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Let $V$ be a vector space of dimension $2n$ and $f$ be a non-degenerate skew-symmetric bilinear form on $V$. $V'$ is a subspace of $V$, and the restriction of $f$ on $V'$ is of rank $2k$ for an integer $k$. Then what is the meaning of the integer $n+k-\operatorname{dim}V'$?

This question might not be clear, but in fact, it comes from my considering on the maximal subalgebra of Lie superalgebras.

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If we let $W$ denote a subspace of $V$ containing $V'$ and of rank $2k$, which is maximal with respect to these two properties, then the dimension of $W$ will be $n + k$, and so $n + k - \dim V'$ will be the codimension of $V'$ in $W$.

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I got it. Thank you very much. – QiWang Jun 24 '12 at 2:32

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