Lie algebra
In mathematics, a Lie algebra (named after Sophus Lie, pronounced "lee") is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds.
Definition
A Lie algebra is a vector space g over some field F (typically the real or complex numbers) together with a binary operation [·, ·] : g × g -> g, called the Lie bracket, which satisfies the following properties:
- it is bilinear, i.e., [a x + b y, z] = a [x, z] + b [y, z] and [z, a x + b y] = a [z, x] + b [z, y] for all a, b in F and all x, y, z in g.
- it satisfies the Jacobi identity, i.e., [[x, y], z] + [[z, x], y] + [[y, z], x] = 0 for all x, y, z in g.
- [x, x] = 0 for all x in g.
Note that the first and third properties together imply [x, y] = − [y, x] for all x, y in g ("anti-symmetry") if the characteristic of F is not 2. Note also that the multiplication represented by the Lie bracket is not in general associative, that is, [[x, y], z] need not equal [x, [y, z]].
Examples
Every vector space becomes an abelian Lie algebra trivially if we define the Lie bracket to be identically zero.
Euclidean space R3 becomes a Lie algebra with the Lie bracket given by the cross-product of vectors.
If an associative algebra A with multiplication * is given, it can be turned into a Lie algebra by defining [x, y] = x * y − y * x. This expression is called the commutator of x and y. Conversely, it can be shown that every Lie algebra can be embedded into one that arises from an associative algebra in this fashion.
Other important examples of Lie algebras come from differential topology: the vector fields on a differentiable manifold form an infinite dimensional Lie algebra; for two vector fields X and Y, the Lie bracket [X, Y] is defined by
- [X, Y] f = (XY − YX) f for every function f on the manifold
(here we view vector fields as operators that turn functions on a manifold into other functions). This is the Lie algebra of the infinite dimensional Lie group of diffeomorphisms of the manifold.
The vector space of left-invariant vector fields on a Lie group is closed under this operation and is therefore a finite dimensional Lie algebra. One may alternatively think of the underlying vector space of the Lie algebra belonging to a Lie group as the tangent space at the group's identity element. The multiplication is the differential of the group commutator, (a,b) |-> aba−1b−1, at the identity element.
As a concrete example, consider the Lie group SL(n,R) of all n-by-n matrices with real entries and determinant 1. The tangent space at the identity matrix may be identified with the space of all real n-by-n matrices with trace 0, and the Lie algebra structure coming from the Lie group coincides with the one arising from commutators of matrix multiplication.
For more examples of Lie groups and their associated Lie algebras, see the Lie group article.
Homomorphisms, Subalgebras and Ideals
A homomorphism φ : g -> h between Lie algebras g and h over the same base field F is an F-linear map such that [φ(x), φ(y)] = φ([x, y]) for all x and y in g. The composition of such homomorphisms is again a homomorphism, and the Lie algebras over the field F, together with these morphisms, form a category. If such a homomorphism is bijective, it is called an isomorphism, and the two Lie algebras g and h are called isomorphic. For all practical purposes, isomorphic Lie algebras are identical.
A subalgebra of the Lie algebra g is a linear subspace h of g such that [x, y] ∈ h for all x, y ∈ h. The subalgebra is then itself a Lie algebra.
An ideal of the Lie algebra g is a subspace h of g such that [a, y] ∈ h for all a ∈ g and y ∈ h. All ideals are subalgebras. If h is an ideal of g, then the quotient space g/h becomes a Lie algebra by defining [x + h, y + h] = [x, y] for all x, y ∈ g. The ideals are precisely the kernels of homomorphisms, and the fundamental theorem on homomorphisms is valid for Lie algebras.
Classification of Lie Algebras
Real and complex Lie algebras can be classified to some extent, and this classification is an important step toward the classification of Lie groups. Every finite-dimensional real or complex Lie algebra arises as the Lie algebra of unique real or complex simply connected Lie group (Ado's theorem), but there may be more than one group, even more than one connected group, giving rise to the same algebra. For instance, the groups SO(3) (3×3 orthogonal matrices of determinant 1) and SU(2) (2×2 unitary matrices of determinant 1) both give rise to the same Lie algebra, namely R3 with cross-product.
A Lie algebra is abelian if the Lie bracket vanishes, i.e. [x, y] = 0 for all x and y. More generally, a Lie algeba g is nilpotent if the lower central series
- g > [g, g] > [[g, g], g] > [[[g, g], g], g] > ...
becomes zero eventually. By Engel's theorem, a Lie algebra is nilpotent iff for every u in g the map ad(u): g -> g defined by
- ad(u)(v) = [u,v]
is nilpotent. More generally still, a Lie algebra g is said to be solvable if the derived series
- g > [g, g] > [[g, g], [g,g]] > [[[g, g], [g,g]],[[g, g], [g,g]]] > ...
becomes zero eventually. A maximal solvable subalgebra is called a Borel subalgebra.
A Lie algebra g is called semi-simple if the only solvable ideal of g is trivial. Equivalently, g is semi-simple if and only if the Killing form K(u,v) = tr(ad(u)ad(v)) is non-degenerate; here tr denotes the trace operator. When the field F is of characteristic zero, g is semi-simple if and only if every representation is completely reducible, that is for every invariant subspace of the representation there is an invariant complement (Weyl's theorem).
A Lie algebra is simple if it has no non-trivial ideals. In particular, a simple Lie algebra is semi-simple, and more generally, the semi-simple Lie algebras are the direct sums of the simple ones.
Semi-simple complex Lie algebras are classified through their root systems.