Linear algebraic group

In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation MTM = I where MT is the transpose of M.

Lie groups that aren't algebraic

There are several classes of examples of Lie groups that aren't the real or complex points of an algebraic group.

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• Any Lie group with an infinite group of components G/G^o 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.

See also:

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• Differential Galois theory