# Lie Groups lecture 2

This lecture was given on september 1st 2009.

This lecture concerned important examples of matrix lie groups: . Most of the material is taken from chapter 1 of Brian Hall’s book. Also a classification of bilinear forms was presented.

The notation is that is the group of invertible matrices that preserve : . In general, it suffices to study symmetric and anti symmetric bilinear forms because it can be written as a sum of symmetric and antisymmetric forms

### Symmetric Bilinear Forms

Assume is symmetric. Then its normal so the spectral thm says there exists a basis under which its orthogonal. Its also Hermitian so its eigenvalues are real. So after scaling can assume where $p + n + m = N$ is the dimension of the vector space. Let this basis be given by . Define

The inertial thm says if is another basis with then . If not, say then so there subspaces have to intersect nontrivially giving a such that and ; gives contradiction.

### Anti Symmetric Bilinear forms

If is antisymmetric ( ) then define as . Then can be decomposted . By retricting to can assume is nondegernate. Choose arbirary, by assumption there exists s.t. . Let and decompose where . Repeating the argument (and noting that B is nondegenerate) it follows that where each are two dimensional and (afer scaling) . It follows that is even dimensional and the contruction gives a matrix for which is a direct sum of ‘s.

Its a fac that also has a matrix of the form .

It can be checked that by identitifying it with the . If are decomposed as and taking the usual norm on gives

Which separate it into a symmetric and antisymmetric part. Consequently .

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- September 2, 2009 / 8:11 am

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