# Highlights in Linear Algebraic Groups 12: Radical, Reductive

To analyse the structure of a group G
you will need the radical and a torus T.
The group of Weyl may also may also suit
to prevent the scattering of many a root.
Functors are nice including the one of Lie
Parabolics bring in the ge-o-metry!
The theory of weights may seem oh so eerie
Until you start representation theory!

The structure of reductive and semisimple groups over an algebraically closed field will be pinnacle of this post series. After we have finished with this, this series will end and we will start to learn about algebraic groups from the perspective of group schemes, and we shall use some of the results we have seen so far by using that we really have just been studying the $\overline{k}$-points of group schemes (classical algebraic geometry).

The topic for today is the radical and unipotent radical, that will allow us to define the concept of semisimple and reductive group. We will then use the roots, which are certain characters of a maximal torus. These will give us a root system, so we will take a break to study these, and classification of root systems will enable us to classify algebraic groups.
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