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Related concepts
Given a field and an algebraic group over , and given a field extension , then a -form of is an algebraic group over such that its base change to yields :
Typically one requires that is also defined over , hence one considers -forms of the -ification of a given -form.
Consider the inclusion of the real numbers into the complex numbers and let be the multiplicative group hence the group of units of .
Then one -form, hence a real form of is given by , and another is given by the circle group .
To see this, realize as the group of 2x2 matrices with entries in the complex numbers which are diagonal and of unit determinant:
The same prescription over the real numbers yields and exhibits it as a real form of .
On the other hand, realize the circle group as the group of 2x2 real matrices of the form
One checks that over the complex numbers this is isomorphic to the previous group of diagonal matrices, with the isomorphism being given by
(e.g. eom)
Created on July 2, 2014 at 00:30:26. See the history of this page for a list of all contributions to it.