Why are linear algebraic groups defined the way they are? What is the intuition?

Question

Definitions

A subgroup $G \subset GL(n, \mathbb{C})$ is a linear algebraic group, if there is a set $A$ of polynomial functions on $Mn(\mathbb{C})$ such that $G = \{g \in GL(n, \mathbb{C})| f(g) = 0 \forall f \in A \}$. ($Mn(\mathbb{C})$ is a set of all $n \times n$ complex matrices).

Let $V$ be a finite-dimensional vector space over $\mathbb{C}$. Let us choose any basis of $V$ and denote by µ the resulting isomorphisms $\mu : V \rightarrow \mathbb{C}^n$ and $\mu : End(V ) → M_n(\mathbb{C})$. A subgroup $G \subset GL(V )$ is called a linear algebraic group, if $\mu(G)$ is a linear algebraic subgroup of $GL(n, \mathbb{C})$. It is easy to check that the definition does not depend on the choice of the basis for $V$.

1. My main question: Why are these group defined like this? What is the interpretation of such definitions? I know that linear algebraic groups are interesting, for example Hilbert´s Finiteness Theorem claims that any of them can be defined by finite number of polynomial equations. I am just interested, what the definition says exactly that makes such groups that extraordinary. Is it somehow similar to algebraic varieties, that are also sets of something that is mapped to zero by a set of functions? What exactly the linearity tells us? And why exactly is called "linear" and "algebraic"?

2. Second priority question: In the definitions, do polynomial functions taking matrices as arguments look like this? (It is taken from Wikipedia). I am sorry if it is a really easy questions, I just never came accross this, so I just want a confirmation that this is what we work with.

EDIT: I have found another definition of linear algebraic group - such that it is a subgroup of $GL_n$, for some fixed $n$, that is defined by polynomials. In other wors, it has to be an affine variety.

Answer 1

Linear algebraic groups (also called affine algebraic groups, these two notions are equivalent!) relate to affine varieties like Lie groups relate to smooth manifolds and topological groups relate to topological spaces: they are the group objects in the category of affine varieties. So if you have some intuition for what a topological group or a Lie group is and some intuition for affine algebraic varieties, you can infer some intuition for affine algebraic groups by analogy.

Concretely, this means that an affine algebraic group is an affine variety $G$ together with a multiplication morphism $m:G \times_\Bbb C G \to G$ (Here we‘re taking the product in the category of affine varieties), an inversion morphism $i:G \to G$ and a unit morphism $e:* \to \Bbb C$ such that the identities corresponding to the group axioms are satisfied, i.e. we have:

• $m \circ (m \times \mathrm{id})=m\circ (\mathrm{id} \times m)$ (Associativity)
• $m\circ (e \times \mathrm{id})=m\circ (\mathrm{id}\times e)=\mathrm{id}$ (Neutral element)
• $m \circ (\mathrm{id} \times i)=m\circ (i \times \mathrm{id})=e$ (Inverse elements)

If one takes this as a definition, it is a theorem that every affine algebraic group is an algebraic subgroup of $\mathrm{GL}_n$ for some $n$, i.e. is a linear algebraic group. That theorem shows that affine and linear algebraic groups are the same.

Actually one can show that affine algebraic groups over $\Bbb C$ are always smooth and thus, if one considers the analytification, one gets a Lie group. The functor from complex linear algebraic groups to complex Lie groups is faithful, so one may think of affine algebraic groups over $\Bbb C$ as certain special complex Lie groups, those which may be defined by polynomial equations. (And morphisms of affine algebraic groups are also those Lie group homomorphisms that are defined by polynomial equations in a precise sense, i.e. the exponential map $\exp:(\Bbb C,+) \to (\Bbb C^\times,\cdot)$ is not algebraic, i.e. not a morphism of algebraic groups.)

Because the category of affine varieties is dual to the category of reduced commutative $\Bbb C$-algebras of finite type, one may also derive a corresponding algebraic characterization of affine algebraic groups: these correspond to reduced commutative finitely generated Hopf algebras over $\Bbb C$

Using category theory, especially the Yoneda lemma, one may derive a different characterization of linear algebraic groups that is sometimes useful, it is called the “functor of points” perspective: Let $\mathbf{Alg}_\Bbb C$ be the category of commutative $\Bbb C$-algebras and $\mathbf{Grp}$ be the category of groups, then an algebraic group is equivalently a functor $\mathbf{Alg}_\Bbb C \to \mathbf{Grp}$ such that the underlying set-valued functor $\mathbf{Alg}_\Bbb C \to \mathbf{Grp}$ is representable by a reduced commutative $\Bbb C$-algebra of finite type. For example, consider the algebraic group $\mathrm{GL}_1$ this corresponds to the functor $\mathbf{Alg}_\Bbb C \to \mathbf{Grp}$ that sends each algebra to its group of units. This is represented by $\Bbb C[x,x^{-1}]$.