🧩Representation Theory Unit 10 – Introduction to Lie Algebras

Key Concepts and Definitions

• Lie algebra defined as a vector space $V$ over a field $F$ with a bilinear operation called the Lie bracket $[x, y]$ satisfying antisymmetry and the Jacobi identity
• Lie bracket operation $[x, y]$ measures the extent to which the multiplication in a Lie group fails to be commutative near the identity element
• Dimension of a Lie algebra is its dimension as a vector space over its base field
• Abelian Lie algebra has a Lie bracket that always vanishes, i.e., $[x, y] = 0$ for all elements $x$ and $y$
• Subalgebra is a vector subspace that is closed under the Lie bracket operation
  • Proper subalgebra is a subalgebra that is strictly smaller than the whole Lie algebra
• Ideal is a subalgebra $I$ satisfying $[x, y] \in I$ for all $x \in I$ and $y \in V$
• Simple Lie algebra contains no non-trivial ideals and is not abelian
• Semisimple Lie algebra is a direct sum of simple Lie algebras
• Nilpotent Lie algebra has a central series that terminates in zero
  • Lower central series defined by $L_1 = L$ and $L_{i+1} = [L, L_i]$

Historical Context and Motivation

• Sophus Lie introduced Lie algebras in the late 19th century to study symmetries of differential equations
• Lie algebras capture the local structure of Lie groups near the identity element
• Infinitesimal approach to studying Lie groups by linearizing the group operation
• Representation theory of Lie algebras used to construct representations of Lie groups
• Lie algebras play a crucial role in various areas of mathematics and physics
  • Symmetries in quantum mechanics and particle physics
  • Differential geometry and topology
  • Integrable systems and mathematical physics
• Classification of simple Lie algebras over $\mathbb{C}$ by Killing and Cartan in the late 19th and early 20th centuries
• Real forms of complex Lie algebras studied by Élie Cartan

Fundamental Properties of Lie Algebras

• Antisymmetry of the Lie bracket: $[x, y] = -[y, x]$ for all elements $x$ and $y$
• Jacobi identity: $[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$ for all elements $x$, $y$, and $z$
• Bilinearity of the Lie bracket over the base field $F$
  • $[ax + by, z] = a[x, z] + b[y, z]$ for all $a, b \in F$ and $x, y, z \in V$
  • $[x, ay + bz] = a[x, y] + b[x, z]$ for all $a, b \in F$ and $x, y, z \in V$
• Adjoint representation $\text{ad}_x: V \to V$ defined by $\text{ad}_x(y) = [x, y]$ is a derivation of the Lie algebra
• Engel's theorem states that a Lie algebra is nilpotent if and only if $\text{ad}_x$ is nilpotent for all $x \in V$
• Derived series of a Lie algebra defined by $L^{(0)} = L$ and $L^{(i+1)} = [L^{(i)}, L^{(i)}]$
  • Lie algebra is solvable if its derived series terminates in zero

Classification of Simple Lie Algebras

• Killing-Cartan classification of simple Lie algebras over $\mathbb{C}$
  • Four infinite families: $A_n$ ($n \geq 1$), $B_n$ ($n \geq 2$), $C_n$ ($n \geq 3$), and $D_n$ ($n \geq 4$)
  • Five exceptional cases: $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$
• Each simple Lie algebra corresponds to a connected and simply-connected simple Lie group
• Dynkin diagrams encode the structure of simple Lie algebras
  • Nodes represent simple roots and edges represent angles between them
• Weyl group is the reflection group generated by reflections in the hyperplanes orthogonal to the roots
• Cartan matrix encodes the angles between simple roots and determines the Lie algebra up to isomorphism
• Real forms of complex simple Lie algebras classified using Satake diagrams and Vogan diagrams

Representation Theory Basics for Lie Algebras

• Representation of a Lie algebra $L$ is a vector space $V$ together with a Lie algebra homomorphism $\rho: L \to \mathfrak{gl}(V)$
  • $\rho([x, y]) = [\rho(x), \rho(y)]$ for all $x, y \in L$
• Irreducible representation has no non-trivial invariant subspaces
• Completely reducible representation decomposes into a direct sum of irreducible representations
• Adjoint representation $\text{ad}: L \to \mathfrak{gl}(L)$ defined by $\text{ad}_x(y) = [x, y]$ is a representation of $L$
• Weights are eigenvalues of the action of a Cartan subalgebra on a representation
  • Weight space is the eigenspace corresponding to a weight
• Highest weight theory classifies irreducible representations of semisimple Lie algebras
  • Highest weight vector is annihilated by the action of positive root spaces
  • Irreducible representation generated by a highest weight vector

Important Examples and Applications

• $\mathfrak{sl}_n(\mathbb{C})$, the special linear Lie algebra, consists of $n \times n$ matrices with trace zero
  • Corresponds to the special linear group $SL_n(\mathbb{C})$
• $\mathfrak{so}_n(\mathbb{R})$, the special orthogonal Lie algebra, consists of skew-symmetric $n \times n$ matrices
  • Corresponds to the special orthogonal group $SO(n)$
• $\mathfrak{su}_n$, the special unitary Lie algebra, consists of skew-Hermitian $n \times n$ matrices with trace zero
  • Corresponds to the special unitary group $SU(n)$
• Heisenberg Lie algebra is a nilpotent Lie algebra with basis $\{x, y, z\}$ and non-zero brackets $[x, y] = z$
• Virasoro algebra is an infinite-dimensional Lie algebra important in conformal field theory and string theory
• Kac-Moody algebras are infinite-dimensional generalizations of semisimple Lie algebras
  • Affine Lie algebras are a special case related to loop groups and central extensions

Computational Techniques and Exercises

• Computation of Lie brackets using the antisymmetry and bilinearity properties
• Verifying the Jacobi identity for specific examples of Lie algebras
• Finding subalgebras and ideals of given Lie algebras
• Determining the derived series and lower central series of Lie algebras
• Classifying low-dimensional Lie algebras up to isomorphism
• Computing the Killing form and Cartan matrix for simple Lie algebras
• Constructing irreducible representations using highest weight theory
• Decomposing representations into irreducible components
• Calculating weights and weight spaces for representations of Lie algebras
• Exploring the structure of root systems and Weyl groups

Connections to Other Mathematical Fields

• Relationship between Lie algebras and Lie groups via the exponential map
  • Baker-Campbell-Hausdorff formula expresses the group operation in terms of Lie brackets
• Lie algebra cohomology and its applications in representation theory and geometry
• Poisson algebras as a generalization of Lie algebras with a compatible associative multiplication
• Quantum groups as deformations of universal enveloping algebras of Lie algebras
• Lie algebroids as a generalization of Lie algebras with a vector bundle structure
  • Lie groupoids integrate Lie algebroids analogously to Lie groups integrating Lie algebras
• Representation theory of Lie algebras applied to physics
  • Symmetries in quantum mechanics and particle physics
  • Gauge theories and Yang-Mills equations
• Connections to algebraic geometry through the study of algebraic groups and their Lie algebras
• Relationship to differential geometry via Lie algebra valued differential forms and connections