2025-07-29

Algebraic groups

If unfamiliar with algebraic geometry, can just work over with classical topology.

A group is :

  • ,
  • ,
  • ,

with axioms:

  • associativity
G£G£GG£GG£GGm£idid£mmm
  • identity (and the other one)
GG£¤G£GG»=idid£em
  • inverses (and the other one)
G¤G£GGid£iem

A group object in any (nice: has products and terminal objects) category is the above.

Examples

  • Topological spaces topological groups
  • Manifolds + smooth maps Lie groups
  • Algebraic varieties over algebraic groups
  • (Schemes of finite type over algebraic groups)

( is a field)

Actual examples

  • where invertible matrices with entries in : coordinate ring
  • : a vector space and is a non-degenerate symplectic bilinear form, i.e., , then
    More concretely (via Theorem/Exercise),
    So the condition is .

Some one-dimensional examples:

  • (field with addition) additive group, aka
  • multiplicative group, aka , with group operation multiplication.
  • elliptic curve:
    • complex analytically is , a lattice,
    • in algebraic geometry: a smooth cubic where three points sum to zero iff they’re on the same line.

Above is classification for algebraically closed. For example in there is the circle .

  • the th roots of unity (), i.e., solutions to .
    • Interesting: is not reduced if . For example, if and then .
  • the elements in . (centre).

Theorem/Exercise

For a vector space with a non-degenerate symplectic bilinear, we must have:

  • is even,
  • there exists a basis of such that
    ⋰⋰

What we think about

These have (interesting) representation theory.

Definition (Unipotent element)

  • is unipotent if all its eigenvalues (over ) are .
  • ( a linear algebraic group) is unipotent if is unipotent for some , or
  • equivalently: if is unipotent for all .

Definition (Unipotent group)

An algebraic group is unipotent if contains only unipotent entries.

Example: is unipotent

Recall :

All elements here are unipotent. So is unipotent.

In , every unipotent group has a composition series where the subquotients are .

Theorem

Let be a linear algebraic group. Then contains a maximal connected normal unipotent subgroup .

Example

Note that is normal here. This is not reductive.

Definition (Reductive group)

If , we say that is reductive.

Some authors include is connected in the definition of reductive.

Examples

, , are all reductive. is not reductive.

Sketch of proof

Suppose and unipotent with . Take the following set (subspace)

Because is unipotent, this implies that

First follows from unipotent (have to prove), second follows from . Now is -stable, i.e., for all , , because is normal.

In examples: for all , there exists such that . There is a contradiction if one takes and .

Definition (Borel subgroup)

A Borel subgroup of is a maximal connected solvable subgroup of .

Remark

Solvable means the same as usual: taking commutators reaches .

Example

  • :
  • :
    Warning: This requires the “correct” choice of .