X$ un $G$-insieme. Sia $x \in X$. $H = \Stab_G(x)$ e sia + $O_x$ l'orbita di $x$. Allora esiste un'applicazione bigettiva + naturale $G/H \to O_x$. + \end{proposition} + + \begin{proof} + Sia $\varphi$ tale che $\varphi(gH) = g.x$. Si mostra che + $\varphi$ è ben definita: $g' = gh$, $\varphi(g'H) = (gh).x = + g.(h.x) = g.x$. Chiaramente $\varphi$ è anche surgettiva. + Inoltre, $g.x = g'.x \implies x = (g\inv g').x \implies g\inv g' = h \in H \implies gH = g'H$, e pertanto $\varphi$ è iniettiva. + Allora $\varphi$ è bigettiva. + \end{proof} + + \begin{definition} + Si dice che $G$ opera \textit{liberamente} su $X$ se + $\forall x \in X$, l'applicazione $G \to O_x$ tale che + $g \mapsto g.x$, ossia se $\Stab_G(x) = \{e\}$: + \end{definition} + + \begin{definition} + $G$ opera \textit{transitivamente} su $X$ se $x \sim_G y$ $\forall x$, $y \in X$, cioè se c'è un'unica orbita, che coincide con $X$. In + tal caso si dice che $X$ è \textbf{omogeneo} per l'azione di $G$. + \end{definition} + + \begin{example} + \begin{enumerate}[(i)] + \item $O_n$ opera su $S^{n-1} \subseteq \RR^n$ transitivamente. + %TODO: aggiunge che lo stabilizzatore è isomorfo alle ortogonali + %TODO: di dimensioni n-1 + + \item $\Gr_k(\RR^n) = \{ W \subseteq \RR^n \mid \dim W = k \}$ (Grassmanniana). $O_n$ opera transitivamente su $\Gr_K(\RR^n)$. + \end{enumerate} + \end{example} + + \begin{definition} + $G$ opera in maniera \textit{semplicemente transitiva} su $X$ + se $\exists x \in X$ tale che $g \mapsto g.x$ è una bigezione, + ossia se $G$ opera transitivamente e liberamente. + \end{definition} + + \begin{definition} + Un insieme $X$ con un'azione semplicemente transitiva di $G$ è + detto un $G$-insieme omogeneo \textit{principale}. + \end{definition} + + \begin{example} + \begin{enumerate}[(i)] + \item $X = G$. L'azione naturale di $G$ su $X$ per moltiplicazione + è semplicemente transitivo (per $g$, $g' \in G$, esiste un + unico $h \in G$ tale che $g = h.g' = hg'$). Quindi $X$ + è $G$-omogeneo principale. + + \item Se $X$ è $G$-omogeneo principale, l'azione è fedele. + + \item Se $X$ è omogeneo per un gruppo $G$ commutativo, allora + $G$ agisce fedelmente su $X$ $\implies$ $X$ è un $G$-insieme + omogeneo principale. + \end{enumerate} + \end{example} + + \begin{definition} [spazio affine] + Sia $V$ uno spazio vettoriale su un campo $\KK$ qualsiasi. + Allora uno spazio affine $E$ associato a $V$ è un qualunque + $V$-insieme omogeneo principale. + \end{definition} + + Pertanto, $\forall P, Q \in E$, esiste un unico vettore $\v \in V$ + tale che $Q = \v . P $, denotato come $Q = P + \v = \v + P$. Si + osserva che $\v + (\w + P) = (\v + \w) + P$. Essendo $\v$ unico, + si scrive $\v = Q - P = \vvec{PQ}$. + + %TODO: aggiunge applicazione bigettiva + + Fissato $O \in E$, l'applicazione $\v \mapsto \v + O$, $V \to E$ + è una bigezione. + + \begin{remark}\nl + \li $P-P = \vec 0 \in V$, $P-Q = -(Q-P)$, $(P_3 - P_2) + (P_2 - P_1) = P_3 - P_1$. \\ + + \li $O \in E$ l'applicazione $P \mapsto P-O$ è una bigezione di $E$ + su $V$. + \end{remark} + + Siano $P_1$, ..., $P_n \in E$. $\forall \lambda_1$, ..., $\lambda_k \in \KK$. $\forall O \in E$ possiamo individuare il punto $P = O + \sum_{i=1}^n \lambda_i (P_i - O)$. + + $P = P' = \iff O + \sum_{i=1}^n \lambda_i (P_i - O) = O' + \sum_{i=1}^n \lambda_i (P_i - O') \iff O + \sum_{i=1}^n \lambda_i (O' - O) = O' \iff + (\sum \lambda_i) (O' - O) = O' - O \iff \sum \lambda_i = 1$. + + \begin{definition} + Un punto $P \in E$ è \textbf{combinazione affine} dei punti + $P_1$, ..., $P_k$ se $P = O + \sum \lambda_i (P_i - O)$ se + $\sum \lambda_i = 1$. Si scriverà, in particolare, che + $P = \sum \lambda_i P_i$. + \end{definition} + + Si chiama retta affine l'insieme dei punti che sono combinazione affine di + due punti. Analogamente si fa per un piano e uno spazio. + + \begin{definition} + Un sottoinsieme $D \subseteq E$ si dirà \textbf{sottospazio affine} + se è chiuso per combinazioni affini (finite). + \end{definition} + + \begin{definition} + Il sottospazio affine $D \subseteq E$ generato da un sottoinsieme $S \subseteq E$ è l'insieme delle combinazioni affini (finite) di punti + di $S$, detto $D = \Aff(S)$. %TODO: mostrare che è chiuso per combinazioni affini. + \end{definition} +\end{document} diff --git a/tex/latex/style/personal_commands.sty b/tex/latex/style/personal_commands.sty index 120f0ac..5c22b36 100644 --- a/tex/latex/style/personal_commands.sty +++ b/tex/latex/style/personal_commands.sty @@ -94,6 +94,11 @@ \newcommand{\RRbar}{\overline{\RR}} % Spesso utilizzati al corso di Geometria 1. +\DeclareMathOperator{\Aff}{Aff} +\DeclareMathOperator{\Stab}{Stab} +\DeclareMathOperator{\Gr}{Gr} +\newcommand{\vvec}[1]{\overrightarrow{#1}} + \newcommand{\conj}[1]{\overline{#1}} \let\imm\Im