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zR2g>+Xq>Da|As)RSQ%IoSv%M#GE7WnM;VGAuG8Rq7RCMe7E{Bat89%S^_LL_o37@P zpA*e>B)lK7|CA~4)J-(bOZV=;U1z!r wxC(U-s!QjtBz*Ye|KB7);o)H#9uXK8F*7V^oGF-M@Ybfrty*>O+r#w#04Nd`WB>pF diff --git a/skript.tex b/skript.tex index 34be5c6..e4dc0a2 100644 --- a/skript.tex +++ b/skript.tex @@ -1516,4 +1516,97 @@ Nach 5.2.12 ist $\pi = Tr_1^G(\tilde{\pi})$ $KG$-linear, $\pi(u) = \frac{1}{\abs Sei $v \in V \leq U$. Dann ist $\pi(v) = \frac{1}{\abs{G}} \sum_g g \tilde{\pi}(g^{-1} v) = \frac{1}{\abs{G}} \sum_g g g^{-1} v = \frac{1}{\abs{G}} \sum_g v = v$. \\ $\Rightarrow U = V \oplus \ker \pi$ +% -2 Vorlesungen + +5.4.16 Theorem: Sei $\Delta$ Divisionsalgebra, $n \in \N$. Dann ist $\Delta^n$ (bis auf Isomorphie) der einzige einfache $A$-Modul, $A = M_{n \times n}(\Delta)$. \\ +Es gilt: $\lsub{A}{A} = \bigoplus\limits_{n \text{ Kopien}} \Delta^n$ \\ +Beweis: Sei $0 \neq x = (\alpha_1, \ldots, \alpha_n)^t \in \Delta^n \Rightarrow \exists 1 \leq i \leq n: \alpha_i \neq 0$. \\ +$\alpha_i ^{-1} E_{ii} x = e_i \in \xi_n = $ natürliche Basis von $\Delta^n$ \\ +$\Rightarrow E_{ji} e_i = e_j $, also $ e_j \in A \cdot x \forall 1 \leq j \leq n \Rightarrow \Delta^n = A \cdot x$. \\ +Also ist $\Delta^n$ einfach. \\ +Sei für $1 \leq i \leq n ~ S_i $ die Menge der Matrizen aus $A$, die 0 sind bis auf die $i.$ Spalte. \\ +Dann ist $S_i \leq \lsub{A}{A}$. \\ +$S_i E_{ij} = S_j.$ $\varphi_{E_{ij}} = $ Rechtsmodul mit $E_{ij}$ ist $A$-Modul Isomorphismus von $S_i$ auf $S_j$. \\ +$S_i \cong \Delta^n$. Also ist $\lsub{A}{A} \cong S_1 \oplus \ldots \oplus S_n \cong \bigoplus\limits_{n \text{ Kopien}} \Delta^n$. + +Definition: Eine $K$-Algebra $A$ heißt einfach ("`simple"'), falls $(0)$ und $A$ die einzigen Ideale von $A$ sind. + +5.4.17 Lemma: Einfache Algebren sind halbeinfach (semisimple). \\ +Beweis: Sei $A$ eine einfache Algebra, $\Sigma = $ Summe aller einfachen Untermoduln von $\lsub{A}{A}$. \\ +Klar: $\Sigma$ ist Linksideal von $A$. Sei $S$ einfacher Untermodul von $\lsub{A}{A}$, und sei $a \in A$. \\ +$\rho_a : S \rightarrow A: s \mapsto sa$ ist $A$-Modulhomomorphismu, und daher ist $S \cdot a = \im \rho_a = (0)$ oder $S \cdot a = $ einfacher $A$-Modul. \\ +In beiden Fällen haben wir $Sa \subseteq \Sigma$, daher ist $\Sigma$ auch Rechtsideal von $A$, also ist $\Sigma$ Ideal von $A$. \\ +$\lsub{A}{A}$ hat einfache Untermoduln ($\leq (0)$), daher ist $\Sigma \leq (0)$. Da $A$ einfach ist, ist $\Sigma = A$ und daher $A$ semisimple wegen Lemma 5.4.2 (Charakterisierung vo "`ss"') +und 5.4.13 (Algebra ss. $\Leftrightarrow \lsub{A}{A}$ ss.). \qed + +5.4.18 Theorem: Sei $\Delta$ Divisionsalgebra und sei $n \in N$. Dann ist $A = M_{n \times n}(\Delta)$ einfache Algebra. \\ +Beweis: Sei $(0) \neq J \trianglelefteq A$. Sei $0 \neq a \in J \Rightarrow \exists \alpha_{ij} \in \Delta: a = \sum \alpha_{ij}E_{ij} (1 \leq i,j \leq n)$. \\ +$\exists 1 \leq r,s \leq n: \alpha_{rs} \neq 0$. Sei $1 \leq l, k \leq n$. Dann ist $J \ni b = \alpha_{rs}^{-1} E_{lr} a E_{sk} = \sum \alpha_{rs}^{-1} E_{lr} \alpha_{ij} E_{ij} E_{sk} += \sum \alpha_{rs}^{-1} \alpha_{ij} E_{lr} E_{ij} E_{sk} = \alpha_{rs}^{-1} \alpha_{rs} E_{lk} = E_{lk}$ \\ +$ \Rightarrow E_{lk} \in J \forall 1 \leq l,k \leq n \Rightarrow J = A$. \qed + +$B_1, \ldots, B_r$ = Algebren \\ +$B = B_1 \oplus B_2 \oplus \ldots \oplus B_r$ "`ringdirekte"' Summe, d.h. als $\Lambda$-Modul ist Summe direkt mit Multiplikation komponentenweise. \\ +$\tilde{B_i} = \set{(b_1, \ldots, b_r) \mid b_j \in B_j, b_j = 0 \text{ für } j \neq i} \Rightarrow \tilde{B_i} \cong B_i$ als $\Lambda$-Algebra. \\ +$\tilde{B_i} \trianglelefteq B, b_1 \in \tilde{B_i}, b_2 \in \sum\limits_{i \neq j} \tilde{B_j} \Rightarrow b_1 b_2 = b_2 b_1 = 0$ \\ +$\tilde{B_i} \cap \tilde{B_j} = (0) ~~ (i \neq j)$. \\ +$\tilde{B_i}$ heißen "`Blöcke"' von $B$ (Blockideale). + +5.4.19 Lemma: $B = B_1 \oplus \ldots \oplus B_r$ ringdirekte Summe von Algebren $B_i$. Dann sind die zweiseitigen Ideale von $B$ genau die Mengen der Form +$ J = J_1 \oplus \ldots \oplus J_r$ mit $J_i \trianglelefteq B_i$ ($J_i = J \cap \tilde{B_i}$). \\ +Beweis: Setze $J_i = J \cap B_i \trianglelefteq B$. Klar: $\sum\limits_{i=1}^r J_i = \bigoplus\limits_{i=1}^r JJ-i \subseteq J.$ Sei $b \in J, b = b_1 + \ldots b_r$ mit $b_i \in B_i$ +und $1 = e_1 + \ldots + e_r, e_j = 1_{B_j}$. \\ +Dann ist $b cdot e_1 = b_1 e_1 + b_2 e_1 + \ldots + b_r e_r = b_1 \in J_1$, analog $b_i \in B_i \Rightarrow $ Behauptung. + +5.4.20 Theorem: Sei $r \in \N, \Delta_i$ Divisionsalgebra über $K$ endlicher Dimension, $n_i \in N, B_i = M_{n_i \times n_i}(\Delta_i) \forall 1 \leq i \leq r$. \\ +Sei $B = B_1 \oplus \ldots \oplus B_r$ ringdirekte Summe. ($\dim_K B = \sum_{i=1}^r \dim_K(\Delta_i) n_i^2 < \infty).$ Dann ist $B$ semisimple, und es gibt exakt $r$ viele nicht isomorphe einfache $B$-Moduln, +und gneau $2^4$ zweiseitige Ideale von $B$, nämlich $\bigoplus_{j \in J} B_j$ mit $J \subseteq \set{1, \ldots, r}$ (Ideale), einfachen $B$-Moduln $S_1, \ldots, S_r$ mit $S_i \leq \lsub{B_i}{B_i}$ einfach. \\ +Beweis: Ist $S \leq \lsub{B_i}{B_i}$ einfacher Untermodul, so ist auch $S$ ein einfacher $B$-Modul mit $b \cdot x = 0 \forall b \in B_j$ mit $j \neq i$. ($U \leq \lsub{B_i}{B_i} \leq_{\text{Liid}} \lsub{B}{B}$ immer). \\ +Haben gesehen (5.4.16): $\forall 1 \leq i \leq r$ gibt es genau einen einfachen $B_i$-Modul $S_i$, $B_i \cong \bigoplus\limits_{n_i \text{ Kopien}} S_i$. +$ \Rightarrow \lsub{B}{B} \cong \bigoplus\limits_{i=1}^r \bigoplus\limits_{n_i \text{ Kopien}} S_i$. \\ +Insbesondere ist $\lsub{B}{B}$ (direkte) Summe einfacher Untermoduln, und daher nach 5.4.13 semisimple. Jeder einfache $B$-Modul ist daher isomorph zu einem $S_i$ mit $1 \leq i \leq r$ (5.4.14). \\ +Behauptung folgt sofort aus 5.4.18 und 5.4.19. \qed + +5.4.21 Satz: Sei $0 \neq e^2 = e \in B \Rightarrow \End_B(Be) \cong eBe$. + +5.4.22 Lemma:Sei $B$ Algebra. Dann ist $M_{n \times n}(B)^{op} \cong M_{n \times n}(B^{op})$. \\ +Beweis Idee: $M_{n \times n}(B)^{op} \rightarrow M_{n \times n}(B^{op}): a \mapsto a^t$ ist Isomorphismus. + +5.4.23 Theorem (Wedderburn): Eine $K$-Algebra $A$ ($\dim_K A < \infty$) ist semisimple $\Leftrightarrow A = \bigoplus\limits_{i=1}^r M_{n_i \times n_i}(\Delta_i)$ mit $\Delta_i$ = Divisionsalgebra über $K$ mit $\dim_K \Delta_i < \infty$. \\ +Beweis: "`$\Leftarrow$"' ist 5.4.20. \\ +Sei also $A$ semisimple. Sei $S_1, \ldots, S_r$ eine vollständige Menge paarweiser nicht isomorpher einfacher $A$-Moduln und sei $\Delta_i = \End_A(S_i)$ Schiefkörper ($ \dim_K \Lambda_i = k$), $n_i = \dim_{\Delta_i}(S_i)$ \\ +Da $\lsub{A}{A}$ semisimple gibt es $v_1, \ldots, v_r \in \N: \lsub{A}{A} = S_1^{\oplus v_1} \oplus \ldots \oplus S_r^{\oplus v_r}$.\\ +5.4.21 $\Rightarrow \End_A(\lsub{A}{A}) = 1 \cdot A \cdot 1 \cong A$ \\ +$\End_A(S_1^{\oplus v_1} \oplus \ldots \oplus S_r{\oplus v_r}) \cong_{\text{Satz 5.4.12}} M_{v_1 \times v_1}(\Delta_1) \oplus \ldots \oplus M_{v_r \times v_r}(\Delta_r)$ \\ +5.4.20 $\Rightarrow v_i = n_i$ \qed. + +5.4.25 Spezialfall: $A$ semisimple $K$-Algebra, $K$ algebraisch abgeschlossen. Dann $\exists n_1, \ldots, n_r \in \N: A \cong M_{n_1 \times n_1}(K) \oplus \ldots M_{n_r \times n_r}(K)$ \\ +$\set{S_1, \ldots, S_r}$ = vollständige Menge nicht isomorpher einfacher $A$-Moduln, $\dim_K S_i = n_i$. \\ + +\chapter{Charaktere} + +$G$ = endliche Gruppe, alle $\C G$-Moduln sind endlich erzeugt. + +6.1.1 Anmerkung: $G = $ endliche Gruppe $\Rightarrow^{\text{Maschke}} \C G$ ist semisimple.\\ +$\Rightarrow \exists n_i \in \N: \C G \cong M_{n_1 \times n_1}(\C) \oplus \ldots M_{n_r \times n_r}(\C)$ +6.1.2 : $\sum n_i^2 = \abs{G}$ \\ + +Bemerkung: $\Char K \nmid \abs{G} \Rightarrow KG$ semisimple. $KG \cong M_{n_1 \times n_1}(\Lambda_1) \oplus \ldots \oplus M_{n_r \times n_r}(\Lambda_r)$ \\ +Fakten: 1) $\Lambda_i$ kommutativ \\ +2) $K = \C: n_i$ Teiler von $\abs{G}$ + +6.1.3 Theorem: $\abs{\set{\text{einfache }\C G\text{-Moduln}}} = \abs{\set{\text{Konjugationsklassen von }G}} = \dim_{\C} Z(\C G)$ \\ +Beweis: $\C G = M_{n_1 \times n_1}(\C) \oplus \ldots \oplus M_{n_r \times n_r}(\C) \exists r, n_1, \ldots, n_r \in \N$ (6.1.1) \\ +$\Rightarrow_{\text{5.4.23}}$ Anzahl nicht isomorpher irreduzibler $\C G$-Moduln $ = r$ \\ +Klar: $B = B_1 \oplus \ldots \oplus B_r$ ringdirekte Summe $\Rightarrow Z(B) = Z(B_1) \oplus \ldots \oplus Z(B_r)$ \\ +$Z(M_{n \times n}(\C)) = \set{ \alpha 1_M \mid \alpha \in \C }$. Also ist $\dim_K(Z(\C G)) = r$. \\ +Sei $x = \sum \alpha_g g$ mit $\alpha_g \in \C$ \\ +$x \in Z(\C G) \Leftrightarrow hx = xh \forall h \in G \Leftrightarrow x = hxh^{-1} \forall h \in G$ \\ +d.h. $x \in Z(\C G) \Leftrightarrow \sum \alpha_g g = \sum \alpha_g h g h^{-1} \forall h \in G$ \\ +$\sum \alpha_g g = \sum \alpha_{hgh^{-1}} g \Rightarrow \forall h \in G: \alpha_{hgh^{-1}} = \alpha_g.$ \\ +Sei $C_1, \ldots, C_m$ Konjugationsklassen von $G$, $g_i \in C_i, \hat{C_i} = \sum_{h \in C_i} h$. \\ +$\Rightarrow x = \sum_{i=1}^m \alpha_g \hat{C_i}$. \\ +Klar: $(\hat{C_1}, \ldots, \hat{C_m}) \subseteq \C G$ lineare unabhängig$/ \C$. \\ +$\Rightarrow (\hat{C_1}, \ldots, \hat{C_m})$ ist $\C$-Basis von $Z(\C G) \Rightarrow \dim_{\C}(Z(\C G)) = m = $ Anzahl der Konjugationsklassen $ = r = $ Anzahl der irreduziblen $\C G$-Moduln. + \end{document}