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Wikipedia / Lie algebra

In mathematics, a Lie algebra (pronounced ) is a vector space together with an operation called the Lie bracket, an alternating bilinear map, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator.

Lie algebras are closely related to Lie groups, which are groupss that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-dimensional Lie algebra over the real or complex numbers, there is a corresponding connected Lie group, unique up to covering spaces (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras, which are simpler objects of linear algebra.

In more detail: for any Lie group, the multiplication operation near the identity element 1 is commutative to first order. In other words, every Lie group G to G at the identity. To second order, the group operation may be non-commutative, and the second-order terms describing the non-commutativity of G the structure of a Lie algebra. It is a remarkable fact that these second-order terms (the Lie algebra) completely determine the group structure of G near the identity. They even determine G globally, up to covering spaces.

In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.

with Lie bracket defined by the cross product, and instead of associativity it satisfies the Jacobi identity:

This is the Lie algebra of the Lie group of rotations of space.

A fundamental example of a Lie algebra is the space of all linear maps from a vector space to itself, as discussed below. When the vector space has dimension n, this Lie algebra is called the general linear matrices.

History

Lie algebras were introduced to study the concept of infinitesimal transformations by Sophus Lie in the 1870s, and independently discovered by Wilhelm Killing in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group was used.

Definition

over a field together with a binary operation called the Lie bracket, satisfying the following axioms:

Given a Lie group, the Jacobi identity for its Lie algebra follows from the associativity of the group operation.

. Thus bilinearity and the alternating property together imply

It is customary to denote a Lie algebra by a lower-case fraktur. If a Lie algebra is associated with a Lie group, then the algebra is denoted by the fraktur version of the group's name: for example, the Lie algebra of SU(n).

Generators and dimension

The dimension of a Lie algebra over a field means its dimension as a vector space. In physics, a vector space basis may be called a set of generators of generators.

Basic examples

Abelian Lie algebras

A Lie algebra is called abelian endowed with the identically zero Lie bracket becomes a Lie algebra. Every one-dimensional Lie algebra is abelian, by the alternating property of the Lie bracket.

The Lie algebra of matrices

Basic constructions

Subalgebras, ideals and homomorphisms

The Lie bracket is not required to be associative. Nonetheless, much of the terminology for associative rings and algebras (and also for groups) has analogs for Lie algebras. A Lie subalgebra which is closed under the Lie bracket. An ideal is a linear subspace that satisfies the stronger condition

In the correspondence between Lie groups and Lie algebras, subgroups correspond to Lie subalgebras, and normal subgroups correspond to ideals.

A Lie algebra homomorphism is a linear map compatible with the respective Lie brackets:

An isomorphism of Lie algebras is a bijective homomorphism.

As with normal subgroups in groups, ideals in Lie algebras are precisely the kernels in it, the quotient Lie algebra of Lie algebras. The first isomorphism theorem.

are said to commute.

The centralizer is the set of elements commuting with itself is the center. Similarly, for a subspace S, the normalizer subalgebra of ..

Example

is an abelian Lie subalgebra. (It is a Cartan subalgebra, analogous to a maximal torus in the theory of compact Lie groups).

is an abelian Lie subalgebra, but it need not be an ideal.

Product and semidirect product

, the product, with Lie bracket

This is the product in the category

is said to be a semidirect product. See also semidirect sum of Lie algebras.

Derivations

For an algebra A over a field F, a derivation of A over F that satisfies the Leibniz rule

. (The definition makes sense for a possibly non-associative algebra of all derivations of A over F into a Lie algebra.

Informally speaking, the space of derivations of A is the Lie algebra of the automorphism group of A. (This is literally true when the automorphism group is a Lie group, for example when F is the real numbers and A has finite dimension as a vector space.) For this reason, spaces of derivations are a natural way to construct Lie algebras: they are the "infinitesimal automorphisms" of A. Indeed, writing out the condition that

(where 1 denotes the identity map on A) gives exactly the definition of D being a derivation.

Example: the Lie algebra of vector fields. Let A of smooth functions on a smooth manifold X. Then a derivation of A is equivalent to a vector field on X. (A vector field v gives a derivation of the space of smooth functions by differentiating functions in the direction of v of vector fields into a Lie algebra (see Lie bracket of vector fields). is the Lie algebra of the diffeomorphism group of X. So the Lie bracket of vector fields describes the non-commutativity of the diffeomorphism group. An action of a Lie group G on a manifold X. (An example is illustrated below.)

over a field F such that

  1. .

The inner derivation, and the Lie algebra of outer derivations. (This is exactly analogous to the outer automorphism group of a group.) For a semisimple Lie algebra (defined below) over a field of characteristic zero, every derivation is inner. This is related to the theorem that the outer automorphism group of a semisimple Lie group is finite.

.

Examples

Matrix Lie algebras

A matrix group, where the group operation of G is the space of matrices which are tangent vectors to G: this consists of derivatives of smooth curves in G at the identity matrix:

, one can recover the Lie group as the subgroup generated by the matrix exponential. (To be precise, this gives the identity component of G, if G.

(defined by the same formula).

Here are some matrix Lie groups and their Lie algebras.

Two dimensions

Some Lie algebras of low dimension are described here. See the classification of low-dimensional real Lie algebras for further examples.

  • of dimension 2 over any field F, up to isomorphism. of affine transformations.

  • The affine group G can be identified with the group of matrices

  • consisting of all matrices

  • is given by the matrices

  • can be broken into abelian "pieces", meaning that it is solvable (though not nilpotent), in the terminology below.

Three dimensions

  • The Heisenberg algebra over a field F such that

    1. .

    It can be viewed as the Lie algebra of 33 strictly upper-triangular matrices, with the commutator Lie bracket and the basis

  • is the Lie algebra of the Heisenberg group, that is, the group of matrices

  • under matrix multiplication.

  • For any field F is nilpotent (though not abelian).

  • of the rotation group SO(3). A basis is given by the three matrices

  • The commutation relations among these generators are

  • is equivalent to the Spin (physics) angular-momentum component operators for spin-1 particles in quantum mechanics.

    cannot be broken into pieces in the way that the previous examples can: it is simple.

  • of 2 x 2 matrices of trace zero. A basis is given by the three matrices

  • The Lie bracket is given by:

    is simple, and classify its finite-dimensional representations (defined below). In the terminology of quantum mechanics, one can think of E and F as raising and lowering operators, the relations above imply that E maps the c-eigenspace of H (for a complex number c-eigenspace, while F maps the c-eigenspace.

    is isomorphic to the complexification, meaning the tensor product.

Infinite dimensions

Representations

Definitions

Given a vector space V denote the Lie algebra consisting of all linear maps from V. A representation on V is a Lie algebra homomorphism

to a linear map from V corresponds to the commutator of linear maps.

A representation is said to be faithful if its kernel is zero. Ado's theorem states that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful representation on a finite-dimensional vector space. Kenkichi Iwasawa extended this result to finite-dimensional Lie algebras over a field of any characteristic. Equivalently, every finite-dimensional Lie algebra over a field F for some positive integer n.

Adjoint representation

, the adjoint representation is the representation

by the Jacobi identity.)

Goals of representation theory

. For a semisimple Lie algebra over a field of characteristic zero, Weyl's theorem says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The finite-dimensional irreducible representations are well understood from several points of view; see the representation theory of semisimple Lie algebras and the Weyl character formula.

Universal enveloping algebra

The functor that takes an associative algebra A over a field F to A) has a left adjoint, called the universal enveloping algebra over F, let

be the tensor algebra denotes the tensor product of F-vector spaces. Let I be the two-sided ideal. It satisfies the Poincaré–Birkhoff–Witt theorem as an F is injective.

are equivalent to modules. This also shows that every Lie algebra is contained in the Lie algebra associated to some associative algebra.

Representation theory in physics

The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations. These commutation relations typically come from a symmetry of the problem—specifically, they are the relations of the Lie algebra of the relevant symmetry group. An example is the angular momentum operators. Typically, the space of states is far from being irreducible under the pertinent operators, but one can attempt to decompose it into irreducible pieces. In doing so, one needs to know the irreducible representations of the given Lie algebra. In the study of the hydrogen atom.

Structure theory and classification

Lie algebras can be classified to some extent. This is a powerful approach to the classification of Lie groups.

Abelian, nilpotent, and solvable

Analogously to abelian, nilpotent, and solvable groups, one can define abelian, nilpotent, and solvable Lie algebras.

is abelian if the Lie bracket vanishes; that is, [x,y] = 0 for all x and y under addition or the torus group, meaning an n-dimensional vector space with Lie bracket zero.

A more general class of Lie algebras is defined by the vanishing of all commutators of given length. First, the commutator subalgebra (or derived subalgebra, in fact the smallest ideal such that the quotient Lie algebra is abelian. It is analogous to the commutator subgroup of a group.

is nilpotent if the lower central series

,

for each j. By Engel's theorem, a Lie algebra over any field is nilpotent if and only if for every u the adjoint endomorphism

is nilpotent.

is said to be solvable if the derived series:

is solvable if there is a finite sequence of Lie subalgebras,

abelian for each j.

Every finite-dimensional Lie algebra over a field has a unique maximal solvable ideal, called its radical. Under the Lie correspondence.

For example, for a positive integer n and a field F.

Simple and semisimple

is called simple is called semisimple.

and every field F of characteristic zero (or just of characteristic not dividing n..)

The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground field F has characteristic zero, every finite-dimensional representation of a semisimple Lie algebra is semisimple (that is, a direct sum of irreducible representations).

A finite-dimensional Lie algebra over a field of characteristic zero is called reductive if its adjoint representation is semisimple. Every reductive Lie algebra is isomorphic to the product of an abelian Lie algebra and a semisimple Lie algebra.

is reductive for F, it is isomorphic to the product

where F contains few ideals: only 0, the center F.

Cartan's criterion

Cartan's criterion (by Élie Cartan) gives conditions for a finite-dimensional Lie algebra of characteristic zero to be solvable or semisimple. It is expressed in terms of the Killing form defined by

is semisimple if and only if the Killing form is nondegenerate

Classification

The Levi decomposition asserts that every finite-dimensional Lie algebra over a field of characteristic zero is a semidirect product of its solvable radical and a semisimple Lie algebra. Moreover, a semisimple Lie algebra in characteristic zero is a product of simple Lie algebras, as mentioned above. This focuses attention on the problem of classifying the simple Lie algebras.

The simple Lie algebras of finite dimension over an algebraically closed field F of characteristic zero were classified by Killing and Cartan in the 1880s and 1890s, using root systems. Namely, every simple Lie algebra is of type An, Bn, Cn, Dn, E6, E7, E8, F4, or G2. Here the simple Lie algebra of type An, Bn, Cn, and Dn. The other five are known as the exceptional Lie algebras.

is more complicated, but it was also solved by Cartan (see simple Lie group.

were classified by Richard Earl Block, Robert Lee Wilson, Alexander Premet, and Helmut Strade. (See restricted Lie algebra#Classification of simple Lie algebras.) It turns out that there are many more simple Lie algebras in positive characteristic than in characteristic zero.

Relation to Lie groups

Although Lie algebras can be studied in their own right, historically they arose as a means to study Lie groups.

. This is Lie's third theorem; see the Baker–Campbell–Hausdorff formula. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are locally isomorphic, and more strongly, they have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) have isomorphic Lie algebras, but SU(2) is a simply connected double cover of SO(3).

For simply connected Lie groups, there is a complete correspondence: taking the Lie algebra gives an equivalence of categories.

The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the representation theory of Lie groups. For finite-dimensional representations, there is an equivalence of categories between representations of a real Lie algebra and representations of the corresponding simply connected Lie group. This simplifies the representation theory of Lie groups: it is often easier to classify the representations of a Lie algebra, using linear algebra.

Every connected Lie group is isomorphic to its universal cover modulo a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the Lie algebra is known. For example, the real semisimple Lie algebras were classified by Cartan, and so the classification of semisimple Lie groups is well understood.

For infinite-dimensional Lie algebras, Lie theory works less well. The exponential map need not be a local homeomorphism (for example, in the diffeomorphism group of the circle, there are diffeomorphisms arbitrarily close to the identity that are not in the image of the exponential map). Moreover, in terms of the existing notions of infinite-dimensional Lie groups, some infinite-dimensional Lie algebras do not come from any group.

can usually not be differentiated to produce a representation of its Lie algebra on the same space, or vice versa. The theory of Harish-Chandra modules is a more subtle relation between infinite-dimensional representations for groups and Lie algebras.

Real form and complexification

Given a complex Lie algebra is said to be a real form.

, a split form of it is a real form that splits; i.e., it has a Cartan subalgebra which acts via an adjoint representation with real eigenvalues. A split form exists and is unique (up to isomorphism). A compact form is a real form that is the Lie algebra of a compact Lie group. A compact form exists and is also unique up to isomorphism.

Lie algebra with additional structures

A Lie algebra may be equipped with additional structures that are compatible with the Lie bracket. For example, a graded Lie algebra is a Lie algebra (or more generally a Lie superalgebra) with a compatible grading. A differential graded Lie algebra also comes with a differential, making the underlying vector space a chain complex.

For example, the homotopy groups of a simply connected topological space form a graded Lie algebra, using the Whitehead product. In a related construction, Daniel Quillen used differential graded Lie algebras over the rational numbers to describe rational homotopy theory in algebraic terms.

Lie ring

The definition of a Lie algebra over a field extends to define a Lie algebra over any commutative ring R over R is an R-module with an alternating R of integers is sometimes called a Lie ring. (This is not directly related to the notion of a Lie group.)

Lie rings are used in the study of finite p-groups (for a prime number p) through the Lazard correspondence. The lower central factors of a finite p-group are finite abelian p-groups. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the commutator of two coset representatives; see the example below.

p-adic Lie groups of p-adic numbers of p-adic integers. Part of Claude Chevalley's construction of the finite groups of Lie type involves showing that a simple Lie algebra over the complex numbers comes from a Lie algebra over the integers, and then (with more care) a group scheme over the integers.

Examples

  • be a filtration. (For the Lazard correspondence, one takes the filtration to be the lower central series of G.) Then

  • given by commutators in the group:

  • For example, the Lie ring associated to the lower central series on the dihedral group.

Definition using category-theoretic notation

The definition of a Lie algebra can be reformulated more abstractly in the language of category theory. Namely, one can define a Lie algebra in terms of linear maps—that is, morphisms in the category of vector spaces—without considering individual elements. (In this section, the field over which the algebra is defined is assumed to be of characteristic different from 2.)

For the category-theoretic definition of Lie algebras, two braiding isomorphisms are needed. If is a vector space, the interchange isomorphism is defined by

The cyclic-permutation braiding is defined as

is defined by

in the category of vector spaces together with a morphism

that satisfies the two morphism equalities

and

Generalization

Several generalizations of a Lie algebra have been proposed, many from physics. Among them are graded Lie algebras, Lie superalgebras, Lie n-algebras,

See also

Remarks

References

Sources