Linear algebraic group

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Definition

In terms of underlying variety

A linear algebraic group or affine algebraic group is an algebraic group where the underlying algebraic variety is an affine variety.

By a basic theorem of algebraic geometry, any affine algebraic geometry has a faithful linear representation, and can hence be realized as a linear algebraic group. Thus, we often view affine algebraic group and linear algebraic group as synonyms.

In terms of embedding into general linear group

A linear algebraic group or affine algebraic group over a field k is defined as an algebraic group G such that there exists an embedding of G as a closed subgroup of the general linear group GL(n,k) for some choice of n (and this embedding is a morphism of algebraic varieties).

Note that such an embedding as a closed subgroup of GL(n,k) automatically gives an algebraic group structure, so if we provide an embedding, we do not need to specify an algebraic group structure separately. This is because GL(n,k) comes naturally equipped with an algebraic group structure, and any closed subgroup of algebraic group inherits algebraic group structure.