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Monday, 12 January 2015

L1d. Generalized linear algebra: the symmetries.


Once we have a vector space VV 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 VV we have
O(VV)={AGL(VV):Av,Aw=v,w for v,wVV},
i.e., bijections preserving the vector space structure that also preserve the metric. The groups GL(V) and O(VV) 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(VV). What does it satisfy? How can we describe its elements?


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