An algebraic group is a mathematical structure that combines concepts from both algebra and geometry. It consists of a set of points that can be defined by polynomial equations, along with a group operation that is compatible with the algebraic structure. This means that the group operation and the inverse operation can also be expressed as polynomial functions. Algebraic groups are important in various areas of mathematics, including number theory and representation theory. They can be classified into different types, such as linear algebraic groups, which can be represented by matrices, and abelian varieties, which are more complex structures related to elliptic curves.