\documentclass[11pt]{article} \usepackage{url} %version 1.0,\ \ 13.03.2018, taken from pde-en-11.tex %version 1.1,\ \ 13.03.2018, couple of misprints from EA %version 1.2,\ \ 17.03.2018, last exercise slightly prettified \newcommand{\version}{version 1.2,\ \ 17.03.2018} \newcommand{\firstdate}{14.03.2018} \addtolength{\topmargin}{-10mm} \addtolength{\textheight}{20mm} \addtolength{\oddsidemargin}{-5mm} \addtolength{\textwidth}{10mm} \input{defs-listki-en.tex} \setlength{\headheight}{15pt} \pagestyle{fancy} \lhead{\tiny Hogde theory, HSE} \lfoot{\tiny Issued \firstdate} \cfoot{-- \thepage \ -- } \rfoot{\tiny \sc\version} \rhead{{\tiny Misha Verbitsky}} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \listok{8}{Hodge theory 8: Cartan's formula and Poincar\'e lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\scriptsize {\bf Rules:} You may choose to solve only ``hard'' exercises (marked with !, * and **) or ``ordinary'' ones (marked with ! or unmarked), or both, if you want to have extra stuff to work. To have a perfect score, a student must obtain (in average) a score of 10 points per week. If you have got credit for 2/3 of ordinary problems or 2/3 of ``hard'' problems, you receive $6t$ points, where $t$ is a number depending on the date when it is done. Passing all ``hard'' or all ``ordinary'' problems brings you $10t$ points. Solving of ``**'' (extra hard) problems is not obligatory, but each such problem gives you a credit for 2 ``*'' or ``!'' problems in the ``hard'' set. The first 3 weeks after giving a handout, $t=1.5$, between 21 and 35 days, $t=1$, and afterwards, $t=0.7$. The scores are not cumulative, only the best score for each handout counts. } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Lie derivative} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition An associative algebra $A^* = \oplus_{i\in \Z}A^i$ is called {\bf a graded algebra} if for all $a\in A^i$, $b\in A^j$, the product $ab$ lies in $A^{i+j}$. \ed \definition Let $A^* = \oplus_{i\in \Z}A^i$ be a graded algebra over a field. It is called {\bf graded commutative}, or {\bf supercommutative}, if $ab = (-1)^{ij} ba$ for all $a\in A^i, b \in A^j$. \ed \remark Grassmann algebra $\Lambda^* V$ is clearly supercommutative. \er %\exercise %Let $A^*, B^*$ be graded commutative algebras, %and $A^* \otimes B^*$ their tensor product, %with a grading %$(A^* \otimes B^*)^p := \oplus_{i+j=p} A^i\otimes B^j$, %and multiplication, defined as %$a \otimes b \cdot a' \otimes b' = (-1)^{ij}aa' \otimes bb'$, %where $a'\in A^i, b \in B^j$. %Prove that it is supercommutative. %\ez % %\exercise %Let $V, W$ be vector spaces, and %$A^*:= \Lambda^* V, B^*:=\Lambda^* W$ their Grassmann algebras. %Prove that $\Lambda^*(V\oplus W)$ is isomorphic %to the tensor product $A^* \otimes B^*$, %defined as above. %\ez \definition Let $A^*$ be a graded commutative algebra, and $D:\; A^* \arrow A^{*+i}$ a map which shifts grading by $i$. It is called a {\bf graded derivation} if $D(ab) = D(a) b + (-1)^{ij} a D(b)$, for each $a \in A^j$. \ed \remark If $i$ is even, graded derivation is just a derivation. If it is odd, it is called {\bf odd derivation}. \er \remark De Rham differential is an odd derivation, by definition. \er \definition Let $M$ be a smooth manifold, and $X\in TM$ a vector field. Consider an operation of {\bf convolution with a vector field} $i_X:\; \Lambda^i M \arrow \Lambda^{i-1}M$, mapping an $i$-form $\alpha$ to an $(i-1)$-form $v_1, ..., v_{i-1} \arrow \alpha(X, v_1, ..., v_{i-1})$ \ed \exercise Prove that $i_X$ is an odd derivation. \ez \exercise[*] Let $D:\; A^* \arrow A^{*+i}$ be a linear map such that for all $x\in A$ there exists $N$ such that $D^N(x)=0$. Prove that $e^D:= 1 + D + \frac {D^2}{2} + ... + \frac{D^i}{i!} + ... $ is an automorphism of $A^*$ if and only if $D$ is a derivation. \ez \definition Let $A^*$ be a graded vector space, and $E:\; A^*\arrow A^{*+i}$, $F:\; A^*\arrow A^{*+j}$ operators shifting the grading by $i, j$. Define {\bf the supercommutator} by the formula \[ \{E, F\}:= EF - (-1)^{ij} FE. \] \ed \remark An endomorphism which shifts a grading by $i$ is called {\bf even} if $i$ is even, and {\bf odd} otherwise. \er \remark Supercommutator satisfies {\bf graded Jacobi identity,} \[ \{ E, \{F, G\}\} = \{\{ E, F\}, G\} + (-1)^{\tilde E \tilde F} \{ F, \{E, G\}\} \] where $\tilde E$ and $\tilde F$ are 0 if $E, F$ are even, and 1 otherwise. \er \remark There is a simple mnemonic rule which allows one to remember a superidentity, if you know the commutative analogue. Each time when in commutative case two letters $E$, $F$ are exchanged, in supercommutative case one needs to multiply by $(-1)^{\tilde E\tilde F}$. \er \exercise Let $A^*$ be a graded commutative algebra and $a\in A$. Denote by $L_a:\; A\arrow A$ the operation of multiplication by $a$: $L_a(b) = ab$. Prove that a map $D:\; A^*\arrow A^*$ is a superderivation if and only if $D(1)=0$ and for each $a\in A^i$, the supercommutator $\{D, L_a\}$ is equal to $L_b$ for some $b\in A^*$. \ez \exercise[!] Prove that a supercommutator of superderivations is again a superderivation. \ez \hint Use the Jacobi identity and apply the previous exercise. \eh \definition Let $B$ be a smooth manifold, and $v\in TM$ a vector field. An endomorphism $\Lie_v:\; \Lambda^* M \arrow \Lambda^* M$, preserving the grading is called {\bf a Lie derivative along $v$} if it satisfies the following conditions. \begin{description} \item[(i)] On functions $\Lie_v$ is equal to a derivative along $v$. \item[(ii)] $[\Lie_v, d]=0$ \item[(iii)] $\Lie_v$ is a derivation on the de Rham algebra. \end{description} \ed \exercise \label{_derivations_gene_Exercise_} Let $\nu_1, \nu_2:\; \Lambda^*(M) \arrow \Lambda^*(M)$ be derivations of the de Rham algebra. Suppose that $\nu_1$ is equal to $\nu_2$ on $C^\infty M=\Lambda^0(M)$ and on $d(C^\infty M)$. Prove that $\nu_1=\nu_2$. \ez \hint $\Lambda^*(M)$ is generated (multiplicatively) by $C^\infty M=\Lambda^0(M)$ and $d(C^\infty M)$. \eh \exercise Prove that the Lie derivative is uniquely determined by the properties (i)-(iii). \ez \hint Use the previous exercise. \eh \exercise Prove that $\{d, \{d, E\}\}=0$, for each $E\in \End(\Lambda^* M)$. \ez \hint Use the graded Jacobi identity. \eh \exercise Prove that $\{d, i_v\}$ commutes with $d$, where $i_v:\; \Lambda^* M \arrow \Lambda^{*-1}M$ is a convolution with $v$. \ez \hint Use the previous exercise. \eh \exercise[!] (Cartan formula) Prove that $\{d, i_v\}$ is a Lie derivative along $v$. \ez %\exercise[*] %Let $v, v'\in TM$ be two vector fields, and %$i_{v\otimes v'}:\; \Lambda^* M \arrow \Lambda^{*-2}M$ %is a substitution of $v, v'$ into a 2-form, %$i_{v\otimes v'}= i_v i_{v'}$. %Consider an $i$-form $\alpha \in \Lambda^* M$, %and let $L_\alpha$ be an operator of multiplication by $\alpha$. %Prove that an operator %\[ %x\arrow [i_{v\otimes v'}, L_\alpha](x) - L_{i_{v\otimes %v'}}(\alpha)\wedge x %\] %is a derivation. %\ez \exercise[*] Let $\tau:\; \Lambda^*(M) \arrow \Lambda^{*-1}(M)$ be a derivation shifting grading by $-1$. Prove that there exists a vector field $v\in TM$ such that $\tau=i_v$, or find a counterexample. \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Poincar\'e lemma} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \exercise Let $t$ be the coordinate function on a real line, $f(t)\in C^\infty \R$ a smooth function, and $v:= t \frac d {dt}$ a vector field. Define \[ R(f)(t):=\int^1_0 \frac{f(\lambda t)}{\lambda} d\lambda \] Prove that this integral converges whenever $f(0)=0$, and satisfies $\Lie_v R(f)=f$ in this case. \ez \exercise Let $t_1, ..., t_n$ be coordinate functions in $\R^n$, and $\vec{r}:=\sum_i t_i \frac d {dt_i}$ a radial vector field. Consider a function $f\in C^\infty \R^n$ satisfying $f(0)=0$, and let $x=(x_1, ..., x_n)$ be any point in $\R^n$. Prove that an integral \[ R(f)(x):=\int_0^1 \frac{f(\lambda x)}{\lambda} d\lambda \] converges, and satisfies $\Lie_{\vec{r}} R(f)=f$. \ez \hint Use the previous exercise. \eh \definition An open subset $U\subset \R^n$ is called {\bf starlike} if for any $x\in U$ the interval $[0,x]$ belongs to $U$. \ed \exercise[!] Let $U$ be a starlike subset in $\R^n$, and $i>0$. Construct an operator \[ R:\; \Lambda^i U \arrow \Lambda^i U\] which satisfies $\Lie_{\vec{r}} R\alpha=R\Lie_{\vec{r}} \alpha = \alpha$ for each $\alpha \in\Lambda^i U$. \ez \hint Define the integral $R(\alpha)$ as in the previous exercise, and check that it converges. Prove that $\Lie_{\vec{r}} R(\alpha)=\alpha$. \eh \exercise \label{_R_kernel_Zadacha_} Prove that any form $\alpha\in \Lambda^i U$ on a starlike set $U$ satisfying $\Lie_{\vec{r}} \alpha=0$ vanishes if $i>0$. \ez \hint Use the previous exercise. \eh \exercise[!] Prove that $\{R, d\}=0$ \ez \hint Check that \[ \{R, d\} \Lie_{\vec{r}} \alpha= Rd \Lie_{\vec{r}} \alpha + dR \Lie_{\vec{r}} \alpha = - R\Lie_{\vec{r}} d\alpha + d\alpha = 0. \] For any $\beta \in \ker \{R, d\}\cap \Lambda^iM$, satisfying $i>0$ or $\beta(0)=0$ for $i=0$, solve an equation $\Lie_{\vec{r}}\alpha=\beta$. \eh \exercise Prove that $\{ d, i_{\vec{r}}\} R(\alpha) =\alpha$, for any $i$-form $\alpha$ on a starlike set, $i>0$. \ez \definition Let $d$ be de Rham differential. A form in $\ker d$ is called {\bf closed}, a form in $\im d$ is called {\bf exact}. Since $d^2=0$, any exact form is closed. {\bf The group of $i$-th de Rham cohomology of $M$}, denoted $H^i(M)$, is a quotient of a space of closed $i$-forms by exact: $H^*(M)=\frac{\ker d}{\im d}$. \ed \exercise[!] Let $\alpha\in \Lambda^i U$ be a closed $i$-form on a starlike set $U$, with $i>0$. Prove that $\alpha = d i_{\vec{r}} R(\alpha)$. \ez \hint Use the previous exercise. \eh \exercise[!] (Poincar\'e lemma) Let $U$ be a starlike set. Prove that $H^i(U)=0$ for each $i>0$, and $H^0(M)=\R$. \ez \exercise Let $\theta$ be a closed odd form, and $d_\theta(x)=dx+\theta\wedge x$ the corresponding operator on $\Lambda^*M$. Its {\bf cohomology} are defined as $H^*(\Lambda^*(M), d_\theta):=\frac{\ker d_\theta}{\im d_\theta}$ \enum \ite Show that $d_\theta^2=0$. \ite[*] Let $\theta$ be an exact 1-form. Prove that $H^i(\Lambda^*(M), d_\theta)$ are isomorphic to $H^i(M)$. \ite[*] Let $\theta$ be a closed 1-form. Prove that $H^i(\Lambda^*(M), d_\theta)$ are isomorphic to $H^i(M)$, or find a counterexample. \ite[*] Let $\theta$ be a closed 3-form. Prove that $H^*(\Lambda^*(M), d_\theta)$ are isomorphic to $H^*(M)$, or find a counterexample. \ee \ez \end{document}