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Kara386

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## Homework Statement

The group ##G = \{ a\in M_n (C) | aSa^{\dagger} =S\}## is a Lie group where ##S\in M_n (C)##. Find the corresponding Lie algebra.

## Homework Equations

## The Attempt at a Solution

As far as I've been told the way to find these things is to set ##a = exp(tA)##, so:

##exp(tA)exp(tS)exp(tA^{\dagger}) = exp(tA)exp(tA^{\dagger})exp(tS)## Use series expansion:

##= (I+tA+ \frac{t^2A^2}{2}+O(t^3))(I+tA^{\dagger}+\frac{t^2A^{\dagger 2}}{2}+O(t^3))e^{tS}##

##=[I+t(A+A^{\dagger})+t^2(\frac{A^2}{2}+\frac{A^{\dagger 2}}{2}+AA^{\dagger}) + O(t^3)]e^{tS} = e^{tS}##

So require ##A+A^{\dagger}=0## therefore ##A = -A^{\dagger}##. This also means the second bracket is zero and the equation is satisfied. Does that mean the Lie algebra of G is ##g= \{a\in M_n(C)| A+A^{\dagger}=0\}##? If not, where did I go wrong? Thanks for any help!

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