Algebraic groups blend algebra and geometry, combining group structures with algebraic varieties. They're crucial in many math areas, from number theory to representation theory. Understanding their properties and actions on varieties is key to grasping their power and applications.

Group actions on algebraic varieties reveal deep connections between geometry and algebra. These actions help us understand symmetries of geometric objects and classify important mathematical structures. They're essential tools for studying algebraic geometry and related fields.

Algebraic Groups: Definition and Examples

Definition and Basic Properties

• An algebraic group is a group that is an algebraic variety, with the group operations of multiplication and inversion being morphisms of varieties
• The group law on an algebraic group is required to be a morphism of varieties, meaning it is defined by polynomial equations in the coordinate ring of the variety
• The identity element of an algebraic group is a distinguished point, and the inverse map sending an element to its inverse is an automorphism of the variety
• Algebraic groups can be either affine (closed subvarieties of affine space) or projective (closed subvarieties of projective space)

Examples of Algebraic Groups

• The most common examples of algebraic groups are linear algebraic groups, which are subgroups of the general linear group $GL_n(k)$ over an algebraically closed field $k$
  • Examples include the special linear group $SL_n(k)$, the orthogonal group $O_n(k)$, and the symplectic group $Sp_{2n}(k)$
• Abelian varieties, such as elliptic curves and abelian surfaces, form another important class of algebraic groups
  • Elliptic curves are algebraic groups of dimension 1, while abelian surfaces have dimension 2
  • The group law on an elliptic curve is given by the chord-and-tangent process, which can be described by algebraic equations

Group Actions on Algebraic Varieties

Definition and Properties of Group Actions

• An action of an algebraic group $G$ on an algebraic variety $X$ is a morphism of varieties $φ: G × X → X$ satisfying certain compatibility conditions with the group operations
• The action is said to be transitive if for any two points $x, y ∈ X$, there exists a group element $g ∈ G$ such that $φ(g, x) = y$
• An action is faithful if the induced group homomorphism $G → Aut(X)$ is injective, meaning distinct group elements act differently on $X$
• The action of an algebraic group on itself by left or right multiplication is called the regular action and plays a fundamental role in the theory

Homogeneous Spaces and Quotients

• A variety $X$ with a transitive action by an algebraic group $G$ is called a homogeneous space for $G$
• Quotients of algebraic groups by closed subgroups, such as projective spaces and Grassmannians, provide important examples of homogeneous spaces
  • The projective space $ℙ^n$ is a homogeneous space for the general linear group $GL_{n+1}(k)$
  • The Grassmannian $Gr(k, n)$ of $k$-dimensional subspaces of $k^n$ is a homogeneous space for $GL_n(k)$

Orbits and Stabilizers of Group Actions

Orbits and Their Properties

• For a point $x ∈ X$, the orbit of $x$ under the action of $G$ is the set $O(x) = \{φ(g, x) | g ∈ G\}$, which is a subset of $X$
• The orbits partition the variety $X$ into disjoint subsets, each of which is itself a homogeneous space for $G$
• The dimension of an orbit is related to the codimension of the corresponding stabilizer subgroup by the orbit-stabilizer theorem

Stabilizers and Quotient Varieties

• The stabilizer of a point $x ∈ X$ is the subgroup $Stab(x) = \{g ∈ G | φ(g, x) = x\}$, consisting of elements that fix $x$
• For a transitive action, the stabilizers of any two points are conjugate subgroups of $G$, and the variety $X$ is isomorphic to the quotient $G/Stab(x)$ for any $x ∈ X$
• The set of orbits $X/G$ inherits the structure of an algebraic variety, called the quotient variety, under suitable conditions on the action
  • The quotient map $X → X/G$ sends each point to its orbit, and the quotient variety parametrizes the orbits of the action

Classifying Algebraic Groups

Dimension and Simplicity

• Algebraic groups can be classified according to their dimension, which is the dimension of the underlying variety
• An algebraic group is called simple if it has no proper, closed, connected, normal subgroups
  • The classification of simple algebraic groups over algebraically closed fields is a major result in the theory, generalizing the classification of simple Lie algebras
  • Examples of simple algebraic groups include the special linear groups $SL_n(k)$ for $n ≥ 2$ and the symplectic groups $Sp_{2n}(k)$ for $n ≥ 1$

Reductive Groups and Their Invariants

• Reductive algebraic groups, which include semisimple groups and tori, play a central role in representation theory and the study of algebraic varieties
• The character group of an algebraic torus is a key invariant, consisting of algebraic group homomorphisms from the torus to the multiplicative group
  • For the multiplicative group $𝔾_m$, the character group is isomorphic to the integers $ℤ$
  • For a torus $T ≅ 𝔾_m^n$, the character group is isomorphic to $ℤ^n$
• The Weyl group of a reductive algebraic group is a finite reflection group that controls much of the group's structure and representation theory
• The theory of root systems and weights associated with reductive groups provides a powerful tool for their classification and study
  • The root system of a semisimple group encodes its structure and relates to its representation theory
  • The weight lattice of a reductive group classifies its irreducible representations