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zfuSISkRVmBQm(hB9rzBQC%dYul(L!AS@3N@4EU-Tale}z_-+c6=Ap+c3|t-Dz+WJO zkpI$=I-&|3RuAMrEg%^Ry-OFbO+e~tX`%62dT{hE9o=2(QvW}Sm6W G_1 > \ldots > G_r = N > G_{r+1} = (1)$ ist Hauptreihe f \section{Grundlagen} -\paragraph{Gruppenalgebren} +\subsection{Gruppenalgebren} Alle Ringe haben Einselement $1 = 1_R$, aber sind nicht notwendigerweise kommutativ. @@ -1179,7 +1180,7 @@ Fragen: 5.1.3 Konstruktion von $\Lambda G$: Die Gruppenalgebra $\Lambda G$ ist als $\Lambda$-Modul der freie $\Lambda$-Modul über der Menge $G$, d.h. $\Lambda G = \set{ \sum_{g \in G} \alpha_g g \mid \alpha_g \in \Lambda, \text{ fast alle }\alpha_g = 0}$ \\ -$(\sum \lambda_g g) + (\sum \mu_g g) = \sum (\lambda_g + \mu_g) g$ \\ +$(\sum \lambda_g g) + (\sum \mu_g g) = \sum (V + \mu_g) g$ \\ $\beta (\sum \lambda_g g) = \sum (\beta \lambda_g)g$ \\ $(\sum \alpha_g g)(\sum \beta_h h) = \sum_{g,h} \alpha_g \beta_h (g \cdot h) = \sum_x (\sum_{gh=x} \alpha_g \beta_h) x = \sum_x (\sum_g \alpha_g \beta_{g^{-1}x}) x$ @@ -1198,12 +1199,124 @@ lineare Darstellung von $G$ über dem $K$-Vektorraum $V$ ist ein Homomorphismus Klar: \begin{diagram} -G & \rightarrow^\varphi & \Aut_K(V) & \rightarrow^{\sim}_{\text{Wahl der Basis}} & \GL_n(K) \\ -\downarrow & & \downarrow & & \downarrow \\ -KG \rightarrow \End_K(V) \rightarrow^{\sim}_{\text{Wahl der Basis}} & M_{n \times n}(K) \\ +G & \rTo^\varphi & \Aut_K(V) & \rTo^{\sim}_{\text{Wahl der Basis}} & \GL_n(K) & ~~ n = \dim_V (K) \\ +\dInto & & \dInto & & \dInto \\ +KG & \rTo & \End_K(V) & \rTo^{\sim}_{\text{Wahl der Basis}} & M_{n \times n}(K) \\ \end{diagram} -Für eine $K$-Algebra $A$ ist eine Darstellung von $A$ ein $K$-Algebra-Homomorphismus $A \rightarrow \End_K(V) \cong M_{n \times n}(K) (\dim_K V = n)$ \\ -Sei $\varphi: KG \rightarrow \End_K(V)$ Darstellung. Dan wird $V$ zum $KG$-Modul durch $x \cdot m = (\varphi(x))(m)$ für $y \in KG$ und $m \in V$. +Für eine $K$-Algebra $A$ ist eine Darstellung von $A$ über dem $K$-Vektorraum $V$ ein $K$-Algebra-Homomorphismus $A \rightarrow \End_K(V) \cong M_{n \times n}(K) (\dim_K V = n)$ \\ +Sei $\rho: KG \rightarrow \End_K(V)$ Darstellung. Dann wird $V$ zum $KG$-Modul durch $x \cdot m = (\rho(x))(m)$ für $y \in KG$ und $m \in V$. \\ +Umgekehrt: Ist $V$ ein $A$-Modul, so wird durch $\rho: A \rightarrow \End_K(V): a \mapsto \lambda_a, \lambda_a(v) = av$ für $a \in A, v \in V$ eine Darstellung von $A$ über $V$ definiert. \\ +So: Konzept der Darestellungen von $A$ ist äquivalent zum Konzept der $A$-Moduln. (Vgl. Permutationsdarstellungen). \\ +Homomorphismen von $A$-Moduln: Klar. \\ +Homomorphismen von Darstellungen: Seien $\rho : A \rightarrow \End_K(V), \psi : A \rightarrow \End_K(W)$ Darstellungen von $A$. Ein Homomorphismus von $\rho$ nach $\psi$ ist ein $K$-lineare Abbildung $f: V \rightarrow W$, so dass $\forall a \in A: f \circ \rho(a) = \psi(a) \circ f$ +\begin{diagram} +V & \rTo^f & W \\ \dTo^{\rho(a)} & & \dTo_{\psi(a)} \\ V & \rTo^f & W +\end{diagram} + +Beachte: $W = V$ mit $f \in \Aut_K(V)$ Homomorphismus von $\rho$ nach $\psi \Leftrightarrow \psi(a) = f \circ \rho(a) \circ f^{-1} \forall a \in A$ \\ +$\dim_K(V) = n, \tilde{\rho}: A \rightarrow M_{n \times n}(K), \tilde{\psi}: A \rightarrow M_{n \times n}(K)$ zugehörige Matrixdarstellungen nach Wahl einer Basis $\cB$, so +kann man $f$ als Basiswechsel interpretieren: $m_{\id}(\cC, \cB)= m_f(\cB, \cB)$ für Basis $\cC$ von $V$. \\ +$m_{\tilde{\psi}}(\cC, \cC) = m_{\id}(\cC, \cB) m_{\tilde{\rho}}(\cB, \cB) m_{\id}(\cB, \cC)$ (Details: Übung) + +Definition: Seien $A, B$ $\Lambda$-Algebren mit $1$ und sei $M$ ein $A$-Linksmodul und ein $B$-Rechtsmodul. $M$ heißt $A$-$B$-Bimodul falls gilt: \\ +$\forall \alpha \in A, \beta \in B, m \in M: (\alpha m) \beta = \alpha (m \beta)$ \\ +($M$ $A$-Linksmodul und $B$-Linksmodul: $\alpha(\beta m) = \beta(\alpha m)$) + +Beachte: Für die zugehörigen Darstellungen bedeutet das: $\lambda_\alpha: M \rightarrow M: m \mapsto \alpha m, \rho_\beta: M \rightarrow M: m \mapsto m \beta$ \\ +(Abbildung $\rho: B \rightarrow \End_\Lambda(M)$ ist Antihomomorphismus, d.h. $\rho_\beta \rho_\gamma = \rho_{\gamma \beta}$ für $\gamma, \beta \in B$) \\ +Bedingung $\Leftrightarrow \lambda_\alpha \rho_\beta = \rho_\beta \lambda_\alpha \forall \alpha \in A, \beta \in B$ \\ +d.h. $\lambda_\alpha$ und $\rho_\beta$ zentralisieren einander in $\End_\Lambda(M)$ ($\forall \lambda \in \Lambda, m \in m: \lambda m = m \lambda$) + +5.1.3 Beispiel: $M \in A^{mod} = \set{\text{Links-$A$-Moduln}}$. Sei $E = \End_A(M)$. $E$ operiert auf $M$ von rechts. \\ +$E = \set{b \in \End_\Lambda(M) \mid (am) b = a (mb) \forall a \in A, m \in M}$ \\ +Dann ist $E$ $\Lambda$-Algebra und $M$ ein $A$-$E$-Bimodul. \\ +Beweis: $A = \Lambda$-Algebra $\Rightarrow M$ ist $\Lambda$-Modul durch Einschränken, $\lambda m = (\lambda 1_A) m$. +Durch $\lambda m = m \lambda$ wird $M$ ein $\Lambda$-Rechtsmodul, da $\Lambda$ kommutativ ist. \\ +Ein $A$-Endomorphismus ist dann auch ein $\Lambda$-Endomorphismus, d.h. $E = \End_A(M) \subseteq \End_\Lambda(M)$. + +( Beachte: $\set{\text{$\Z$-Moduln}} = \set{\text{$\Z$-Bimoduln}} = \set{\text{abelsche Gruppen}}$ ) + +Für $b \in E$, $m \in M$ und $\lambda \in \Lambda$ sei $\lambda b \in \End_A(M)$ definiert durch $m (\lambda b) = \lambda (m b) = (m b) \lambda = m (b \lambda)$. So ist $E$ eine $\Lambda$-Algebra. Rest klar. + +Bemerkung: $A$ und $B$ seien $\Lambda$-Algebren, $M$ ein $A$-Links- und $B$-Rechtsmodul. $\lambda: A \rightarrow \End_\Lambda(M): a \mapsto \lambda_a, \rho : B \rightarrow \End_\Lambda(M): b \mapsto \rho_b, \rho_b(m) = m b$ \\ +$\rho $ Antihomomorphismus, $\lambda$ Homomorphismus von $\Lambda$-Algebra \\ +$\tilde{B} = \im \rho \subseteq \End_\Lambda(M) \supseteq \tilde{A} = \im \lambda$ + +($R$ Ring, $R^{opp} = $ Ring auf Menge $R$ mit Multiplikation $\ast$ gegeben durch $r \ast s = sr \forall r,s \in R$) + +Dann ist $M$ ein $A$-$B$-Bimodul $\Leftrightarrow$ $\tilde{B} \subseteq \End_A(M) \subseteq \End_\Lambda(M) \Leftrightarrow \tilde{A} \subseteq \End_B(M) \subseteq \End_\Lambda(M) $ + +Man sagt: $M$ erfüllt Schur-Wyl-Dualität oder ist balanced oder erfüllt Bizentralisatoren Eigenschaft, falls gilt: $\tilde{B} = \End_A(M), \tilde{A} = \End_B(M)$. + +$ \rightsquigarrow $ Ausrechnen von $\End_A(M):$ $\Lambda = K $ Körper, $M \in \Lambda^{mod}, \dim_K M = n < \infty, \rho: A \rightarrow M_{n \times n}(K)$ zugehörige Matrixdarstellung. \\ +$G_a: \rho(a)X = X \rho(a), X \in M_{n \times n}K$. Um $E$ auszurechnen genügt es die Gleichunssysteme $G_a$ für $a \in A$ simultan zu lösen. \\ +Angenommen, $a_1, \ldots, a_k \in A$ so, dass $A$ die kleinste Unteralgebra von $A$ ist, die $a_1, \ldots, a_k$ enthält, d.h. $\set{a_1, \ldots, a_k}$ erzeugt die $K$-Algebra $A$, +dann genügt es $\rho(a_i)X = X \rho(a_i)$ simultan zu lösen $\rightsquigarrow$ mühsamer Weg um $E$ auszurechnen. + +\subsection{Tensorprodukte} + +$A, B, C$ seien $\Lambda$-Algebren, $\Lambda$ kommutativer Ring $\ni 1$. $\lsub{A}{\Mod_B} := \set{\text{$A$-$B$-Bimoduln}}$ + +Definition: Sei $M \in \lsub{A}{\Mod_B}, N \in \lsub{B}{\Mod_C}$. Eine Abbildung $f: M \times N \rightarrow U \in \lsub{A}{\Mod_C}$ heißt $A$-$C$-bilinear und $B$-balanced, falls gilt $\forall a \in A, b \in B, c \in C, m, m_1, m_2 \in M, n, n_1, n_2 \in N$: \\ +\begin{enumerate}[i)] + \item $f$ ist bilinear, d.h. $f(m_1 + m_2, n) = f(m_1, n) + f(m_2, n), f(m, n_1 + n_2) = f(m, n_1) + f(m, n_2)$ + \item $f(am, n) = a f(m, n)$ + \item $f(m, nc) = f(m, n) c$ + \item $f(mb, n) = f(m, bn) $ +\end{enumerate} + +Ein \underline{Tensorprodukt} $M \otimes_B N \in \lsub{A}{\Mod_C}$ über $B$ ist ein $A$-$C$-Bimodul zusammen mit einer $A$-$C$-bilinearen, $B$-balanced Abbildung $\eta: M \times N \rightarrow M \otimes_B N$ so, dass folgende universelle Eigenschaft gilt: +\begin{diagram} +M \times N & \rTo^\eta & M \otimes_B N \\ +& \rdTo_{f} & \dTo_{\exists ! \hat{f}} \\ +& & U \in \lsub{A}{\Mod_C} +\end{diagram} +mit $\hat{f} \circ \eta = f$ + +$\cA$ = freie abelsche Gruppe über \\ $M \times N = \set{ z_1 (m_1, n_2) + z_2(m_2, n_2) + \ldots+ z_k(m_k, n_k) \mid m_i \in M, n_i \in N, k \in N, z_i \in \Z}$ \\ +$I = \set{ (m_1+m_2, n) - (m_1, n) - (m_2, n), (m, n_1 + n_2) - (m, n_1) - (m, n_2), (mb, n) - (m, bn)}$ +$(b \in B, m, m_1, m_2 \in M, n, n_1, n_2 \in N)$ \\ + +$\cA$ ist $A$-Linksmodul durch $a(\sum z_i(m_i, n_i)) = (\sum z_i(am_i, n_i))$, $C$-Rechtsmodul analog. \\ +$cA \in \lsub{A}{\Mod_C}$, $I$ ist $A$-$C$-invariant $\Rightarrow $ ist $A$-$C$-Unterbimodul. \\ +$M \otimes_B N = A / \in \lsub{A}{\Mod_C}$ \\ +$\eta: M \times N \rightarrow M \otimes_B N: (m, n) \mapsto (m,n)+ = m \otimes n$ (einfacher Tensor) \\ +Klar: $\set{m \otimes n \mid m \in M, n \in N}$ erzeugt $M \otimes_B N$, d.h. \\ +$M \otimes_B N = \set{\sum_{i=1}^k m_i \otimes n_i \mid m_i \in M, n_i \in N, k \in \N}$ + +Bemerkung: $M \in \lsub{A}{\Mod_B} \subseteq \Mod_B, N \in \lsub{B}{\Mod}$. \\ +Bilde $M \otimes_B N$ mit $M$ nur als $B$-Rechtsmodul. $\tilde{X} = M \otimes_B N$ mit $M$ als $A$-$B$-Bimodul. \\ +Dann ist $X \in \lsub{A}{\Mod}$ und $X \cong \tilde{X}$ als $A$-Linksmodul. + +Für $a \in A$ definiere eine Abbildung $f_a: M \times N \rightarrow M \times N: (m, n) \mapsto (am, n)$ \\ +$A$-$\Z$-bilinear, $B$-balanced (wegen $(am)b = a(mb)$, da $M$ $A$-$B$-Bimodul). + +\begin{diagram} +M \times N & \rTo^\eta & M \otimes_B N \\ +& \rdTo_{f_a} & \dTo{\exists! \hat{f_a}} \\ +& & M \otimes_B N +\end{diagram} +mit $\hat{f_a} (\sum m_i \otimes n_i) = \sum a m_i \otimes n_i$ + +5.2.1 Fakten: +\begin{enumerate}[i)] + \item $M \in \lsub{A}{\Mod}, A \in \lsub{A}{\Mod_A} \Rightarrow A \otimes_A M \cong M$ \\ + Beweis: $f(a,m) = am$ + \begin{diagram} + A \times M & \rTo^\eta & A \otimes_A M \\ + & \rdTo_f & \dTo^{\hat{f}} \\ + & & M + \end{diagram} + $\hat{f}(\sum a_i \otimes m_i) = \sum a_i m_i \in M, \hat{f}^{-1}: M \rightarrow A \otimes_A M: m \mapsto 1_A \otimes m \in A \otimes M$ + \item Tensorprodukte vertauschen mit direkten Summen: $(\oplus_{i \in I}) \otimes_A N \cong \oplus_{i \in I}(M_i \otimes N)$ \\ + $M \cong \oplus_{i\in I} A_i, A_i \cong_A A, N \in \lsub{A}{\Mod}$ \\ + $M \otimes_I N = (\oplus_{i \in I} A_i) \otimes N \cong \oplus_{i \in I}(A_i \otimes N) \cong N = \oplus_{i \in I} N_i$ mit $N_i \cong N$ als $A$-Modul. + \item $(\lsub{A}{M} \otimes_B N) \otimes_C L_D \cong \lsub{A}{M} \otimes_B (N \otimes_C L)_D$ Assoziativität, $N \in \lsub{B}{\Mod_C}$ +\end{enumerate} + +Beispiel: $X := \Z/z\Z \otimes_\Z \Z/3\Z = (0)$ \\ +$ 1 = 2 \cdot 2 - 3$ \\ +$ a \otimes b \in X: a \otimes b = 1 a \otimes b = (4-3)(a \otimes b) = 4(a \otimes b) - 3(a \otimes b) = (4a)\otimes b - a \otimes (3b) = 0 \otimes b - a \otimes 0 = 0$ \end{document} diff --git a/standard.tex b/standard.tex index 617b0cd..a3de4f0 100644 --- a/standard.tex +++ b/standard.tex @@ -43,6 +43,7 @@ \providecommand{\cO}[0]{\mathcal{O}} \providecommand{\cC}[0]{\mathcal{C}} \providecommand{\cB}[0]{\mathcal{B}} +\providecommand{\cA}[0]{\mathcal{A}} \providecommand{\abs}[1]{\lvert#1\rvert} \providecommand{\norm}[1]{\lVert#1\rVert} @@ -90,6 +91,7 @@ \DeclareMathOperator{\GF}{GF} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Syl}{Syl} +\DeclareMathOperator{\Mod}{mod} \DeclareMathOperator{\Reid}{Reid} \DeclareMathOperator{\Liid}{Liid}