| Lie algebra | |
|---|---|
| Name | Lie algebra |
| Type | Algebraic structure |
| Introduced | 19th century |
| Field | Mathematics; Quantum mechanics |
| Related | Lie group, Representation theory, Universal enveloping algebra |
Lie algebra
A Lie algebra is a vector space equipped with a bilinear, antisymmetric product called the Lie bracket that satisfies the Jacobi identity. In the context of Quantum mechanics and broader Quantum Physics, Lie algebras encode infinitesimal generators of continuous symmetry transformations and underlie conserved quantities via Noether's theorem. They provide the algebraic framework for the study of angular momentum, spin, and internal symmetry groups in atomic, nuclear and particle physics.
A Lie algebra over a field F (commonly ℝ or ℂ) is a vector space g with a binary operation [·,·]: g × g → g that is bilinear, skew-symmetric ([X,Y] = −[Y,X]) and satisfies the Jacobi identity: [X,[Y,Z + [Y,[Z,X + [Z,[X,Y = 0 for all X,Y,Z ∈ g. Important elementary constructions include the abelian Lie algebra (where the bracket vanishes) and the commutator Lie algebra associated to an associative algebra A via [X,Y]=XY−YX. Morphisms are linear maps preserving brackets, and ideals, centers, derived subalgebras, solvable and nilpotent chains define the algebra's structural properties. The Killing form, a symmetric bilinear form introduced by Élie Cartan, is central for classification of semisimple Lie algebras.
Key examples arising in physics include the three-dimensional real Lie algebra so(3) of rotations and its complexification so(3,ℂ) ≃ sl(2,ℂ) in relativistic contexts, and the unitary algebra u(n) and special unitary algebra su(n) which govern internal symmetries and gauge theory groups like SU(2) and SU(3). Matrix Lie algebras such as gl(n), sl(n), so(n), and sp(2n) provide concrete models. The classification of complex finite-dimensional simple Lie algebras by Cartan and Killing yields the Dynkin diagrams A_n, B_n, C_n, D_n and exceptional types G2, F4, E6, E7, E8; these are crucial in model building in particle physics and in the study of grand unified theories (e.g., SU(5), SO(10), E6). Infinite-dimensional examples, notably Kac–Moody algebras and the Virasoro algebra, play roles in conformal field theory and string theory.
A representation of a Lie algebra g is a Lie algebra homomorphism ρ: g → gl(V) where V is a vector space (a g-module). Finite-dimensional representation theory for semisimple Lie algebras is governed by highest-weight theory developed by Weyl and Cartan; irreducible representations are labeled by dominant integral weights and constructed via Verma modules and highest-weight modules. In quantum contexts, representations realize physical states: for su(2) the spin-j representations classify angular momentum multiplets, while representations of su(3) describe flavor and color multiplets in quantum chromodynamics (QCD). Tools include Casimir operators (central elements in the universal enveloping algebra), weight lattices, and character formulae such as the Weyl character formula.
In canonical quantization the commutator of observables defines a Lie algebra structure on the space of operators: [A,B] = iħ(C) encodes quantum brackets corresponding to classical Poisson brackets. The angular momentum operators J_x, J_y, J_z form an su(2) Lie algebra with commutation relations [J_i,J_j]=iħ ε_{ijk} J_k; these determine spectra and selection rules in atomic and molecular systems. The Heisenberg algebra generated by position and momentum operators x and p satisfies [x,p]=iħ, forming a central extension of an abelian Lie algebra; its representations via the Schrödinger and Fock spaces are foundational in quantum harmonic oscillator analysis. Symmetry-adapted bases, ladder operators and group-theoretic selection rules arise from representation theory of the relevant Lie algebras.
Lie algebras are the tangent-space, infinitesimal generators of Lie group actions: for a Lie group G acting unitarily on a Hilbert space, its Lie algebra g provides self-adjoint generators via Stone's theorem when represented exponentiated to one-parameter unitary groups. Physical conserved charges correspond to these generators; examples include generators of rotations (angular momentum), translations (momentum), and internal gauge transformations (charges in Noether's theorem). The exponential map links g to G locally, while global topology (e.g., nontrivial fundamental group leading to coverings like SU(2) → SO(3)) influences projective representations and the existence of spinor representations.
In quantum field theory (QFT), local gauge symmetries are modeled by Lie algebras of compact groups: U(1) for electromagnetism, SU(2)×U(1) for the electroweak interaction, and SU(3) for QCD. Lie algebra structure constants appear in interaction vertices and renormalization; anomalies are obstructions in representation-theoretic terms. Spontaneous symmetry breaking involves subalgebras and coset spaces G/H, with associated Goldstone bosons. Grand unified theories employ larger simple Lie algebras (e.g., SO(10), E8) and their representation embeddings to unify fermion multiplets. Conformal and infinite-dimensional algebras such as the Virasoro and affine Kac–Moody algebras appear in two-dimensional QFT and string theory.
Structure theory decomposes Lie algebras via Levi decomposition into semisimple and solvable parts; ideals, root systems and Cartan subalgebras enable analysis of representations and spectra. The universal enveloping algebra U(g) is an associative algebra containing g such that g-modules correspond to U(g)-modules; Poincaré–Birkhoff–Witt theorem describes its basis. Center elements of U(g) (Casimir operators) act by scalars on irreducible modules and are used to label quantum states. Quantization procedures and deformation quantization relate to noncommutative deformations of U(g), while Hopf algebra structures on U(g) underpin quantum group generalizations relevant to integrable systems and topological quantum field theory.
Category:Mathematical physics Category:Lie algebras