Non-algebraic Lie Groups
There are several classes of examples of Lie groups that aren't the real or complex points of an algebraic group.
- Any Lie group with an infinite group of components G/Go cannot be realized as an algebraic group (see identity component).
- The center of a linear algebraic group is again a linear algebraic group. Thus, any group whose center has infinitely many components is not a linear algebraic group. An interesting example is the universal cover of SL2(R). This is a Lie group that maps infinite-to-one to SL2(R), since the fundamental group is here infinite cyclic - and in fact the cover has no faithful matrix representation.
- The general solvable Lie group need not have a group law expressible by polynomials.
Read more about this topic: Linear Algebraic Group
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