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\lhead{\tiny Hyperk\"ahler manifolds, HSE, Fall 2019} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  Class assignment \ \ \sc\version}
\rhead{{\tiny  Hyperk\"ahler manifolds, Misha Verbitsky}}


\begin{document}

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\listok{3}{Class assignment 3: spinors and Clifford algebras}

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\definition
{\bf The Clifford algebra} of a vector space $V$ with
a scalar product $q$ is an algebra generated by $V$ with 
a relation $xy+yx = -2q(x,y) 1$. 
\ed


\exercise
Let $A=A_\even\oplus A_\odd$ be a graded associative algebra.
Let $A^\bot$ be the same vector space 
with new multiplication 
$a \bullet a':= (-1)^{\tilde a \tilde a'} aa'$.
Prove that $\Cl(V,g)^\bot= \Cl(V,-g)$.
\ez

\definition
Let $(V,g)$ be an oriented real vector space with orthogonal basis
 $e_1, ..., e_n$ such that $g(e_i, e_i) =\pm 1$.
{\bf A unit pseudoscalar} in $\Cl(V,g)$ is 
$\epsilon:= e_1e_2e_3 ... e_n$.
\ed

\exercise
Prove that the pseudoscalar is invariant
with respect to the natural $SO(n)$-action.
Prove that it is defined uniquely up to a sign.
\ez

\exercise
Prove the isomorphism  ${\Bbb H} \otimes_\R {\Bbb H} = \Mat(4, \R)$.
\ez

\exercise Let $V$ be a vector space over a field
of characteristic 0.
\enum 
\ite Prove that the automorphism group
$\Aut(\Mat(V))$ is isomorphic to 
$PGL(V)$ (the quotient of $GL(V)$ by its center).
\ite[*] Is it true for all fields?
\ee
\ez



\exercise
Prove that $\Spin(3, \C)\cong SL(2, \C)$.
\ez

\definition
Define {\bf the real spinor group
 $\Spin(n, \R)$} as a subgroup of the cover
$\Spin(n, \C)\arrow SO(n, \C)$,
fixed by the natural anticomplex involution,
obtained from the standard anticomplex involution of
$SO(n,\C)$.
\ed

\exercise
Prove that this involution is always lifted to an involution of
$\Spin(n, \C)$.
\ez

\exercise
Prove that $\Spin(3, \R) \cong SU(2)$.
\ez

\exercise
Prove that $\Spin(n,\R)$ is a non-trivial 2-sheeted
cover of $SO(n)$ for all $n \geq 3$.
\ez




\end{document}
