Algebraic group

Summary

An algebraic group is a group whose elements are the points of an algebraic variety, with multiplication and inversion given by polynomial maps. The two main families are the linear algebraic groups, such as $\mathrm{GL}_n$, and the abelian varieties, such as elliptic curves. Chevalley's structure theorem shows that every smooth connected algebraic group over a perfect field is built from one of each. Algebraic groups play for algebraic geometry the role that Lie groups play for differential geometry.