Once we have a vector space V⊕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, Aut(V) or GL(V), permutations compatible with the sum and the product by scalars. For V⊕V∗ we have
O(V⊕V∗)={A∈GL(V⊕V∗):⟨Av,Aw⟩=⟨v,w⟩ for v,w∈V⊕V∗},
i.e., bijections preserving the vector space structure that also preserve the metric. The groups GL(V) and O(V⊕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 o(V⊕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, Aut(V) or GL(V), permutations compatible with the sum and the product by scalars. For V⊕V∗ we have
O(V⊕V∗)={A∈GL(V⊕V∗):⟨Av,Aw⟩=⟨v,w⟩ for v,w∈V⊕V∗},
i.e., bijections preserving the vector space structure that also preserve the metric. The groups GL(V) and O(V⊕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 o(V⊕V∗). What does it satisfy? How can we describe its elements?
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