🔁Lie Algebras and Lie Groups Unit 12 – Advanced Topics in Lie Theory

Key Concepts and Definitions

• Lie algebras defined as vector spaces equipped with a bilinear operation called the Lie bracket, satisfying skew-symmetry and the Jacobi identity
• Lie groups smooth manifolds with a group structure such that the group operations (multiplication and inversion) are smooth maps
• Exponential map connects Lie algebras and Lie groups, allowing elements of the Lie algebra to be mapped to the corresponding Lie group
  • Defined as $\exp: \mathfrak{g} \to G$, where $\mathfrak{g}$ is the Lie algebra and $G$ is the Lie group
  • For matrix Lie groups, the exponential map is the matrix exponential $\exp(X) = \sum_{n=0}^{\infty} \frac{X^n}{n!}$
• Adjoint representation captures the action of a Lie group on its own Lie algebra, defined as $\mathrm{Ad}_g(X) = gXg^{-1}$ for $g \in G$ and $X \in \mathfrak{g}$
• Root systems and root spaces play a crucial role in the classification and structure of semisimple Lie algebras
• Weyl group generated by reflections associated with the root system, captures the symmetries of the root system and the corresponding Lie algebra

Fundamental Structures

• Semisimple Lie algebras direct sums of simple Lie algebras, which have no non-trivial ideals
  • Characterized by having a non-degenerate Killing form, a symmetric bilinear form defined as $B(X, Y) = \mathrm{tr}(\mathrm{ad}_X \circ \mathrm{ad}_Y)$
• Solvable Lie algebras contain a chain of ideals, each of which is contained in the next, and the last ideal is the entire Lie algebra
• Nilpotent Lie algebras Lie algebras for which the lower central series terminates at zero after a finite number of steps
• Cartan subalgebras maximal abelian subalgebras of a Lie algebra, consisting of semisimple elements
  • Play a crucial role in the classification of semisimple Lie algebras and the construction of root systems
• Universal enveloping algebra associative algebra generated by the elements of a Lie algebra, subject to the relations defined by the Lie bracket
• Casimir elements central elements in the universal enveloping algebra, commute with all other elements and act as scalar multiples of the identity on irreducible representations

Advanced Representation Theory

• Highest weight representations constructed using a highest weight vector, annihilated by the action of positive root spaces
  • Highest weight uniquely determines the irreducible representation, allowing for a classification of irreducible representations
• Verma modules induced representations constructed from a one-dimensional representation of a Borel subalgebra (maximal solvable subalgebra)
  • Contain a unique irreducible quotient, which is the corresponding highest weight representation
• Weyl character formula expresses the character of an irreducible representation in terms of its highest weight and the root system of the Lie algebra
• Tensor product decomposition describes how the tensor product of two irreducible representations decomposes into a direct sum of irreducible representations
  • Governed by the Littlewood-Richardson rules, which provide a combinatorial method for determining the multiplicities of the irreducible components
• Schur-Weyl duality relates the representation theory of the general linear group $\mathrm{GL}(V)$ and the symmetric group $S_n$ acting on the tensor product space $V^{\otimes n}$
• Crystal bases combinatorial objects that encode the structure of representations, providing a canonical basis for the representation space

Algebraic Groups and Lie Groups

• Algebraic groups defined as subgroups of $\mathrm{GL}_n(\mathbb{C})$ that are closed in the Zariski topology
  • Can be studied using tools from algebraic geometry, such as polynomial equations and algebraic varieties
• Chevalley groups constructed as algebraic groups associated with semisimple complex Lie algebras
  • Obtained by exponentiating certain integral combinations of the Chevalley basis elements of the Lie algebra
• Reductive groups algebraic groups whose unipotent radical (maximal connected normal unipotent subgroup) is trivial
  • Include semisimple groups and algebraic tori (products of multiplicative groups of the base field)
• Flag varieties projective varieties that parametrize certain types of subspaces (flags) of a vector space
  • Play a crucial role in the geometry and representation theory of algebraic groups and Lie groups
• Bruhat decomposition describes the double coset decomposition of a reductive group with respect to a Borel subgroup and its opposite
  • Indexed by elements of the Weyl group, provides a cellular decomposition of the flag variety
• Representation theory of compact Lie groups simpler than that of general Lie groups, as all representations are direct sums of irreducible unitary representations
  • Characterized by the highest weight theorem, which states that irreducible representations are uniquely determined by their highest weights

Classification and Structure Theorems

• Cartan-Killing classification classifies simple Lie algebras over the complex numbers into four infinite families ($A_n$, $B_n$, $C_n$, $D_n$) and five exceptional cases ($E_6$, $E_7$, $E_8$, $F_4$, $G_2$)
  • Based on the properties of the root systems and the associated Dynkin diagrams
• Levi decomposition states that any finite-dimensional Lie algebra can be written as a semidirect product of a solvable ideal (its radical) and a semisimple subalgebra (a Levi subalgebra)
• Weyl's theorem on complete reducibility states that any finite-dimensional representation of a semisimple Lie algebra is completely reducible (a direct sum of irreducible representations)
• Borel-Weil theorem realizes irreducible representations of a semisimple Lie group as spaces of sections of line bundles over the flag variety
• Kostant's convexity theorem describes the projection of orbits in the dual of the Lie algebra onto the dual of a Cartan subalgebra
  • States that the projection is a convex polytope, the convex hull of the Weyl group orbit of the highest weight
• Harish-Chandra isomorphism establishes a correspondence between the center of the universal enveloping algebra and the Weyl group invariants in the symmetric algebra of the Cartan subalgebra

Applications in Physics and Geometry

• Gauge theories in physics use Lie groups to describe the symmetries of the fundamental forces (electromagnetic, weak, and strong interactions)
  • Gauge fields are represented by connections on principal bundles, with the Lie group acting as the structure group
• Representation theory of Lie groups plays a crucial role in quantum mechanics, describing the symmetries of physical systems and the states of particles
  • Irreducible unitary representations correspond to elementary particles, with the Casimir operators related to physical observables
• Homogeneous spaces and symmetric spaces naturally arise in geometry as quotients of Lie groups by closed subgroups
  • Provide a rich class of examples in Riemannian geometry and have applications in various areas of mathematics and physics
• Moment maps in symplectic geometry relate the action of a Lie group on a symplectic manifold to the dual of its Lie algebra
  • Play a key role in the study of Hamiltonian group actions and the construction of symplectic quotients
• Lie group methods in numerical analysis used to develop efficient algorithms for solving differential equations while preserving geometric structures
  • Examples include Lie group integrators for ordinary differential equations and Lie group methods for discretizing partial differential equations
• Lie groups in control theory describe the symmetries of control systems and provide a framework for studying controllability, observability, and optimal control problems

Computational Techniques

• Gröbner basis methods applied to the study of Lie algebras and their representations
  • Used to compute bases for invariant polynomials, classify orbits, and study the geometry of nilpotent orbits
• Computational methods for Lie groups include algorithms for matrix exponentiation, logarithms, and decompositions (such as the polar decomposition and the QR decomposition)
• Symbolic software packages (such as LiE, GAP, and SageMath) provide tools for working with Lie algebras, root systems, and representations
  • Allow for the computation of weight multiplicities, tensor product decompositions, and character formulas
• Computational approaches to the representation theory of algebraic groups and Lie groups include algorithms for computing with highest weight representations, crystal bases, and Kazhdan-Lusztig polynomials
• Numerical methods for Lie groups and homogeneous spaces include techniques for discretizing differential equations, interpolating on Lie groups, and computing geodesics and exponential maps
• Machine learning on Lie groups involves the development of neural network architectures and optimization algorithms that respect the group structure and symmetries
  • Applications include computer vision tasks (such as pose estimation and object detection) and the analysis of data with Lie group symmetries

Current Research and Open Problems

• Langlands program seeks to unify various areas of mathematics (including number theory, representation theory, and harmonic analysis) through a web of conjectures relating Galois representations, automorphic forms, and L-functions
  • Langlands correspondence for Lie groups and Lie algebras is an active area of research, with connections to geometric representation theory and the geometric Langlands program
• Categorification aims to lift algebraic structures (such as Lie algebras and their representations) to categorical level, replacing vector spaces with categories and linear maps with functors
  • Leads to the study of higher representation theory, including Khovanov homology and categorified quantum groups
• Infinite-dimensional Lie theory studies Lie algebras and Lie groups in infinite dimensions, such as loop groups, affine Lie algebras, and Kac-Moody algebras
  • Plays a central role in conformal field theory, integrable systems, and the geometric Langlands program
• Quantum groups are deformations of universal enveloping algebras of Lie algebras, depending on a parameter $q$
  • Provide a framework for studying the representation theory of Lie algebras and Lie groups in a deformed setting, with connections to knot theory, low-dimensional topology, and mathematical physics
• Geometric complexity theory seeks to resolve fundamental questions in computational complexity (such as P vs. NP) using tools from algebraic geometry and representation theory
  • Involves the study of orbit closures, the geometry of determinantal varieties, and the representation theory of algebraic groups
• Lie groups and Lie algebras in positive characteristic exhibit new phenomena and challenges compared to the characteristic zero case
  • Active areas of research include the study of modular representation theory, the Lusztig character formula, and the representation theory of algebraic groups over finite fields