Algebraic Geometry Unit 10 Review: Algebraic Groups and Lie Algebras

Key Concepts and Definitions

• Algebraic groups are groups that are also algebraic varieties, combining the structures of group theory and algebraic geometry
• Lie algebras are vector spaces equipped with a bilinear operation called the Lie bracket, which captures the infinitesimal behavior of Lie groups
• Group actions describe how a group can act on a set, such as an algebraic variety, by permuting its elements
• Representations are homomorphisms from a group or Lie algebra to the general linear group $GL(V)$ of a vector space $V$, providing a way to study abstract structures using linear algebra
• Tangent spaces are vector spaces associated with each point of a manifold or algebraic variety, capturing the local linear approximation of the space
• Root systems are combinatorial structures that encode the structure of semisimple Lie algebras and algebraic groups
  • Consist of a set of vectors in a Euclidean space satisfying certain symmetry and integrality conditions
  • Examples include the root systems of classical Lie algebras (An, Bn, Cn, Dn)
• Weyl groups are finite reflection groups associated with root systems, which act on the Lie algebra and algebraic group by permuting the roots

Group Theory Foundations

• Groups are algebraic structures consisting of a set equipped with a binary operation satisfying axioms of closure, associativity, identity, and inverses
• Homomorphisms are structure-preserving maps between groups, preserving the group operation
  • Isomorphisms are bijective homomorphisms, indicating that two groups have the same structure
• Subgroups are subsets of a group that are closed under the group operation and form a group themselves
  • Normal subgroups are subgroups invariant under conjugation by elements of the larger group
• Quotient groups are formed by collapsing elements of a group that differ by an element of a normal subgroup, creating a new group structure
• Direct products of groups are formed by taking the Cartesian product of the underlying sets and defining a componentwise group operation
• Group actions capture the symmetries of an object, such as the rotations and reflections of a regular polygon or the permutations of the roots of a polynomial
• Sylow theorems provide information about the existence and properties of certain subgroups of finite groups, based on the prime factorization of the group order

Introduction to Lie Algebras

• Lie algebras are vector spaces equipped with a bilinear operation called the Lie bracket, satisfying axioms of antisymmetry and the Jacobi identity
  • The Lie bracket captures the commutator of elements in a Lie group, encoding infinitesimal information
• Lie subalgebras are vector subspaces of a Lie algebra that are closed under the Lie bracket
  • Ideals are Lie subalgebras invariant under the adjoint action of the larger Lie algebra
• The adjoint representation is a representation of a Lie algebra on itself, given by the Lie bracket
  • The kernel of the adjoint representation is the center of the Lie algebra
• Semisimple Lie algebras are direct sums of simple Lie algebras, which have no non-trivial ideals
  • Characterized by having a non-degenerate Killing form, a symmetric bilinear form that measures the "angle" between elements
• Solvable Lie algebras are Lie algebras whose derived series terminates at zero, generalizing the notion of solvable groups
• Nilpotent Lie algebras are Lie algebras whose lower central series terminates at zero, generalizing the notion of nilpotent groups
• The Poincaré-Birkhoff-Witt theorem states that the universal enveloping algebra of a Lie algebra has a basis consisting of ordered monomials in a basis of the Lie algebra

Structure of Algebraic Groups

• Algebraic groups are groups that are also algebraic varieties, defined by polynomial equations in some affine or projective space
  • Examples include the general linear group $GL_n$, the special linear group $SL_n$, and the orthogonal group $O_n$
• The Lie algebra of an algebraic group is the tangent space at the identity, equipped with a Lie bracket derived from the group structure
  • The exponential map sends elements of the Lie algebra to elements of the algebraic group, providing a local diffeomorphism near the identity
• Algebraic subgroups are subgroups that are also closed subvarieties of the algebraic group
  • Borel subgroups are maximal connected solvable algebraic subgroups, playing a key role in the structure theory
• Tori are algebraic subgroups isomorphic to products of the multiplicative group $\mathbb{G}_m$, generalizing the notion of diagonalizable matrices
• Unipotent groups are algebraic subgroups consisting of unipotent elements, which have all eigenvalues equal to 1
• Reductive groups are algebraic groups whose unipotent radical (maximal connected normal unipotent subgroup) is trivial
  • Examples include $GL_n$, $SL_n$, and semisimple groups (which have trivial radical)
• Parabolic subgroups are algebraic subgroups containing a Borel subgroup, arising as stabilizers of flags in a representation

Representation Theory

• Representations of algebraic groups are homomorphisms from the group to $GL(V)$ for some vector space $V$, capturing the group action on a linear space
  • Finite-dimensional representations are those where $V$ has finite dimension over the base field
• Irreducible representations are those with no proper non-zero subrepresentations
  • Every representation can be decomposed as a direct sum of irreducible representations
• Characters are functions on the group that encode the trace of a representation, providing a way to study representations using functions on the group
• The regular representation is the action of a group on its group algebra by left multiplication, containing every irreducible representation as a subrepresentation
• Schur's lemma states that homomorphisms between irreducible representations are either zero or isomorphisms, providing a powerful tool for studying representations
• The Weyl character formula gives an explicit expression for the characters of irreducible representations of compact Lie groups and semisimple algebraic groups
• The highest weight theory classifies irreducible representations of semisimple Lie algebras and algebraic groups using dominant integral weights, which are elements of the weight lattice satisfying certain positivity conditions

Applications in Algebraic Geometry

• Flag varieties are projective varieties that parametrize flags (nested sequences of subspaces) in a vector space, providing a geometric realization of parabolic subgroups
  • The Grassmannian variety parametrizes $k$-dimensional subspaces of an $n$-dimensional vector space, arising as a flag variety for $GL_n$
• Homogeneous spaces are varieties that admit a transitive action by an algebraic group, generalizing the notion of coset spaces in group theory
  • Examples include projective spaces (homogeneous under $PGL_n$) and affine spaces (homogeneous under the affine group)
• Quotients of algebraic groups by closed subgroups often inherit the structure of an algebraic variety, providing a way to construct new varieties from group actions
  • GIT (Geometric Invariant Theory) quotients are a way to construct well-behaved quotients of varieties by group actions, using invariant functions to define the quotient
• Algebraic group actions on varieties can be used to study the geometry and topology of the variety
  • Orbit closures, fixed points, and stabilizers provide information about the structure of the variety and the group action
• Equivariant cohomology is a version of cohomology that takes into account the action of a group on a space, providing additional algebraic structure
• Schubert varieties are subvarieties of flag varieties defined by certain incidence conditions, playing a key role in the study of flag varieties and their cohomology
  • Schubert calculus is the study of intersections of Schubert varieties, which can be computed using combinatorial rules

Connections to Other Mathematical Fields

• Lie groups are smooth manifolds that are also groups, with the group operations being smooth maps
  • Many classical Lie groups (e.g., $SO_n$, $SU_n$) arise as the real points of algebraic groups
• Representation theory of Lie groups and Lie algebras is closely connected to the representation theory of algebraic groups over $\mathbb{C}$
  • The Weyl unitary trick relates representations of compact Lie groups to representations of complex semisimple Lie algebras
• Algebraic topology can be used to study the topology of algebraic varieties and algebraic groups
  • The cohomology ring of a flag variety can be described using Schubert classes, which form a basis for the cohomology
• Combinatorics arises in the study of flag varieties, Schubert varieties, and representation theory
  • Young tableaux and Schur functions are used to describe representations of $GL_n$ and the cohomology of Grassmannians
• Number theory and arithmetic geometry can be studied using algebraic groups and their representations
  • The Langlands program relates representations of Galois groups to automorphic forms and representations of algebraic groups over local and global fields
• Mathematical physics uses Lie groups and their representations to describe symmetries in physical systems
  • The Lorentz group and the Poincaré group are used in special and general relativity to describe spacetime symmetries
  • Gauge theories in particle physics use Lie groups to describe the symmetries of fundamental interactions

Problem-Solving Strategies

• Identify the type of algebraic group or Lie algebra involved (e.g., $GL_n$, $SL_n$, semisimple, nilpotent) and use the corresponding structure theory
  • Semisimple Lie algebras and reductive groups can be studied using root systems and highest weight theory
  • Solvable and nilpotent Lie algebras have a simpler structure and can often be understood using the Lie bracket and the adjoint representation
• Relate the problem to known results or techniques in representation theory, such as characters, irreducible representations, or the Weyl character formula
  • Decomposing a representation into irreducible components can simplify the problem and provide insights
• Use the geometry of algebraic groups and their homogeneous spaces to translate the problem into geometric terms
  • Flag varieties and Schubert varieties can be used to study the structure of algebraic groups and their representations
• Consider the action of the algebraic group or Lie algebra on a related object, such as a vector space or an algebraic variety
  • Orbit structures, fixed points, and stabilizers can provide valuable information about the problem
• Break the problem down into smaller subproblems involving subgroups, subalgebras, or subrepresentations
  • Induction and restriction of representations can be used to relate the representation theory of a group to that of its subgroups
• Look for connections to other areas of mathematics, such as algebraic topology, combinatorics, or number theory
  • Techniques from these fields can often be adapted to the study of algebraic groups and Lie algebras
• Use explicit computations and examples to gain intuition and insight into the problem
  • Computing the Lie bracket, the adjoint representation, or the characters of small examples can provide valuable guidance
• Consult the literature and seek out similar problems or techniques that have been used in related contexts
  • The theory of algebraic groups and Lie algebras is vast and well-developed, and many powerful results and techniques are available