Instead of doing geometry, we were doing linear algebra. Let us now generalize this linear algebra before doing any geometry. This means looking at \( V \oplus V^* \).
Notice that we have not added an arbitrary vector space to \( V \), we have added its dual space \( V^* \) and we cannot dismiss this. How do we keep track of this fact in a nice way? By defining a (canonical) pairing.
This is a good moment to introduce some conventions. We will normally use the capital letters \( X,Y \) for elements of \( V \), and the greek letters \( \xi, \eta \) (xi and eta) for elements of \( V^* \). For the elements of \( V\oplus V^* \) we will use \( v, w \). Thus, \( v=X+\xi, w=Y+\eta \) are the usual elements of \( V\oplus V^* \).
And finally the pairing \( \langle \; , \; \rangle \):
$$ \langle X+\xi, Y+\eta \rangle = \frac{1}{2} (i_X \eta + i_Y \xi ).$$
Note that \( \langle X+\xi, X+\xi \rangle = i_X \xi \).
This pairing has signature \( (n,n) \). Given bases \( \{ e_1,\ldots,e_n\} \) and \( \{ e^1, \ldots, e^n \} \) of \( V \) and \( V^* \), the matrix of the pairing is written as
$$ \left( \begin{array}{cc} 0 & \frac{1}{2}\textrm{Id} \\ \frac{1}{2}\textrm{Id} & 0 \end{array} \right) $$
for the basis \( \{ e_1,\ldots,e_n, e^1, \ldots, e^n \} \) of \( V \oplus V^* \), or as
$$ \left( \begin{array}{cc} \textrm{Id} & 0 \\ 0 & -\textrm{Id} \end{array} \right) $$ for the basis \( \{ e_1+e^1,\ldots,e_n+e^n, e^1-e^1,\ldots,e_n-e^n \} \), from where we clearly see the signature.
Note that the pairing between any two elements of V, or any two elements of V^* is zero. This is what we call being isotropic.
Definition: A vector subspace \( L \) of a metric vector space \( (W,\langle\;,\;\rangle ) \) is said to be isotropic if \( \langle x, y \rangle = 0 \) for all \( x,y \in L \).
Notice that we have not added an arbitrary vector space to \( V \), we have added its dual space \( V^* \) and we cannot dismiss this. How do we keep track of this fact in a nice way? By defining a (canonical) pairing.
This is a good moment to introduce some conventions. We will normally use the capital letters \( X,Y \) for elements of \( V \), and the greek letters \( \xi, \eta \) (xi and eta) for elements of \( V^* \). For the elements of \( V\oplus V^* \) we will use \( v, w \). Thus, \( v=X+\xi, w=Y+\eta \) are the usual elements of \( V\oplus V^* \).
And finally the pairing \( \langle \; , \; \rangle \):
$$ \langle X+\xi, Y+\eta \rangle = \frac{1}{2} (i_X \eta + i_Y \xi ).$$
Note that \( \langle X+\xi, X+\xi \rangle = i_X \xi \).
This pairing has signature \( (n,n) \). Given bases \( \{ e_1,\ldots,e_n\} \) and \( \{ e^1, \ldots, e^n \} \) of \( V \) and \( V^* \), the matrix of the pairing is written as
$$ \left( \begin{array}{cc} 0 & \frac{1}{2}\textrm{Id} \\ \frac{1}{2}\textrm{Id} & 0 \end{array} \right) $$
for the basis \( \{ e_1,\ldots,e_n, e^1, \ldots, e^n \} \) of \( V \oplus V^* \), or as
$$ \left( \begin{array}{cc} \textrm{Id} & 0 \\ 0 & -\textrm{Id} \end{array} \right) $$ for the basis \( \{ e_1+e^1,\ldots,e_n+e^n, e^1-e^1,\ldots,e_n-e^n \} \), from where we clearly see the signature.
Note that the pairing between any two elements of V, or any two elements of V^* is zero. This is what we call being isotropic.
Definition: A vector subspace \( L \) of a metric vector space \( (W,\langle\;,\;\rangle ) \) is said to be isotropic if \( \langle x, y \rangle = 0 \) for all \( x,y \in L \).
Hello Roberto. There is a typo in the definition of isotropic subspace above.
ReplyDeleteFixed. Thank you!
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