previous example. A non-singular cubic curve in the projective plane can be endowed with a geometrically defined group law which makes it into an algebraic group (see elliptic curve). Related definitions An algebraic subgroup of an algebraic group is a subvariety of which is also a subgroup of (that is, the maps and defining the group structure map and , respectively, into ). A morphism between two algebraic groups is a regular map which is also a group morphism. Its kernel is an algebraic sugroup of , its image is an algebraic subgroup of . Quotients in the category of algebraic groups are more delicate to deal with. An algebraic subgroup is said to be normal if it is stable under every inner automorphism (which are regular maps). If is a normal algebraic subgroup of then there exists an algebraic group and a surjective morphism such that is the kernel of . Note that if the field is not algebraically closed, the morphism of groups may not be surjective (the default of surjectivity is measured by Galois cohomology). Lie algebra of an algebraic group Similarly to the Lie group–Lie algebra correspondence, to an algebraic group over a field is associated a Lie algebra over . As a vector space the Lie algebra is isomorphic to the tangent space at the identity element. The Lie bracket can be constructed from its interpretation as a space of derivations. Alternative definitions A more sophisticated definition of an algebraic group over a field is that it is that of a group scheme over (group schemes can more generally be defined over commutative rings). Yet another definition of the concept is to say that an algebraic group over is a group object in the category of algebraic varieties over . Affine algebraic groups An algebraic group is said to be affine if its underlying algebraic variety is an affine variety. Among the examples above the additive, multiplicative groups and the general and special linear groups are affine. Using the action of an affine algebraic group on its coordinate ring it can be shown that every affine algebraic group is a linear (or matrix group), meaning that it is isomorphic to an algebraic subgroup of the general linear group. For example the additive group can be embedded in by the morphism . There are many examples of such groups beyond those given previously: orthogonal and symplectic