commit c708e7589668f5f78a3704d70d0776d8b65a494a
parent 9ec2d37b52c524546227dc530e1a41f795083751
Author: miksa <milutin@popovic.xyz>
Date: Tue, 20 Apr 2021 17:48:26 +0200
done some exercise i didn't
get to do this few weeks in week6.pdf
Diffstat:
10 files changed, 626 insertions(+), 475 deletions(-)
diff --git a/pdfs/week5.pdf b/pdfs/week5.pdf
Binary files differ.
diff --git a/pdfs/week6.pdf b/pdfs/week6.pdf
Binary files differ.
diff --git a/src/pres/main.aux b/src/pres/main.aux
@@ -66,4 +66,6 @@
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\abx@aux@defaultrefcontext{0}{ncgshort}{none/global//global/global}
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diff --git a/src/pres/main.log b/src/pres/main.log
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diff --git a/src/pres/main.nav b/src/pres/main.nav
@@ -57,3 +57,7 @@
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diff --git a/src/pres/main.tex b/src/pres/main.tex
@@ -0,0 +1,273 @@
+\documentclass[fleqn]{beamer}
+\beamertemplatenavigationsymbolsempty
+
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+
+\usepackage{amsmath,amssymb}
+\usepackage{graphicx}
+\usepackage{mathptmx}
+\usepackage{subcaption}
+\usepackage{amsthm}
+\usepackage{tikz}
+%\usepackage[colorlinks=true,naturalnames=true,plainpages=false,pdfpagelabels=true]{hyperref}
+\usetikzlibrary{patterns,decorations.pathmorphing,positioning, arrows, chains}
+
+\usepackage[backend=biber, sorting=none]{biblatex}
+\addbibresource{uni.bib}
+
+\setbeamertemplate{endpage}{%
+ \begin{frame}
+ \centering
+ \Large \emph{Thank You!}
+ \end{frame}
+}
+
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+
+% Beamer theme
+\usetheme{UniVienna}
+\usefonttheme[onlysmall]{structurebold}
+\mode<presentation>
+\setbeamercovered{transparent=10}
+
+\title
+{Noncommutative Geometry}
+\subtitle{Bachelor's seminar}
+\author[Popovic Milutin]
+{Popovic Milutin \newline Supervisor: Dr. Lisa Glaser}
+\date{16. April 2021}
+
+\begin{document}
+ \begin{frame}
+ \titlepage
+ \end{frame}
+
+ \begin{frame}{Introduction}
+ \begin{itemize}
+ \item Noncommutative geometry (NCG) brings many\\
+ mathematical fields together (e.g. K-Theory, Differential Geometry)
+ \item Physics application (spectral Standard Model)
+ \item Gelfand-Naimark-Theorem in Functional Analysis in the 1940s \\
+ duality between (classical) geometry and Algebra
+ \end{itemize}
+ \end{frame}
+
+ \begin{frame}{Spaces and Algebras}
+ Introduce:
+ \begin{block}
+ {Algebra}
+ \centering
+ Vectorspace with a multiplication operation\\
+ (associative and possesses an identity element)
+ \end{block}
+ \begin{block}
+ {Finite topological Space $X$ consisting of $N$ points. (discrete topology)}
+ \begin{figure}[h!]
+ \centering
+ \begin{tikzpicture}[
+ dot/.style = {draw, circle, inner sep=0.05cm, fill},
+ smalldot/.style = {draw, circle, inner sep=0.015cm,fill},
+ ]
+ \node[dot] at (-3,0.) [label=below:$1$]{};
+ \node[dot] at (-1.5,0) [label=below:$2$]{};
+ \node[dot] at (2.1,0) [label=below:$N$]{};
+ \node[smalldot] at (-0.4,0) {};
+ \node[smalldot] at (0.1,0) {};
+ \node[smalldot] at (0.6,0) {};
+ \end{tikzpicture}
+ \end{figure}
+
+ \end{block}
+ \begin{block}{Commutative algebra of continuous functions on $X$}
+ \centering
+ $C(X) = \{ f: X \rightarrow \mathbb{C}:\;\;
+ \text{$f$ is continuous}\}$
+ \end{block}
+ \end{frame}
+ \begin{frame}
+ {Spaces and commutative Algebras}
+ Results of the Theorem:
+ \begin{itemize}
+ \item $X$ and $C(X)$ contain the same information (duality)
+ \item Construct $X$, given $C(X)$.
+ \item Translate geometrical properties of $X$ to algebraic data\\
+ (metric, differential forms, vector fields, curvature, etc.)
+ \end{itemize}
+ \end{frame}
+
+ \begin{frame}{Geometry as a Spectral Triple}
+ \begin{block}
+ {\centering The Spectral Triple}
+ \centering
+ $(\;A,\;\; H,\;\; D\;)$
+ \end{block}
+ \begin{itemize}
+ \item $A$ - Algebra
+ \item $H$ - Hilbertspace
+ \item $D$ - self adjoint Operator acting on $H$
+ \end{itemize}
+ \end{frame}
+
+ \begin{frame}{Geometry as a Spectral Triple}
+ \begin{block}
+ {The Spectral Triple of a Circle $\mathbb{S}^1$}
+ \centering
+ $ (\; C^{\infty}(\mathbb{S}^1),\;\; L^2(\mathbb{S}^1),\;\; -i\frac{d}{dt} \;)$
+ \end{block}
+ \end{frame}
+
+ \begin{frame}{Introducing the Metric}
+ \begin{itemize}
+ \item The metric describes distances between points on a space
+ \end{itemize}
+ \begin{columns}[T]
+ \column{0.4\textwidth}
+ \begin{block}{\centering Discrete Metric}
+ \centering
+ $
+ d_{ij} =
+ \begin{cases}
+ 0\;\;\; \text{if}\;\;\; i = j \\
+ 1\;\;\; \text{if}\;\;\; i \neq j
+ \end{cases}
+ $
+ \end{block}
+
+ \column{0.4\textwidth}
+ \begin{block}{\centering Minkowski Metric}
+ \centering
+ $
+ \eta _{\mu \nu} =
+ \begin{pmatrix}
+ -1 & 0 & 0 & 0 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 0 \\
+ 0 & 0 & 0 & 1
+ \end{pmatrix}
+ $
+ \end{block}
+ \end{columns}
+
+ \begin{figure}[h!] \centering
+ \begin{tikzpicture}[
+ dot/.style = {draw, circle, inner sep=0.05cm, fill},
+ smalldot/.style = {draw, circle, inner sep=0.015cm,fill},
+ ]
+ \node[dot](m1) at (-3,0.) [label=left:$1$] {};
+ \node[dot](m2) at (-1.5, 2) [label=above right:$2$] {};
+ \node[dot](m3) at (2.1,0) [label=right:$3$] {};
+
+ \draw[<->, >=stealth](m1) -- ++(m2) node [midway, fill=white] {$d_{12}$};
+ \draw[<->, >=stealth](m2) -- ++(m3) node [midway, fill=white] {$d_{23}$};
+ \draw[<->, >=stealth](m3) -- ++(m1) node [midway, fill=white] {$d_{13}$};
+ \end{tikzpicture}
+ \end{figure}
+
+ \end{frame}
+
+
+ \begin{frame}{Algebraic Formulation of the Metric}
+ \begin{itemize}
+ \item Utilize results of the Gelfand-Naimark Theorem
+ \item Characterize the Metric with\\
+ \begin{itemize}
+ \item[\bullet] commutative Algebra
+ \item[\bullet] finite-dimensional Hilbertspace $H$
+ \item[\bullet] symmetric operator $D$
+ \end{itemize}
+ \end{itemize}
+ \begin{block}{Metric with $(A, H, D)$ on finite Space (commutative case)}
+ \centering
+ $d_{ij} = \sup_{a \in A}\{ |a(i) - a(j)| : ||[D, a]|| \leq 1\}$
+ \end{block}
+ \end{frame}
+
+ \begin{frame}{Algebraic Formulation of the Metric}
+ In the noncommutative Case:
+ \begin{itemize}
+ \item replace Algebra with matrix Algebra (noncommutative)
+ \item define in terms of invariants
+ \end{itemize}
+ \begin{block}{Metric with $(A, H, D)$ on finite Space (noncommutative case)}
+ \centering
+ $d_{ij} = \sup_{a \in A}\{ |\text{Tr}(a(i)) - \text{Tr}(a(j))| :||[D, a]|| \leq 1\}$
+ \end{block}
+ \end{frame}
+
+ \begin{frame}{Algebraic Formulation of the Metric}
+ \begin{itemize}
+ \item describe the Metric on a Manifold $M$
+ \item We need \\
+ \begin{itemize}
+ \item[\bullet] $C^\infty(M)$ - Algebra
+ \item[\bullet] $L^2(S)$ - Hilbertspace
+ \item[\bullet] $D$ - Dirac Operator
+ \end{itemize}
+ \end{itemize}
+ \begin{block}{Metric with $(C^\infty(M),\;\; L^2(S),\;\; D)$ on a Manifod}
+ \centering
+ $d(x, y) = \sup_{f \in C^\infty(M) }\{ |f(x) - f(y)| :
+ ||[D, f]|| \leq 1\}$
+ \end{block}
+ \begin{figure}[h!] \centering
+ \begin{tikzpicture}[
+ dot/.style = {draw, circle, inner sep=0.06cm, fill},
+ smalldot/.style = {draw, circle, inner sep=0.015cm,fill},
+ ]
+ \node[dot](b) at (0,0) [label=below left:$x$] {};
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+ \node[dot](a) at (5, 0) [label=below left:$x$] {};
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+ % \node[dot](m3) at (2.1,0) [label=right:$3$] {};
+
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+
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+ \draw[<->, >=stealth, line width=0.4mm, style=dashed](8, 2) -- (8, 0.1) {};
+ \draw[line width=0.5mm] (5, 0) -- (8, 2) node [pos=.75, label=:$f$] {} ;
+ \end{tikzpicture}
+ \end{figure}
+ \end{frame}
+
+%\begin{frame}{Noncommutative Case}
+% \begin{itemize}
+% \item Introduce a richer geometry
+% \item From finite topological space to a Manifold with noncommutativity
+% \item From finite to general spectral triples with a \\
+% self adjoint Operator (Dirac Operator)
+% \end{itemize}
+% \end{frame}
+
+ \begin{frame}{Applications In Physics}
+ \begin{itemize}
+ \item NCG of the Quantum Hall Effect
+ \item NCG of the Standard Model
+ \begin{itemize}
+ \item[\bullet] going to noncommutative Manifolds
+ \item[\bullet] obtain Standard Model gauge fields (scalar Higgs filed)
+ \item[\bullet] minimal coupling to gravity
+ \item[\bullet] construct the Full Lagrangian
+ \end{itemize}
+ \end{itemize}
+ \end{frame}
+
+ \begin{frame}{Bibliography}
+ \nocite{ncgwalter}
+ \nocite{liealgebra}
+ \nocite{ncg4pages}
+ \nocite{ncgshort}
+ \printbibliography
+ \end{frame}
+\end{document}
+
diff --git a/src/pres/pres.tex b/src/pres/pres.tex
@@ -1,264 +0,0 @@
-\documentclass[fleqn]{beamer}
-\beamertemplatenavigationsymbolsempty
-
-\usepackage[T1]{fontenc}
-\usepackage[utf8]{inputenc}
-
-\usepackage{amsmath,amssymb}
-\usepackage{graphicx}
-\usepackage{mathptmx}
-\usepackage{subcaption}
-\usepackage{amsthm}
-\usepackage{tikz}
-%\usepackage[colorlinks=true,naturalnames=true,plainpages=false,pdfpagelabels=true]{hyperref}
-\usetikzlibrary{patterns,decorations.pathmorphing,positioning, arrows, chains}
-
-\usepackage[backend=biber, sorting=none]{biblatex}
-\addbibresource{uni.bib}
-
-% vertical separator macro
-\newcommand{\vsep}{
- \column{0.0\textwidth}
- \begin{tikzpicture}
- \draw[very thick,black!10] (0,0) -- (0,7.3);
- \end{tikzpicture}
-}
-\setlength{\mathindent}{0pt}
-
-% Beamer theme
-\usetheme{UniVienna}
-\usefonttheme[onlysmall]{structurebold}
-\mode<presentation>
-\setbeamercovered{transparent=10}
-
-\title
-{Noncommutative Geometry}
-\subtitle{Bachelor's seminar}
-\author[Popovic Milutin]
-{Popovic Milutin \newline Supervisor: Dr. Lisa Glaser}
-\date{16. April 2021}
-
-\begin{document}
- \begin{frame}
- \titlepage
- \end{frame}
-
- \begin{frame}{Introduction}
- \begin{itemize}
- \item Noncommutative geometry (NCG) brings many\\
- mathematical fields together (e.g. K-Theory, Differential Geometry)
- \item Physics application (spectral Standard Model)
- \item Gelfand-Naimark-Theorem in Functional Analysis in the 1940s \\
- duality between (classical) geometry and Algebra
- \end{itemize}
- \end{frame}
-
- \begin{frame}{Spaces and Algebras}
- Introduce:
- \begin{block}
- {Algebra}
- \centering
- Vectorspace with a multiplication operation\\
- (associative and possesses an identity element)
- \end{block}
- \begin{block}
- {Finite topological Space $X$ consisting of $N$ points. (discrete topology)}
- \begin{figure}[h!]
- \centering
- \begin{tikzpicture}[
- dot/.style = {draw, circle, inner sep=0.05cm, fill},
- smalldot/.style = {draw, circle, inner sep=0.015cm,fill},
- ]
- \node[dot] at (-3,0.) [label=below:$1$]{};
- \node[dot] at (-1.5,0) [label=below:$2$]{};
- \node[dot] at (2.1,0) [label=below:$N$]{};
- \node[smalldot] at (-0.4,0) {};
- \node[smalldot] at (0.1,0) {};
- \node[smalldot] at (0.6,0) {};
- \end{tikzpicture}
- \end{figure}
-
- \end{block}
- \begin{block}{Commutative algebra of continuous functions on $X$}
- \centering
- $C(X) = \{ f: X \rightarrow \mathbb{C}:\;\;
- \text{$f$ is continuous}\}$
- \end{block}
- \end{frame}
- \begin{frame}
- {Spaces and commutative Algebras}
- Results of the Theorem:
- \begin{itemize}
- \item $X$ and $C(X)$ contain the same information (duality)
- \item Construct $X$, given $C(X)$.
- \item Translate geometrical properties of $X$ to algebraic data\\
- (metric, differential forms, vector fields, curvature, etc.)
- \end{itemize}
- \end{frame}
-
- \begin{frame}{Geometry as a Spectral Triple}
- \begin{block}
- {\centering The Spectral Triple}
- \centering
- $(\;A,\;\; H,\;\; D\;)$
- \end{block}
- \begin{itemize}
- \item $A$ - Algebra
- \item $H$ - Hilbertspace
- \item $D$ - self adjoint Operator acting on $H$
- \end{itemize}
- \end{frame}
-
- \begin{frame}{Geometry as a Spectral Triple}
- \begin{block}
- {The Spectral Triple of a Circle $\mathbb{S}^1$}
- \centering
- $ (\; C^{\infty}(\mathbb{S}^1),\;\; L^2(\mathbb{S}^1),\;\; -i\frac{d}{dt} \;)$
- \end{block}
- \end{frame}
-
- \begin{frame}{Introducing the Metric}
- \begin{itemize}
- \item The metric describes distances between points on a space
- \end{itemize}
- \begin{columns}[T]
- \column{0.4\textwidth}
- \begin{block}{\centering Discrete Metric}
- \centering
- $
- d_{ij} =
- \begin{cases}
- 0\;\;\; \text{if}\;\;\; i = j \\
- 1\;\;\; \text{if}\;\;\; i \neq j
- \end{cases}
- $
- \end{block}
-
- \column{0.4\textwidth}
- \begin{block}{\centering Minkowski Metric}
- \centering
- $
- \eta _{\mu \nu} =
- \begin{pmatrix}
- -1 & 0 & 0 & 0 \\
- 0 & 1 & 0 & 0 \\
- 0 & 0 & 1 & 0 \\
- 0 & 0 & 0 & 1
- \end{pmatrix}
- $
- \end{block}
- \end{columns}
-
- \begin{figure}[h!] \centering
- \begin{tikzpicture}[
- dot/.style = {draw, circle, inner sep=0.05cm, fill},
- smalldot/.style = {draw, circle, inner sep=0.015cm,fill},
- ]
- \node[dot](m1) at (-3,0.) [label=left:$1$] {};
- \node[dot](m2) at (-1.5, 2) [label=above right:$2$] {};
- \node[dot](m3) at (2.1,0) [label=right:$3$] {};
-
- \draw[<->, >=stealth](m1) -- ++(m2) node [midway, fill=white] {$d_{12}$};
- \draw[<->, >=stealth](m2) -- ++(m3) node [midway, fill=white] {$d_{23}$};
- \draw[<->, >=stealth](m3) -- ++(m1) node [midway, fill=white] {$d_{13}$};
- \end{tikzpicture}
- \end{figure}
-
- \end{frame}
-
-
- \begin{frame}{Algebraic Formulation of the Metric}
- \begin{itemize}
- \item Utilize results of the Gelfand-Naimark Theorem
- \item Characterize the Metric with\\
- \begin{itemize}
- \item[\bullet] commutative Algebra
- \item[\bullet] finite-dimensional Hilbertspace $H$
- \item[\bullet] symmetric operator $D$
- \end{itemize}
- \end{itemize}
- \begin{block}{Metric with $(A, H, D)$ on finite Space (commutative case)}
- \centering
- $d_{ij} = \sup_{a \in A}\{ |a(i) - a(j)| : ||[D, a]|| \leq 1\}$
- \end{block}
- \end{frame}
-
- \begin{frame}{Algebraic Formulation of the Metric}
- In the noncommutative Case:
- \begin{itemize}
- \item replace Algebra with matrix Algebra (noncommutative)
- \item define in terms of invariants
- \end{itemize}
- \begin{block}{Metric with $(A, H, D)$ on finite Space (noncommutative case)}
- \centering
- $d_{ij} = \sup_{a \in A}\{ |\text{Tr}(a(i)) - \text{Tr}(a(j))| :||[D, a]|| \leq 1\}$
- \end{block}
- \end{frame}
-
- \begin{frame}{Algebraic Formulation of the Metric}
- \begin{itemize}
- \item describe the Metric on a Manifold $M$
- \item We need \\
- \begin{itemize}
- \item[\bullet] $C^\infty(M)$ - Algebra
- \item[\bullet] $H^2(S)$ - Hilbertspace
- \item[\bullet] $D$ - Dirac Operator
- \end{itemize}
- \end{itemize}
- \begin{block}{Metric with $(C^\infty(M),\;\; H^2(S),\;\; D)$ on a Manifod}
- \centering
- $d(x, y) = \sup_{f \in C^\infty(M) }\{ |f(x) - f(y)| :
- ||[D, f]|| \leq 1\}$
- \end{block}
- \begin{figure}[h!] \centering
- \begin{tikzpicture}[
- dot/.style = {draw, circle, inner sep=0.06cm, fill},
- smalldot/.style = {draw, circle, inner sep=0.015cm,fill},
- ]
- \node[dot](b) at (0,0) [label=below left:$x$] {};
- \node[dot](a) at (2, 0) [label=below right:$y$] {};
- \node[dot](a) at (5, 0) [label=below left:$x$] {};
- \node[dot](a) at (8, 0) [label=below right:$y$] {};
- % \node[dot](m3) at (2.1,0) [label=right:$3$] {};
-
- \draw[<->, >=stealth, line width=0.4mm, style=dashed](0, 0.2) -- ++(2, 0) {};
- \draw[line width=0.5mm] (-0.3, 0) -- (2.3, 0) {};
-
- \draw[line width=0.5mm] (4.7, 0) -- (8.3, 0) {};
- \draw[<->, >=stealth, line width=0.4mm, style=dashed](8, 2) -- (8, 0.1) {};
- \draw[line width=0.5mm] (5, 0) -- (8, 2) node [pos=.75, label=:$f$] {} ;
- \end{tikzpicture}
- \end{figure}
- \end{frame}
-
-%\begin{frame}{Noncommutative Case}
-% \begin{itemize}
-% \item Introduce a richer geometry
-% \item From finite topological space to a Manifold with noncommutativity
-% \item From finite to general spectral triples with a \\
-% self adjoint Operator (Dirac Operator)
-% \end{itemize}
-% \end{frame}
-
- \begin{frame}{Applications In Physics}
- \begin{itemize}
- \item NCG of the Quantum Hall Effect
- \item NCG of the Standard Model
- \begin{itemize}
- \item[\bullet] going to noncommutative Manifolds
- \item[\bullet] obtain Standard Model gauge fields (scalar Higgs filed)
- \item[\bullet] construct the Full Lagrangian
- \item[\bullet] minimal coupling to gravity
- \end{itemize}
- \end{itemize}
- \end{frame}
-
- \begin{frame}{Bibliography}
- \nocite{ncgwalter}
- \nocite{liealgebra}
- \nocite{ncg4pages}
- \nocite{ncgshort}
- \printbibliography
- \end{frame}
-\end{document}
-
diff --git a/src/week5.tex b/src/week5.tex
@@ -325,20 +325,14 @@ Then there is a map $d:A\rightarrow \Omega _D ^1 (A)$, $d = [D, \cdot]$.
\end{align*}
\newline
- % First off we know the algebra is associative then we know that elements
- % in $A$ can be represented faithfully on a Hilbert space $H$. Because of
- % the Hilbert Basis $\{\textbf{n}_i\}_{i\in \mathbb{N}}$ of the Hilbert space we can decompose these elements
- % in therms of the basis elements.
- % \begin{align*}
- % aa_k &= \sum _{\textbf{n}}(\langle a, \textbf{n} \rangle) a_k \\
- % &= \sum _{k} a'_{k}
- % \end{align*}
- % Which would than be the same as the sum of some elements
- % $a'_{k} \in A$. Then we calculate the commutator:
- % \begin{align*}
- % [D, b_k] b = d(b_k)b = d(b_kb) - b_kd(b)\\
- % \end{align*}
- %I don't think this is correct I'll try it again
+ Begin
+ \begin{align*}
+ a(a_k[D, b_k])b &= a_k'(Db_k - b_k D) b = \\
+ &= a_k'(Db_k b - b_k D b) = a_k(Db_k b - b_k Db -Db_k +Db_kb)=\\
+ &= a_k'([D, b_kb] - b_k D b + D b_k + \cdots - \cdots) = \\
+ &=\sum_k a_k' [D, b_k']
+ \end{align*}
+
\end{MyExercise}
\begin{lemma}
diff --git a/src/week6.tex b/src/week6.tex
@@ -149,6 +149,7 @@ vector space with the opposite product
\end{align}
\end{definition}
+
\begin{example}
Matrix algebra $M_N(\mathbb{C})$ acting on $H=M_N(\mathbb{C})$ by left
matrix multiplication with the Hilbert Schmidt inner product.
@@ -158,6 +159,19 @@ vector space with the opposite product
Then we define $\gamma (a) = a$ and $J(a) = a^*$ with $a\in H$.
Since $D$ mus be odd with respect to $\gamma$ it vanishes identically.
\end{example}
+
+\begin{definition}
+ We call $\xi \in H$ \textbf{cyclic vector} in $A$ if:
+ \begin{align}
+ A\xi := { a\xi:\;\; a\in A} = H
+ \end{align}
+
+ We call $\xi \in H$ \textbf{separating vector} in $A$ if:
+ \begin{align}
+ a\xi = 0\;\; \Rightarrow \;\; a=0;\;\;\; a\in A
+ \end{align}
+\end{definition}
+
\begin{MyExercise}
\textbf{
In the previous example, show that the right action on $M_N(\mathbb{C})$
@@ -166,10 +180,65 @@ vector space with the opposite product
}\newline
\begin{align*}
- a^\circ \xi = J a^* J^{-1} = Ja^* \xi^* = J\xi a=\xi^* a
+ a^\circ \xi = J a^* J^{-1}\xi = Ja^* \xi^* = J\xi a=\xi^* a
\end{align*}
\end{MyExercise}
-
+\begin{MyExercise}
+ \textbf{
+ Let $A= \bigoplus _i M_{n_i}(\mathbb{C})$, represented on $H = \bigoplus_i \mathbb{C}^{n_i}
+ \otimes \mathbb{C}^{m_i}$, meaning that the irreducible representation $\textbf{n}_i$ has
+ multiplicity $m_i$.
+ \begin{enumerate}
+ \item Show that the commutant $A'$ of $A$ is $A'\simeq \bigoplus_i M_{m_i} (\mathbb{C})$. As a consequence show $A'' \simeq A$.
+ \item Show that if $\xi$ is a separating vector for $A$ than it is cyclic for $A'$.
+ \end{enumerate}
+ }\newline
+
+
+ \begin{enumerate}
+ \item We know the multiplicity space is $V_i = \mathbb{C}^{m_i}$. We know that
+ for $T\in H$ and
+ $a\in A'$ to work we need $aT=Ta$ by laws of matrix multiplication we need
+ $A' \simeq \oplus _i M_{m_i}(\mathbb{C})$ for this to work since $H = \bigoplus_i
+ \mathbb{C}^{n_i}
+ \otimes \mathbb{C}^{m_i}$
+
+ \item Suppose $\xi$ is cyclic for $A$ then $A'\xi = \{0\}$. Under the action of $A$ we
+ then have $A'A\xi = AA' \xi = 0 \Rightarrow A' = 0$.\\
+ Suppose now $\xi$ is separating for $A'$, we have $A'\xi = \{0\}$. We can define a
+ projection in $A'$, $A\xi = P'$. With this projection we have $(1-P')\xi = 0
+ \Rightarrow 1-P' = 0 \Rightarrow A\xi = H$.
+ \end{enumerate}
+\end{MyExercise}
+\begin{MyExercise}
+ \textbf{ Suppose $(A, H, D = 0)$ is a finite spectral triple such that $H$ possesses a
+ cyclic and separating vector for $A$.
+ \begin{enumerate}
+ \item Show that the formula $S(a \xi) = a* \xi$ defines a anti-linear operator\\
+ $S: H \rightarrow H$.
+ \item Show that $S$ is invertible
+ \item Let $J: H \rightarrow H$ be the operator in $S = J \Delta ^{1/2}$ with
+ $\Delta = S*S$. Show that $J$ is anti-unitary
+ \end{enumerate}
+ }\newline
+
+
+ \begin{enumerate}
+ \item By composition $S(a\xi) = a*\xi$ this is literally anti-linearity. Does this mean
+ $S\xi = \xi$?
+ \item Let $\xi \in H$ be cyclic then: $S(A\xi) = A*\xi = A\xi = H$. The same has to work
+ for $S^{-1}$ if not then $\xi$ wouldn't exist. $S^{-1}(A*\xi) = S^{-1}(H) = H$.
+ \item Since $S$ is bijective then $\Delta ^{1/2}$ and $J$ need to be bijective.\\
+ Now let $\xi _1 , \xi _2 \in H$.\\
+ \begin{align*}
+ <J \xi _1 , J \xi _2 > &= < J^*J\xi_1 , \xi_2>^* =\\
+ &= <(\Delta ^{1/2})^* S^* S \Delta ^{1/2} \xi_1, \xi_2>^* = \\
+ &= <(SS^*)^{1/2}S^*S(SS^*) \xi_1, \xi_2>^* =\\
+ &= <(SS^*SS^*)^{1/2} \xi_1, \xi_2>^* = \\
+ &= <\xi _1, \xi_2>^* = <\xi_2 , \xi_1>.
+ \end{align*}
+ \end{enumerate}
+\end{MyExercise}
\subsection{Morphisms Between Finite Real Spectral Triples}
Extend unitary equivalence of finite spectral triples to real ones (with $J$
and $\gamma$)
@@ -210,6 +279,38 @@ and linearity in $A$:
\langle a \bar{e}_1, \bar{e}_2 \rangle = a \langle \bar{e}_1, \bar{e}_2
\rangle \;\;\;\; \forall a \in A.
\end{align}
+
+\begin{MyExercise}
+ \textbf{Show that $E^\circ$ is a Hilbert bimodule $(B^{\circ}, A^{\circ})$
+ }\newline
+
+
+ Straightforward show properties of the Hilbert bimodule and its $B^{\circ}$
+ valued inner product. Let $\bar{e}_1, \bar{e}_2 \in E^{\circ}$ and $a^\circ \in A,
+ b^\circ \in B$. \\
+ \begin{align*}
+ <\bar{e}_1, a^\circ \bar{e}_2> &= <\bar{e}_1, Ja^*J^{-1} \bar{e}_2>=\\
+ &= <\bar{e}_1 , J a^* e_2> = \\
+ &= <J^{-1} e_1, a^* e_2> =\\
+ & = <a^* e_1, e_2>= <J^{-1}(a^\circ)^* J e_1, e_2> = \\
+ & = <J^{-1} (a^\circ)^* \bar{e}_1, e_2> =\\
+ & = <(a^\circ)^* \bar{e}_1 , \bar{e}_2>.
+ \end{align*}
+
+ Next $<\bar{e}_1, \bar{e}_2 b^\circ> = <\bar{e}_1, \bar{e_2}> b^\circ$.
+ \begin{align*}
+ <\bar{e}_1, \bar{e}_2 b^\circ> &= <\bar{e}_1, \bar{e}_2 Jb^*J^{-1}> =\\
+ &= <\bar{e}_1, \bar{e_2}> Jb^*J^{-1} = \\
+ &= <\bar{e}_1, \bar{e}_2> b^\circ.
+ \end{align*}
+ Then:
+ \begin{align*}
+ (<\bar{e}_1, \bar{e}_2)>_{E^\circ})^* &= (<e_2, e_1>_E)^* =\\
+ &= <e_1, e_2>_E^* = <\bar{e}_2, \bar{e}_2>_{E^\circ}
+ \end{align*}
+ And of course $<\bar{e}, \bar{e}> = <e, e> \geq 0$
+\end{MyExercise}
+
\subsubsection{Construction of a Finite Real Spectral Triple from a Finite
Real Spectral Triple}
Given a Hilbert bimodule $E$ for $(B, A)$ we construct a spectral triple
@@ -236,6 +337,34 @@ $E^\circ$
with $b^\circ = J' b^* (J')^{-1}$, $b^* \in B$ action on $H'$.
\newline
+
+\newpage
+\begin{MyExercise}
+ \textbf{ Let $\nabla : E \Rightarrow E \otimes _A \Omega _d^1 (A)$ be a right connection on $E$
+ consider the following anti-linear map:
+ \begin{align}
+ \tau : E \otimes_A \Omega _D^1 (A) &\rightarrow \Omega _D^1 (A) \otimes_A E^\circ\\
+ e \otimes \omega &\mapsto -\omega ^* \otimes \bar{e}
+ \end{align}
+ Show that the map $\bar{\nabla} : E^\circ \righarrow \Omega _D^1(A) \otimes E^\circ$
+ with $\bar{\nabla}(\bar{e}) = \tau \circ \nabla(e)$ is a left connection, that means
+ show that it satisfied the left Leibniz rule:
+ \begin{equation}
+ \bar{\nabla}(a\bar{e}) = [D, a] \otimes \bar{e} + a \bar{\nabla}(\bar{e})
+ \end{equation}
+ }\newline
+
+
+ Hagime:
+ \begin{align*}
+ &\text{For one:}\\
+ &\tau \circ \nabla(ae) = \bar{\nabla}(a\bar{e}) = \bar{\nabla}(a^* \bar{e})\\
+ &\text{For two:}\\
+ &\tau \circ \nabla(ae) = \tau(\nabla(e)a) + \tau \circ(e \otimes d(a))=\\
+ &=a^*\bar{\nabla}(\bar{e}) - d(a)^* \otimes \bar{e}. \\
+ &= a^*\bar{\nabla}(\bar{e}) + d(a^*) \otimes \bar{e}.
+ \end{align*}
+\end{MyExercise}
Then the connections
\begin{align}
&\nabla: E \rightarrow E\otimes _A \Omega _D ^1(A) \\
