Lie algebra

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In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie, pronounced "lee") was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used.

Definition

A Lie algebra is a type of an algebra over a field; it is a vector space g over some field F together with a binary operation [·, ·] : g × gg, called the Lie bracket, which satisfies the following properties:

for all a, bF and all x, y, zg.
  • For all x in g
for all x, y, z in g.

Note that the first and second properties together imply

for all x, y in g ("anti-symmetry"). Conversely, the antisymmetry property implies property 2 above as long as F is not of characteristic 2.

Also note that the multiplication represented by the Lie bracket is not in general associative, that is, need not equal . Therefore, Lie algebras are not rings or associative algebras in the usual sense, although much of the same language is used to describe them.

Examples

1. Every vector space becomes an abelian Lie algebra trivially if we define the Lie bracket to be identically zero.

2. Euclidean space R3 becomes a Lie algebra with the Lie bracket given by the cross product of vectors.

3. If an associative algebra A with multiplication * is given, it can be turned into a Lie algebra by defining [xy] = 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. See universal enveloping algebra.

4. Another important example of a Lie algebra comes from differential topology: the smooth vector fields on a differentiable manifold form an infinite dimensional Lie algebra when equipped with the Lie derivative as the Lie bracket. The Lie derivative identifies a vector field X with a partial differential operator acting on any smooth scalar field f by letting X(f) be the directional derivative of f in the direction of X. Then in the expression (YX)(f), the juxtaposition YX represents composition of partial differential operators. Then the Lie bracket [X, Y] is defined by

[X, Y] f = (XYYX) f

for every smooth function f on the manifold.

This is the Lie algebra of the infinite-dimensional Lie group of diffeomorphisms of the manifold.

5. 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.

6. 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 : gh between Lie algebras g and h over the same base field F is an F-linear map such that [φ(x), φ(y)] = φ([xy]) 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 [xy] ∈ h for all x, yh. The subalgebra is then itself a Lie algebra.

An ideal of the Lie algebra g is a subspace h of g such that [ay] ∈ h for all ag and yh. 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] + h for all x, yg. The ideals are precisely the kernels of homomorphisms, and the fundamental theorem on homomorphisms is valid for Lie algebras.

Relation to Lie groups

Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups. Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the identity with the differential of the adjoint map, or by considering the left-invariant vector fields as mentioned in the examples. This association is functorial, meaning that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, quotients and cokernels of Lie algebras, respectively.

The functor which takes each Lie group to its Lie algebra and each homomorphism to its differential is a full and faithful exact functor. This functor is not invertible; different Lie groups may have the same Lie algebra, for example SO(3) and SU(2) have isomorphic Lie algebras. Even worse, some Lie algebras need not have any associated Lie group. Nevertheless, when the Lie algebra is finite-dimensional, there is always at least one Lie group whose Lie algebra is the one under discussion, and a preferred Lie group can be chosen. Any finite-dimensional connected Lie group has a universal cover. This group can be constructed as the image of the Lie algebra under the exponential map. More generally, we have that the Lie algebra is homeomorphic to a neighborhood of the identity. But globally, if the Lie group is compact, the exponential will not be injective, and if the Lie group is not connected, simply connected or compact, the exponential map need not be surjective.

If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a homeomorphism. Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.

The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the related matter of the representation theory of Lie groups. Any representation of a Lie algebra lifts uniquely to a representation of the corresponding connected simply connected group, and conversely any representation of the group induces a representation of corresponding Lie algebras; the representations are in one to one correspondence. Therefore, knowing the representations of a Lie algebra settles the question of representations of the group. As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the classification of Lie algebras is known (solved by Cartan et al. in the semisimple case).

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 algebra 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 if and only if for every u in g the map

ad(u): gg

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 and is not abelian. 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.

Category theoretic definition

Using the language of category theory, a Lie algebra can be defined as an object A in the category of vector spaces together with a morphism such that

where and σ is the cyclic permutation braiding . In diagrammatic form:

File:Liealgebra.png

See also

References

  • Humphreys, James E. Introduction to Lie Algebras and Representation Theory, Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. ISBN 0-387-90053-5
  • Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4

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