Linear Algebraic Group - Non-algebraic Lie Groups

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.