12.1 Algebraic groups and group actions

Algebraic groups blend algebra and geometry, giving us powerful tools to study symmetries and transformations. They're like mathematical Swiss Army knives, helping us understand everything from simple shapes to complex geometric structures.

In this part, we'll see how these groups work and how they act on varieties. We'll explore their structure, learn about important examples, and see how they're used to solve real-world problems in math and beyond.

Algebraic groups and properties

Definition and basic structure

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• An algebraic group is a variety G equipped with morphisms for multiplication μ: G × G → G and inversion ι: G → G, satisfying the group axioms
• The identity element of an algebraic group is a distinguished point e in G
• The group axioms (associativity, identity, inverses) hold in the category of varieties, meaning they are satisfied by the morphisms μ and ι
• The multiplication map μ and the inversion map ι are required to be morphisms of varieties, ensuring compatibility between the group structure and the algebraic geometry of G

Examples of algebraic groups

• The multiplicative group Gm is the variety $\mathbb{A}^1 \setminus \{0\}$ with multiplication $(x, y) \mapsto xy$ and inversion $x \mapsto x^{-1}$
• The additive group Ga is the variety $\mathbb{A}^1$ with addition $(x, y) \mapsto x + y$ and inversion $x \mapsto -x$
• The general linear group GLn is the variety of invertible $n \times n$ matrices with matrix multiplication and inversion
• The special linear group SLn is the subgroup of GLn consisting of matrices with determinant 1
• Elliptic curves and abelian varieties are examples of projective algebraic groups

Structure of algebraic groups

Subgroups and quotients

• A subgroup H of an algebraic group G is a closed subvariety that is also a subgroup in the group-theoretic sense
• The identity component G0 of an algebraic group G is the connected component containing the identity element e
• The quotient G/H of an algebraic group G by a normal subgroup H has a natural structure of an algebraic group induced by the group operations on G
• The quotient morphism π: G → G/H is a morphism of algebraic groups with kernel H

Lie algebras and the exponential map

• The Lie algebra g of an algebraic group G is the tangent space at the identity $T_e G$, equipped with a Lie bracket operation $[\cdot, \cdot]: g \times g \to g$
• The exponential map exp: g → G is a local isomorphism from a neighborhood of 0 in g to a neighborhood of e in G, relating the Lie algebra to the algebraic group
• The differential of the multiplication map μ at (e, e) induces the Lie bracket on g, making it compatible with the group structure
• The adjoint representation Ad: G → GL(g) describes the action of G on its Lie algebra by conjugation

Classification of algebraic groups

• An algebraic group is solvable if it has a composition series with solvable quotients (successive extensions by $\mathbb{G}_a$ or $\mathbb{G}_m$)
• An algebraic group is unipotent if it has a composition series with unipotent quotients (successive extensions by $\mathbb{G}_a$)
• An algebraic group is semisimple if it has no non-trivial solvable normal subgroups
• The Jordan decomposition expresses an algebraic group as an extension of a semisimple group by a solvable group
• The classification of semisimple algebraic groups is related to the classification of root systems and Dynkin diagrams

Group actions on varieties

Definition and orbits

• An action of an algebraic group G on a variety X is a morphism α: G × X → X satisfying the usual axioms: $\alpha(e, x) = x$ and $\alpha(g, \alpha(h, x)) = \alpha(gh, x)$
• The orbit of a point x in X under the G-action is the set G·x = {α(g,x) | g in G}, which is a locally closed subvariety of X
• Orbits partition X into disjoint subvarieties, each isomorphic to a quotient of G by a stabilizer subgroup
• The stabilizer subgroup Gx of a point x is the subgroup {g in G | α(g,x) = x}, which is a closed subgroup of G

Quotients and fibers

• There is a bijective correspondence between orbits G·x and cosets of stabilizers G/Gx, given by $gG_x \mapsto g \cdot x$
• A G-action is transitive if it has only one orbit, meaning every point can be reached from any other point by the action of G
• A G-action is free if all stabilizers are trivial, i.e., $G_x = \{e\}$ for all x in X
• A G-action is faithful if the map G → Aut(X) given by $g \mapsto (x \mapsto g \cdot x)$ is injective
• A geometric quotient of X by a G-action is a variety Y with a morphism π: X → Y constant on orbits, such that π induces a bijection between orbits and points of Y
• Fibers of the quotient map π are the orbits of the G-action, and Y parameterizes the set of orbits

Algebraic groups for geometry

Symmetries and automorphisms

• Algebraic groups can be used to study symmetries and automorphisms of algebraic varieties
• The automorphism group Aut(X) of a variety X is an algebraic group, often with a rich structure
• Symmetries of X correspond to elements of Aut(X) or its subgroups, and can be used to simplify the study of X
• The structure theory of algebraic groups helps classify certain types of varieties with large automorphism groups, e.g., toric varieties, spherical varieties, flag varieties

Homogeneous spaces and bundles

• Group actions encode the geometry of homogeneous spaces, fiber bundles, and principal bundles in algebraic geometry
• A homogeneous space is a variety X with a transitive action of an algebraic group G, e.g., projective spaces, Grassmannians, flag varieties
• A G-equivariant bundle over X is a variety E with a G-action and a G-invariant morphism π: E → X, such that π is locally trivial with fibers isomorphic to a fixed G-variety F
• A principal G-bundle is a G-equivariant bundle where the fibers are isomorphic to G acting on itself by multiplication
• The quotient of E by the G-action is isomorphic to X, and E can be recovered from X and the cocycle defining the bundle

Representation theory and invariant theory

• Representation theory of algebraic groups provides tools for studying linear actions on vector spaces and sheaves
• A representation of an algebraic group G is a morphism of algebraic groups ρ: G → GL(V) for some finite-dimensional vector space V
• Representations can be used to construct G-equivariant sheaves and study their cohomology, which often has additional structure coming from the representation
• Invariant theory describes polynomials and rational functions invariant under a group action, and their relations to quotients
• The ring of invariants $\mathbb{C}[V]^G$ of a G-representation V is the subring of G-invariant polynomials, which is finitely generated by the Hilbert Basis Theorem
• The GIT quotient $V // G$ is the variety Spec($\mathbb{C}[V]^G$), which parameterizes closed orbits of the G-action on V

Applications over finite fields

• Algebraic groups over finite fields are used in coding theory, cryptography, and other applications
• The Lang-Steinberg theorem states that the fixed points of a Frobenius endomorphism on a connected algebraic group over a finite field form a finite subgroup
• Finite subgroups of algebraic groups over finite fields are used to construct error-correcting codes, such as Goppa codes and algebraic-geometric codes
• Cryptographic protocols based on the hardness of the discrete logarithm problem or the Diffie-Hellman problem can be formulated using algebraic groups over finite fields
• Algebraic groups over finite fields also appear in the study of zeta functions, L-functions, and other arithmetic invariants of varieties