Once we have a vector space \( V \oplus V^* \) that comes with a pairing of signature \( (n,n) \), we would like to know what its symmetries are.
By symmetries we mean bijections preserving the structure. For a set without any structure we have that the symmetries are all the possible permutations of that set. For a vector space, the symmetries are the (linear) automorphisms, \( \textrm{Aut}(V) \) or \( \textrm{GL}(V) \), permutations compatible with the sum and the product by scalars. For \( V\oplus V^* \) we have
$$ \textrm{O}(V\oplus V^*) = \{ A\in GL(V\oplus V^*) : \langle Av,Aw \rangle = \langle v,w \rangle \textrm{ for } v,w \in V\oplus V^* \}, $$
i.e., bijections preserving the vector space structure that also preserve the metric. The groups \( \textrm{GL}(V) \) and \( \textrm{O}(V\oplus V^*)\) are Lie groups: they are both group and manifold in a nice compatible way. Neither a group or a manifold are objects easy to handle, so we will do something similar to what we do in geometry: linearize, look at the tangent space (in this case of the identity of the group)!
Let's look then at the Lie algebra \( \mathfrak{o}(V\oplus V^* ) \). What does it satisfy? How can we describe its elements?
By symmetries we mean bijections preserving the structure. For a set without any structure we have that the symmetries are all the possible permutations of that set. For a vector space, the symmetries are the (linear) automorphisms, \( \textrm{Aut}(V) \) or \( \textrm{GL}(V) \), permutations compatible with the sum and the product by scalars. For \( V\oplus V^* \) we have
$$ \textrm{O}(V\oplus V^*) = \{ A\in GL(V\oplus V^*) : \langle Av,Aw \rangle = \langle v,w \rangle \textrm{ for } v,w \in V\oplus V^* \}, $$
i.e., bijections preserving the vector space structure that also preserve the metric. The groups \( \textrm{GL}(V) \) and \( \textrm{O}(V\oplus V^*)\) are Lie groups: they are both group and manifold in a nice compatible way. Neither a group or a manifold are objects easy to handle, so we will do something similar to what we do in geometry: linearize, look at the tangent space (in this case of the identity of the group)!
Let's look then at the Lie algebra \( \mathfrak{o}(V\oplus V^* ) \). What does it satisfy? How can we describe its elements?
No comments:
Post a Comment
Please, use the comments to share any thoughts or concerns, and also to leave some anonymous feedback, which will not be published.