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\lhead{\tiny Complex geometry, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{6}{Complex geometry handout 6: Lie superalgebras}

\definition
Let $A^*$ be a graded commutative algebra, 
and $D:\; A^* \arrow A^{*+i}$ a map which
shifts the grading by $i$. 
It is called a
{\bf graded derivation}, or {\bf superderivation}, if
$D(ab) = D(a) b + (-1)^{ij} a D(b)$,
for each $a \in A^j$. 
\ed

\exercise
Prove that a supercommutator of superderivations
is again a superderivation.
\ez


\exercise
Let $\tau:\; \Lambda^*(M) \arrow \Lambda^{*-1}(M)$
be an odd derivation shifting the grading by $-1$.
Prove that there exists a vector field $v\in TM$ such that
$\tau=i_v$ (convolution with a vector field), 
or find a counterexample.
\ez 

\exercise
Let $\tau:\; \Lambda^*(M) \arrow \Lambda^{*-2}(M)$
be a derivation shifting the grading by $-2$.
Prove that $\tau=0$.
\ez

\definition
Let $A^*$ be a graded commutative algebra over a field $k$.
{\bf Differential operators}
on $A^*$ are $k$-linear operators $D:\; A^*\arrow A^*$ (even or odd), defined
inductively as follows. {\bf Differential operators of order
0} are maps $L_a(x)=ax$, where $a\in A^*$ (also even or odd). 
{\bf Differential operators of order
$p$} are maps $u:\; A^*\arrow A^*$ such that $\{L_a, u\}$
is a differential operator of order $p-1$ for all $a\in A^*$.
\ed


\exercise
Let $D:\; A^*\arrow A^*$ be a differential operator of
order 1, and $a=D(1)$. Prove that $D-L_a$ is a 
super-derivation of $A^*$.
\ez

\exercise
Let $\omega\in \Lambda^2 V^*$ be a 2-form on a vector space $V$,
$\nu\in \Lambda^2 V$ a bivector, $L_\omega(\eta):=
\omega\wedge\eta$ and $\Lambda_\nu:\; \Lambda^i(V^*) \arrow \Lambda^{i-2}(V^*)$
the convolution of a differential form and a bivector.
Let $A\in \End(\Lambda^*(V^*))$ 
be the multiplication by a constant $-\Lambda_\nu(\omega)$.
Prove that $[L_\omega, \Lambda_\nu]-A$ 
is an even derivation of $\Lambda^*(V^*)$.
\ez

\exercise
Let $V$ be the fundamental representation
of $\Sp(V)$, and $A\in \End(V)$ its automorphism commuting
with $\Sp(V)$-action. Prove that 
$A$ is a constant.
\ez 

\exercise
Let $\omega\in \Lambda^2 V^*$ be a Hermitian 2-form on a $n$-dimensional
complex vector space $V$, and $L, \Lambda$ the corresponding Lefschetz operators
(Lecture 11).
\enum
\ite 
Prove that $[L, \Lambda]\restrict{\Lambda^1 V^*}=\alpha\Id$
for some scalar $\alpha$.
\ite Prove that $\Lambda(\omega)=n$. 
Deduce from this that $[L, \Lambda]+n$ is a derivation of $\Lambda^*(V^*)$.

\ite Deduce that $n-[L, \Lambda]$ acts on $k$-forms
as a multiplication by $k\alpha$, where $\alpha$ is a constant
given by $[L, \Lambda]\restrict{\Lambda^1 V^*}+n=\alpha\Id$.

\ite Prove that $[L, \Lambda]\restrict{\Lambda^{2n} V^*}=n\Id$.

\ite Deduce that $[L, \Lambda]\restrict{\Lambda^k V^*}=(k\alpha-n)\Id$,
where $\alpha=1$.
\ee
\ez
\remark This gives another proof of
the identity $[L, \Lambda]\restrict{\Lambda^k V^*}=(n-k)\Id$ .
\er


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