Algebra
Lie Algebras
Цели урока
- Understand the axioms of a Lie algebra: antisymmetry and the Jacobi identity
- Know the structure of sl(2,C): generators H, E, F and their relations
- Apply the Killing form and Cartan's criterion for semisimplicity
- Understand the adjoint representation and the concept of an ideal
Предварительные знания
- Linear algebra
- Matrix groups
- Abstract algebra basics
Three numbers describe the angular momentum of the entire universe. The structure constants of so(3) are all of rotational mechanics, quantum spin, and elementary particle symmetry. One small algebra, infinite consequences.
- Google DeepMind: robot motion learning via SO(3) rotation groups - the Lie algebra so(3) encodes physical constraints
- Quantum mechanics: particle spin is a representation of su(2), selection rules follow from [H,E]=2E
- Particle physics: the Standard Model is based on su(3) x su(2) x u(1) - a direct sum of Lie algebras
- String theory: exceptional algebra E8 describes the symmetry of 10-dimensional spacetime
From Lie to Cartan's classification
Sophus Lie (1842-1899) studied symmetries of differential equations and found that continuous groups are determined by their linear approximations - Lie algebras. Wilhelm Killing in 1888-1890 published the complete classification of simple Lie algebras, including exceptional types G2, F4, E6, E7, E8 - long before physics found any application for them. Elie Cartan in 1894 corrected Killing's errors and provided rigorous proofs. The Killing form ironically bears his name.
Lie Bracket and the Definition of a Lie Algebra
2023. Google DeepMind trains robots to walk on uneven terrain. The key object: rotation group SO(3). Its Lie algebra so(3) has exactly 3 basis elements and 3 structure constants. Three numbers encode all rotational mechanics of the universe.
A Lie algebra is a vector space V over a field k equipped with a bilinear bracket [x,y] satisfying antisymmetry and the Jacobi identity. This is not an associative product - it is something structurally different.
The prototype: gl(n)
Matrices as a Lie algebra
The space of all n x n matrices with commutator [A,B] = AB - BA is the Lie algebra gl(n). The Jacobi identity follows from associativity of matrix multiplication. The subalgebra sl(n) consists of trace-zero matrices.
Any associative algebra A becomes a Lie algebra via [a,b] = ab - ba. This is the main source of examples. Physicists obtain Lie algebras from matrix groups precisely this way.
Besides antisymmetry, which identity must the Lie bracket satisfy?
The Jacobi identity together with antisymmetry and bilinearity completely defines a Lie algebra.
The Algebra sl(2,C) and Its Structure
sl(2,C) is the smallest nontrivial simple Lie algebra. Three generators, three relations. From them grows all of quantum spin mechanics, particle classification, and the Dirac equation. Three numbers rule physics.
sl(2) over R and over C are different Lie algebras. su(2) is the compact real form of sl(2,C). Physicists use su(2) for spin, mathematicians use sl(2,C) for representation theory. Over R there is no basis {H,E,F} with real matrices satisfying the same relations with the same eigenvalue structure.
What is the commutator [E,F] in sl(2,C)?
The three relations [H,E]=2E, [H,F]=-2F, [E,F]=H completely determine the structure of sl(2,C).
Killing Form and Semisimple Algebras
1888. Wilhelm Killing publishes the classification of simple Lie algebras over C. Besides four infinite series (A, B, C, D), he finds five exceptional ones: G2, F4, E6, E7, E8. String theorists in 2024 use E8 to describe the symmetry of 10-dimensional spacetime. Killing found these without knowing what physics would need them for.
| Series | Algebra | dim | Physical application |
|---|---|---|---|
| A_n | sl(n+1) | n(n+2) | SU(n+1), quarks (A_2 = su(3)) |
| B_n | so(2n+1) | n(2n+1) | Orthogonal groups, spinors |
| C_n | sp(2n) | n(2n+1) | Hamiltonian mechanics, symplectic |
| D_n | so(2n) | n(2n-1) | Rotations in even dimensions |
| G_2 | g_2 | 14 | Octonions, M-theory |
Cartan-Killing theorem: any semisimple Lie algebra over C decomposes uniquely into a direct sum of simple ones. Simple algebras are completely classified. This is the most complete structural theorem in algebra.
The Killing form B(x,y) = tr(ad_x circ ad_y). What does it characterize?
Cartan's criterion: semisimplicity is equivalent to nondegeneracy of the Killing form. For sl(2,C) the form is nondegenerate - sl(2,C) is semisimple and simple.
Adjoint Representation and Ideals
Every Lie algebra acts on itself. This is not a tautology - it is structural information. The adjoint representation ad_x(y) = [x,y] defines a linear operator on g. Its image is everything that x can 'shift'.
Matrix of ad_H in sl(2,C)
Computing the adjoint operator
In basis {H,E,F}: ad_H(H)=[H,H]=0, ad_H(E)=[H,E]=2E, ad_H(F)=[H,F]=-2F. Matrix of ad_H = diag(0,2,-2). Eigenvalues 0, 2, -2 are the roots of sl(2,C).
Roots of a Lie algebra are eigenvalues of ad_h for h in the Cartan subalgebra. For sl(2,C), the Cartan subalgebra is one-dimensional: h = span{H}. Roots: +2 and -2.
What is an ideal of a Lie algebra g?
An ideal is an invariant subspace under all operators ad_x. For simple Lie algebras there are no nontrivial ideals.
Connections to other topics
Lie algebras are the language of symmetry in physics and geometry.
- Quantum mechanics — Related topic
- Representation theory — Related topic
- Quantum groups — Related topic
- Universal enveloping algebra — Related topic
Итоги
- Lie algebra = vector space + bracket [x,y] antisymmetric and satisfying Jacobi identity
- sl(2,C) = span{H,E,F} with [H,E]=2E, [H,F]=-2F, [E,F]=H - prototype of the whole theory
- Killing form B(x,y) = tr(ad_x ad_y): g is semisimple iff B is nondegenerate
- Adjoint representation ad_x(y) = [x,y] defines the action of g on itself
- Simple Lie algebras over C: four infinite series A,B,C,D plus five exceptional types
Вопросы для размышления
- Why is the Jacobi identity an analogue of associativity rather than associativity itself?
- How does the Killing form distinguish a semisimple algebra from a nilpotent one?
- Why do the exceptional Lie algebras E6, E7, E8 not fit into any infinite series?
Связанные уроки
- alg-25-universal-enveloping — U(g) is built on top of the Lie algebra
- alg-27-lie-repr — Representation theory uses the sl(2) structure
- alg-26-quantum-groups — Quantum groups are q-deformations of Lie algebras