🧮Non-associative Algebra Unit 2 – Lie algebras

What are Lie Algebras?

• Lie algebras are vector spaces equipped with a bilinear operation called the Lie bracket, denoted as $[x, y]$, satisfying specific properties
• The Lie bracket operation is non-associative, meaning $[x, [y, z]] \neq [[x, y], z]$ in general, which distinguishes Lie algebras from associative algebras
• Lie algebras are named after Sophus Lie, a Norwegian mathematician who introduced them in the late 19th century while studying continuous transformation groups
• Lie algebras capture the infinitesimal behavior of Lie groups, which are smooth manifolds with a group structure compatible with the manifold structure
• The Lie bracket of two elements in a Lie algebra can be interpreted as the commutator of the corresponding elements in the associated Lie group
• Lie algebras are used to study the structure and properties of Lie groups, as they often have a simpler and more tractable algebraic structure
• Examples of Lie algebras include the space of $n \times n$ matrices with the commutator bracket $[A, B] = AB - BA$ (general linear Lie algebra $\mathfrak{gl}(n)$) and the space of vector fields on a smooth manifold with the Lie bracket given by the Jacobi-Lie bracket

Key Definitions and Terminology

• Vector space: A set equipped with addition and scalar multiplication operations satisfying certain axioms, serving as the underlying structure for a Lie algebra
• Lie bracket: A bilinear operation $[·, ·]: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}$ on a Lie algebra $\mathfrak{g}$ satisfying skew-symmetry and the Jacobi identity
• Skew-symmetry: The property of the Lie bracket where $[x, y] = -[y, x]$ for all $x, y \in \mathfrak{g}$
• Jacobi identity: The identity $[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$ for all $x, y, z \in \mathfrak{g}$, ensuring the Lie bracket is compatible with itself
• Derivation: A linear map $D: \mathfrak{g} \to \mathfrak{g}$ satisfying the Leibniz rule $D([x, y]) = [D(x), y] + [x, D(y)]$ for all $x, y \in \mathfrak{g}$
• Ideal: A subspace $\mathfrak{i} \subseteq \mathfrak{g}$ satisfying $[x, y] \in \mathfrak{i}$ for all $x \in \mathfrak{g}$ and $y \in \mathfrak{i}$
• Simple Lie algebra: A non-abelian Lie algebra with no non-trivial ideals
• Semisimple Lie algebra: A direct sum of simple Lie algebras

Fundamental Properties of Lie Algebras

• The Lie bracket is bilinear, meaning $[ax + by, z] = a[x, z] + b[y, z]$ and $[z, ax + by] = a[z, x] + b[z, y]$ for all $x, y, z \in \mathfrak{g}$ and $a, b \in \mathbb{F}$ (the underlying field)
• The Lie bracket satisfies skew-symmetry, i.e., $[x, y] = -[y, x]$ for all $x, y \in \mathfrak{g}$
  • As a consequence, $[x, x] = 0$ for all $x \in \mathfrak{g}$
• The Jacobi identity holds: $[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$ for all $x, y, z \in \mathfrak{g}$
  • This identity ensures the Lie bracket is compatible with itself and can be viewed as a generalization of the Leibniz rule for derivations
• The center of a Lie algebra $\mathfrak{g}$, denoted as $Z(\mathfrak{g})$, is the set of elements that commute with all elements in $\mathfrak{g}$, i.e., $Z(\mathfrak{g}) = \{x \in \mathfrak{g} : [x, y] = 0 \text{ for all } y \in \mathfrak{g}\}$
• The derived series of a Lie algebra $\mathfrak{g}$ is defined recursively as $\mathfrak{g}^{(0)} = \mathfrak{g}$ and $\mathfrak{g}^{(i+1)} = [\mathfrak{g}^{(i)}, \mathfrak{g}^{(i)}]$, measuring the non-commutativity of $\mathfrak{g}$
• A Lie algebra is nilpotent if its lower central series, defined as $\mathfrak{g}_0 = \mathfrak{g}$ and $\mathfrak{g}_{i+1} = [\mathfrak{g}, \mathfrak{g}_i]$, terminates at zero after finitely many steps
• A Lie algebra is solvable if its derived series terminates at zero after finitely many steps

Types and Classification of Lie Algebras

• Abelian Lie algebras: Lie algebras where the Lie bracket is identically zero, i.e., $[x, y] = 0$ for all $x, y \in \mathfrak{g}$
• Simple Lie algebras: Non-abelian Lie algebras with no non-trivial ideals
  • Examples include the special linear Lie algebra $\mathfrak{sl}(n)$ and the orthogonal Lie algebra $\mathfrak{so}(n)$
• Semisimple Lie algebras: Direct sums of simple Lie algebras
  • Characterized by having no non-zero solvable ideals
• Nilpotent Lie algebras: Lie algebras whose lower central series terminates at zero after finitely many steps
  • Examples include the Heisenberg Lie algebra and the upper triangular matrices with zero diagonal
• Solvable Lie algebras: Lie algebras whose derived series terminates at zero after finitely many steps
  • Examples include the Borel subalgebra of upper triangular matrices in $\mathfrak{gl}(n)$
• Classical Lie algebras: The infinite series of simple Lie algebras $\mathfrak{sl}(n)$, $\mathfrak{so}(n)$, $\mathfrak{sp}(n)$, and the exceptional Lie algebras $\mathfrak{g}_2$, $\mathfrak{f}_4$, $\mathfrak{e}_6$, $\mathfrak{e}_7$, $\mathfrak{e}_8$
• Real and complex Lie algebras: Lie algebras over the field of real numbers $\mathbb{R}$ or complex numbers $\mathbb{C}$, respectively

Representation Theory

• Representation: A linear map $\rho: \mathfrak{g} \to \mathfrak{gl}(V)$ from a Lie algebra $\mathfrak{g}$ to the general linear Lie algebra of a vector space $V$, preserving the Lie bracket
  • In other words, $\rho([x, y]) = [\rho(x), \rho(y)]$ for all $x, y \in \mathfrak{g}$
• Irreducible representation: A representation with no non-trivial invariant subspaces
• Completely reducible representation: A representation that can be decomposed into a direct sum of irreducible representations
• Adjoint representation: The representation $\text{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})$ defined by $\text{ad}_x(y) = [x, y]$ for all $x, y \in \mathfrak{g}$
• Weights: Eigenvalues of a maximal toral subalgebra (a subalgebra consisting of semisimple elements) acting on a representation
• Highest weight: The unique weight of an irreducible representation that is maximal with respect to a certain partial order
• Weyl's theorem: Every finite-dimensional representation of a semisimple Lie algebra is completely reducible
• Schur's lemma: Morphisms between irreducible representations are either zero or isomorphisms

Applications in Physics and Mathematics

• Quantum mechanics: Lie algebras are used to describe symmetries of quantum systems, with the Lie bracket corresponding to the commutator of observables
• Particle physics: Lie algebras, such as $\mathfrak{su}(3)$ and $\mathfrak{su}(2) \oplus \mathfrak{u}(1)$, describe the symmetries of fundamental particles and interactions in the Standard Model
• General relativity: The Lorentz and Poincaré Lie algebras capture the symmetries of spacetime in special and general relativity, respectively
• Differential geometry: Lie algebras of vector fields on a manifold are used to study the infinitesimal symmetries and geometric properties of the manifold
• Integrable systems: Lie algebras and their representations play a crucial role in the study of integrable systems, such as the Korteweg-de Vries equation and the Toda lattice
• Representation theory: Lie algebras provide a rich source of examples and applications in the study of group representations and character theory
• Algebraic topology: Lie algebras and their cohomology are used to construct and study invariants of topological spaces, such as the Chevalley-Eilenberg complex

Connections to Lie Groups

• Lie's third theorem: Every finite-dimensional Lie algebra over $\mathbb{R}$ or $\mathbb{C}$ is the Lie algebra of some Lie group
• Exponential map: A map $\exp: \mathfrak{g} \to G$ from a Lie algebra $\mathfrak{g}$ to its associated Lie group $G$, given by $\exp(x) = e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$
  • The exponential map is a local diffeomorphism near the identity and provides a way to relate the Lie algebra to the Lie group
• Baker-Campbell-Hausdorff formula: A formula expressing the product of two exponentials in a Lie group in terms of the Lie bracket of their Lie algebra elements
• Adjoint representation of a Lie group: The representation $\text{Ad}: G \to \text{GL}(\mathfrak{g})$ defined by $\text{Ad}_g = d(\text{conj}_g)_e$, where $\text{conj}_g(h) = ghg^{-1}$ is the conjugation map
• Correspondence between Lie subgroups and Lie subalgebras: Connected Lie subgroups of a Lie group $G$ correspond bijectively to Lie subalgebras of its Lie algebra $\mathfrak{g}$
• Covering groups: The universal covering group of a connected Lie group $G$ has the same Lie algebra as $G$, providing a way to study $G$ through its Lie algebra

Advanced Topics and Current Research

• Infinite-dimensional Lie algebras: Lie algebras of infinite dimension, such as the Virasoro algebra and the Kac-Moody algebras, which have applications in conformal field theory and string theory
• Quantum groups: Deformations of universal enveloping algebras of Lie algebras, which have applications in knot theory, topological quantum field theory, and integrable systems
• Lie superalgebras: Generalizations of Lie algebras that include both even and odd elements, with applications in supersymmetric quantum mechanics and supergravity
• Lie algebroids: Generalizations of Lie algebras that are vector bundles equipped with a Lie bracket and an anchor map, with applications in Poisson geometry and mathematical physics
• Vertex algebras: Algebraic structures that combine features of Lie algebras and complex analysis, with applications in conformal field theory and the geometric Langlands program
• Categorification of Lie algebras: The study of higher categorical structures that give rise to Lie algebras, such as 2-groups and 2-categories, with applications in topological quantum field theory and geometric representation theory
• Lie algebra cohomology: The study of the cohomology of Lie algebras and its relation to the geometry and topology of homogeneous spaces and flag varieties
• Geometric Langlands program: A far-reaching generalization of the classical Langlands program that relates representations of Lie algebras to geometric objects on algebraic curves, with connections to number theory, algebraic geometry, and mathematical physics