Lie algebra

Lie groups
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In mathematics, a Lie algebra (pronounced /ˈliː/ ("lee"), not /ˈlaɪ/ ("lye")) is an algebraic structure whose main use is 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) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used.

Contents

Definition and first properties

A Lie algebra is a vector space \,\mathfrak{g} over some field F together with a binary operation [·, ·]

[\cdot,\cdot]: \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}

called the Lie bracket, which satisfies the following axioms:

[a x + b y, z] = a [x, z] + b [y, z], \quad [z, a x + b y] = a[z, x] + b [z, y]
for all scalars a, b in F and all elements x, y, z in \mathfrak{g}.
[x,x]=0\
for all x in \mathfrak{g}. This implies anticommutativity, or skew-symmetry (in fact the conditions are equivalent for any Lie algebra over any field whose characteristic is not 2):
[x,y]=-[y,x]\,
for all elements x, y in \mathfrak{g}.
[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 \quad
for all x, y, z in \mathfrak{g}.

For any associative algebra A with multiplication *, one can construct a Lie algebra L(A). As a vector space, L(A) is the same as A. The Lie bracket of two elements of L(A) is defined to be their commutator in A:

[a,b]=a * b-b * a.\

The associativity of the multiplication * in A implies the Jacobi identity of the commutator in L(A). In particular, the associative algebra of n × n matrices over a field F gives rise to the general linear Lie algebra \mathfrak{gl}_n(F). The associative algebra A is called an enveloping algebra of the Lie algebra L(A). It is known that every Lie algebra can be embedded into one that arises from an associative algebra in this fashion. See universal enveloping algebra.

Homomorphisms, subalgebras, and ideals

The Lie bracket is not an associative operation in general, meaning that [[x,y],z] need not equal [x,[y,z]]. Nonetheless, much of the terminology that was developed in the theory of associative rings or associative algebras is commonly applied to Lie algebras. A subspace \mathfrak{h} \subseteq \mathfrak{g} that is closed under the Lie bracket is called a Lie subalgebra. If a subspace I\subseteq\mathfrak{g} satisfies a stronger condition that

[\mathfrak{g},I]\subseteq I,

then I is called an ideal in the Lie algebra \mathfrak{g}.[1] A Lie algebra in which the commutator is not identically zero and which has no proper ideals is called simple. A homomorphism between two Lie algebras (over the same ground field) is a linear map that is compatible with the commutators:

f: \mathfrak{g}\to\mathfrak{g'}, \quad f([x,y])=[f(x),f(y)],

for all elements x and y in \mathfrak{g}. As in the theory of associative rings, ideals are precisely the kernels of homomorphisms, given a Lie algebra \mathfrak{g} and an ideal I in it, one constructs the factor algebra \mathfrak{g}/I, and the first isomorphism theorem holds for Lie algebras. Given two Lie algebras \mathfrak{g} and \mathfrak{g'}, their direct sum is the vector space \mathfrak{g}\oplus\mathfrak{g'} consisting of the pairs \mathfrak{}(x,x'), \,x\in\mathfrak{g}, x'\in\mathfrak{g'}, with the operation

[(x,x'),(y,y')]=([x,y],[x',y']), \quad x,y\in\mathfrak{g},\, x',y'\in\mathfrak{g'}.

Examples

x = \left( \begin{array}{ccc} 0&1&0\\ 0&0&0\\ 0&0&0 \end{array}\right),\quad y = \left( \begin{array}{ccc} 0&0&0\\ 0&0&1\\ 0&0&0 \end{array}\right),\quad z = \left( \begin{array}{ccc} 0&0&1\\ 0&0&0\\ 0&0&0 \end{array}\right),\quad
whose commutation relations are
[x,y]=z,\quad [x,z]=0, \quad [y,z]=0.\,
It is explicitly exhibited as the space of 3×3 strictly upper-triangular matrices.
\exp(tX)\in G\,
for all real numbers t. The Lie bracket of \mathfrak{g} is given by the commutator of matrices. As a concrete example, consider the special linear group SL(n,R), consisting of all n × n matrices with real entries and determinant 1. This is a matrix Lie group, and its Lie algebra consists of all n × n matrices with real entries and trace 0.
L_{[X,Y]}f=L_X(L_Y f)-L_Y(L_X f).\,
This Lie algebra is related to the pseudogroup of diffeomorphisms of M.
[L_x, L_y] = i \hbar L_z
[L_y, L_z] = i \hbar L_x
[L_z, L_x] = i \hbar L_y

Structure theory and classification

Every finite-dimensional real or complex Lie algebra has a faithful representation by matrices (Ado's theorem). Lie's fundamental theorems describe a relation between Lie groups and Lie algebras. In particular, any Lie group gives rise to a canonically determined Lie algebra (concretely, the tangent space at the identity), and conversely, for any Lie algebra there is a corresponding connected Lie group (Lie's third theorem). This Lie group is not determined uniquely, however, any two connected Lie groups with the same Lie algebra are locally isomorphic, and in particular, have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) give rise to the same Lie algebra, which is isomorphic to R3 with the cross-product, and SU(2) is a simply-connected twofold cover of SO(3). Real and complex Lie algebras can be classified to some extent, and this is often an important step toward the classification of Lie groups.

Abelian, nilpotent, and solvable

Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras.

A Lie algebra \mathfrak{g} is abelian if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in \mathfrak{g}. Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups such as vector spaces K^n or tori T^n, and are all of the form \mathfrak{k}^n, meaning an n-dimensional vector space with the trivial Lie bracket.

A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra \mathfrak{g} is nilpotent if the lower central series

\mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],\mathfrak{g}] > [[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] > \cdots

becomes zero eventually. By Engel's theorem, a Lie algebra is nilpotent if and only if for every u in \mathfrak{g} the adjoint endomorphism

\operatorname{ad}(u):\mathfrak{g} \to \mathfrak{g}, \quad \operatorname{ad}(u)v=[u,v]

is nilpotent.

More generally still, a Lie algebra \mathfrak{g} is said to be solvable if the derived series:

\mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]] > [[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]],[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]] > \cdots

becomes zero eventually.

Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its radical. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.

Simple and semisimple

A Lie algebra is "simple" if it has no non-trivial ideals and is not abelian. A Lie algebra \mathfrak{g} is called semisimple if its radical is zero. Equivalently, \mathfrak{g} is semisimple if it does not contain any non-zero abelian ideals. In particular, a simple Lie algebra is semisimple. Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras.

The concept of semisimplicity for Lie algebras is closely related with the complete reducibility of their representations. When the ground field F has characteristic zero, semisimplicity of a Lie algebra \mathfrak{g} over F is equivalent to the complete reducibility of all finite-dimensional representations of \mathfrak{g}. An early proof of this statement proceeded via connection with compact groups (Weyl's unitary trick), but later entirely algebraic proofs were found.

Classification

In many ways, the classes of semisimple and solvable Lie algebras are at the opposite ends of the full spectrum of the Lie algebras. The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way. Semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems. The classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general.

Cartan's criterion gives conditions for a Lie algebra to be nilpotent, solvable, or semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on \mathfrak{g} defined by the formula

K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),

where tr denotes the trace of a linear operator. A Lie algebra \mathfrak{g} is semisimple if and only if the Killing form is nondegenerate. A Lie algebra \mathfrak{g} is solvable if and only if K(\mathfrak{g},[\mathfrak{g},\mathfrak{g}])=0.

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 Lie 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 (for example, in Diff(S1), one may find diffeomorphisms arbitrarily close to the identity which are not in the image of exp). 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. Every representation of a Lie algebra lifts uniquely to a representation of the corresponding connected, simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; 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).

Category theoretic definition

Using the language of category theory, a Lie algebra can be defined as an object A in Vec, the category of vector spaces together with a morphism [.,.]: A ⊗ A → A, where ⊗ refers to the monoidal product of Vec, such that

where τ (a ⊗ b) := b ⊗ a and σ is the cyclic permutation braiding (id ⊗ τA,A) ° (τA,A ⊗ id). In diagrammatic form:

Liealgebra.png

See also

Notes

  1. ↑ Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.
  2. ↑ Humphreys p.2

References