Algebraic group

• The additive group: the affine line ${\displaystyle \mathbb {A} ^{1}}$ endowed with addition and opposite as group operations is an algebraic group. It is called the additive group (because its ${\displaystyle k}$-points are isomorphic as a group to the additive group of ${\displaystyle k}$), and usually denoted by ${\displaystyle \mathrm {G} _{a}}$.
• The multiplicative group: Let ${\displaystyle \mathrm {G} _{m}}$ be the affine variety defined by the equation ${\displaystyle xy=1}$ in the affine plane ${\displaystyle \mathbb {A} ^{2}}$. The functions ${\displaystyle ((x,y),(x',y'))\mapsto (xx',yy')}$ and ${\displaystyle (x,y)\mapsto (x^{-1},y^{-1})}$ are regular on ${\displaystyle \mathrm {G} _{m}}$, and they satisfy the group axioms (with neutral element ${\displaystyle (1,1)}$). The algebraic group ${\displaystyle \mathrm {G} _{m}}$ is called multiplicative group, because its ${\displaystyle k}$-points are isomorphic to the multiplicative group of the field ${\displaystyle k}$ (an isomorphism is given by ${\displaystyle x\mapsto (x,x^{-1})}$ ; note that the subset of invertible elements does not define an algebraic subvariety in ${\displaystyle \mathbb {A} ^{1}}$).
• The special linear group ${\displaystyle \mathrm {SL} _{n}}$ is an algebraic group: it is given by the algebraic equation ${\displaystyle \det(g)=1}$ in the affine space ${\displaystyle \mathbb {A} ^{n^{2}}}$ (identified with the space of ${\displaystyle n}$-by-${\displaystyle n}$ matrices), multiplication of matrices is regular and the formula for the inverse in terms of the adjugate matrix shows that inversion is regular as well on matrices with determinant 1.
• The general linear group ${\displaystyle \mathrm {GL} _{n}}$ of invertible matrices over a field ${\displaystyle k}$ is an algebraic group. It can be realised as a subvariety in ${\displaystyle \mathbb {A} ^{n^{2}+1}}$ in much the same way as the multiplicative group in the previous example.[3]
• A non-singular cubic curve in the projective plane ${\displaystyle \mathbb {P} ^{2}}$ can be endowed with a geometrically defined group law that makes it into an algebraic group (see elliptic curve).

An algebraic subgroup of an algebraic group ${\displaystyle \mathrm {G} }$ is a subvariety ${\displaystyle \mathrm {H} }$ of ${\displaystyle \mathrm {G} }$ that is also a subgroup of ${\displaystyle \mathrm {G} }$ (that is, the maps ${\displaystyle \mathrm {G} \times \mathrm {G} \to \mathrm {G} }$ and ${\displaystyle \mathrm {G} \to \mathrm {G} }$ defining the group structure map ${\displaystyle \mathrm {H} \times \mathrm {H} }$ and ${\displaystyle \mathrm {H} }$, respectively, into ${\displaystyle \mathrm {H} }$).

A morphism between two algebraic groups ${\displaystyle \mathrm {G} ,\mathrm {G} '}$ is a regular map ${\displaystyle \mathrm {G} \to \mathrm {G} '}$ that is also a group morphism. Its kernel is an algebraic sugroup of ${\displaystyle \mathrm {G} }$, its image is an algebraic subgroup of ${\displaystyle \mathrm {G} '}$.[4]

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 ${\displaystyle \mathrm {H} }$ is a normal algebraic subgroup of ${\displaystyle \mathrm {G} }$ then there exists an algebraic group ${\displaystyle \mathrm {G} /\mathrm {H} }$ and a surjective morphism ${\displaystyle \pi$ :\mathrm {G} \to \mathrm {G} /\mathrm {H} } such that ${\displaystyle \mathrm {H} }$ is the kernel of ${\displaystyle \pi }$.[5] Note that if the field ${\displaystyle k}$ is not algebraically closed, the morphism of groups ${\displaystyle \mathrm {G} (k)\to \mathrm {G} (k)/\mathrm {H} (k)}$ may not be surjective (the default of surjectivity is measured by Galois cohomology).

Similarly to the Lie group–Lie algebra correspondence, to an algebraic group over a field ${\displaystyle k}$ is associated a Lie algebra over ${\displaystyle k}$. 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.[6]

A more sophisticated definition of an algebraic group over a field ${\displaystyle k}$ is that it is that of a group scheme over ${\displaystyle k}$ (group schemes can more generally be defined over commutative rings).

Yet another definition of the concept is to say that an algebraic group over ${\displaystyle k}$ is a group object in the category of algebraic varieties over ${\displaystyle k}$.