Lie algebra

In, a Lie algebra (pronounced "Lee") is a  $$\mathfrak g$$ together with a   called the Lie bracket, an  $$\mathfrak g \times \mathfrak g \rightarrow \mathfrak g, \  (x, y) \mapsto [x, y]$$, satisfying the.

Lie algebras are closely related to s, which are s that are also : any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding Lie group unique up to finite coverings. This allows one to study the structure and  of Lie groups in terms of Lie algebras.

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 and particle physics.

An elementary example is the space of three dimensional vectors $$\mathfrak{g}=\mathbb{R}^3$$ with the bracket operation defined by the $$[x,y]=x\times y.$$ This is skew-symmetric since $$x\times y = -y\times x$$, and instead of associativity it satisfies the Jacobi identity: "$ x\times(y\times z) \ =\ (x\times y)\times z \ +\ y\times(x\times z). $"This is the Lie algebra of the Lie group of, and each vector $$v\in\R^3$$ may be pictured as an infinitesimal rotation around the axis v, with velocity equal to the magnitude of v. The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property $$[x,x]=x\times x = 0$$.

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

Definition of a Lie algebra
A Lie algebra is a $$\,\mathfrak{g}$$ over some  $$\mathbb{F}$$ together with a  $$[\,\cdot\,,\cdot\,]: \mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$$ called the Lie bracket satisfying the following axioms:


 * $$ [a x + b y, z] = a [x, z] + b [y, z], $$
 * $$ [z, a x + b y] = a[z, x] + b [z, y] $$
 * for all scalars a, b in F and all elements x, y, z in $$\mathfrak{g}$$.
 * for all scalars a, b in F and all elements x, y, z in $$\mathfrak{g}$$.


 * $$ [x,x]=0\ $$
 * for all x in $$\mathfrak{g}$$.
 * for all x in $$\mathfrak{g}$$.


 * The ,
 * $$ [x,[y,z}} + [z,[x,y}} + [y,[z,x}} = 0 \ $$
 * for all x, y, z in $$\mathfrak{g}$$.

Using bilinearity to expand the Lie bracket $$ [x+y,x+y] $$ and using alternativity shows that $$ [x,y] + [y,x]=0\ $$ for all elements x, y in $$\mathfrak{g}$$, showing that bilinearity and alternativity together imply
 * $$ [x,y] = -[y,x],\ $$
 * for all elements x, y in $$\mathfrak{g}$$. If the field's is not 2 then anticommutativity implies alternativity.
 * for all elements x, y in $$\mathfrak{g}$$. If the field's is not 2 then anticommutativity implies alternativity.

It is customary to denote a Lie algebra by a lower-case letter such as $$\mathfrak{g, h, b, n}$$. If a Lie algebra is associated with a, then the algebra is denoted by the fraktur version of the group: for example the Lie algebra of is $$\mathfrak{su}(n)$$.

Generators and dimension
Elements of a Lie algebra $$\mathfrak{g}$$ are said to it if the smallest subalgebra containing these elements is $$\mathfrak{g}$$ itself. The dimension of a Lie algebra is its dimension as a vector space over F. The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension.

See the for other small examples.

Subalgebras, ideals and homomorphisms
The Lie bracket is not, meaning that $$$$. (However, it is .) Nonetheless, much of the terminology of associative and s is commonly applied to Lie algebras. A Lie subalgebra is a subspace $$\mathfrak{h} \subseteq \mathfrak{g}$$ which is closed under the Lie bracket. An ideal $$\mathfrak i\subseteq\mathfrak{g}$$ is a subalgebra satisfying the stronger condition:


 * $$[\mathfrak{g},\mathfrak i]\subseteq \mathfrak i.$$

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


 * $$ \phi: \mathfrak{g}\to\mathfrak{g'}, \quad \phi([x,y])=[\phi(x),\phi(y)] \ \text{for all}\

x,y \in \mathfrak g. $$

As for associative rings, ideals are precisely the kernels of homomorphisms; given a Lie algebra $$\mathfrak{g}$$ and an ideal $$\mathfrak i$$ in it, one constructs the factor algebra or quotient algebra $$\mathfrak{g}/\mathfrak i$$, and the holds for Lie algebras.

Since the Lie bracket is a kind of infinitesimal of the corresponding Lie group, we say that two elements $$x,y\in\mathfrak g$$ commute if their bracket vanishes: $$[x,y]=0$$.

The subalgebra of a subset $$S\subset \mathfrak{g}$$ is the set of elements commuting with S: that is, $$\mathfrak{z}_{\mathfrak g}(S) = \{x\in\mathfrak g\ \mid\ [x, s] = 0 \ \text{ for all } s\in S\}$$. The centralizer of $$\mathfrak{g}$$ itself is the center $$\mathfrak{z}(\mathfrak{g})$$. Similarly, for a subspace S, the subalgebra of S is $$\mathfrak{n}_{\mathfrak g}(S) = \{x\in\mathfrak g\ \mid\ [x,s]\in S \ \text{ for all}\ s\in S\}$$. Equivalently, if S is a Lie subalgebra, $$\mathfrak{n}_{\mathfrak g}(S)$$ is the largest subalgebra such that $$S$$ is an ideal of $$\mathfrak{n}_{\mathfrak g}(S)$$.

Direct sum and semidirect product
For two Lie algebras $$\mathfrak{g^{}}$$ and $$\mathfrak{g'}$$, their Lie algebra is the vector space $$\mathfrak{g}\oplus\mathfrak{g'}$$consisting of all pairs $$\mathfrak{}(x,x'), \,x\in\mathfrak{g}, \ x'\in\mathfrak{g'}$$, with the operation


 * $$ [(x,x'),(y,y')]=([x,y],[x',y']),$$

so that the copies of $$\mathfrak g, \mathfrak g'$$ commute with each other: $$[(x,0), (0,x')] = 0.$$ Let $$\mathfrak{g}$$ be a Lie algebra and $$\mathfrak{i}$$ an ideal of $$\mathfrak{g}$$. If the canonical map $$\mathfrak{g} \to \mathfrak{g}/\mathfrak{i}$$ splits (i.e., admits a section), then $$\mathfrak{g}$$ is said to be a of $$\mathfrak{i}$$ and $$\mathfrak{g}/\mathfrak{i}$$, $$\mathfrak{g}=\mathfrak{g}/\mathfrak{i}\ltimes\mathfrak{i}$$. See also.

says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra.

Derivations
A on the Lie algebra $$\mathfrak{g}$$ (or on any ) is a  $$\delta\colon\mathfrak{g}\rightarrow \mathfrak{g}$$ that obeys the, that is,
 * $$\delta ([x,y]) = [\delta(x),y] + [x, \delta(y)]$$

for all $$x,y\in\mathfrak g$$. The inner derivation associated to any $$x\in\mathfrak g$$ is the adjoint mapping $$\mathrm{ad}_x$$ defined by $$\mathrm{ad}_x(y):=[x,y]$$. (This is a derivation as a consequence of the Jacobi identity.) If $$\mathfrak{g}$$ is, every derivation is inner.

The derivations form a vector space $$\mathrm{Der}(\mathfrak g)$$, which is a Lie algebra under the commutator bracket $$[\delta_1,\delta_2] \ =\ \delta_1\circ\delta_2-\delta_2\circ\delta_1$$, where $$\circ$$denotes composition of mappings. The inner derivations form a Lie subalgebra of $$\mathrm{Der}(\mathfrak g)$$.

Split Lie algebra
Let V be a finite-dimensional vector space over a field F, $$\mathfrak{gl}(V)$$ the Lie algebra of linear transformations and $$\mathfrak{g} \subseteq \mathfrak{gl}(V)$$ a Lie subalgebra. Then $$\mathfrak{g}$$ is said to be split if the roots of the characteristic polynomials of all linear transformations in $$\mathfrak{g}$$ are in the base field F. More generally, a finite-dimensional Lie algebra $$\mathfrak{g}$$ is said to be split if it has a Cartan subalgebra whose image under the $$\operatorname{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak g)$$ is a split Lie algebra. A of a complex semisimple Lie algebra (cf. ) is an example. See also for further information.

Vector spaces
Any vector space $$V$$ endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called, cf. below. Any one-dimensional Lie algebra over a field is abelian, by the alternating property of the Lie bracket.

Associative algebra with commutator bracket

 * On an $$A$$ over a field $$F$$ with multiplication $$(x, y) \mapsto xy$$, a Lie bracket may be defined by the  $$[x,y] = xy - yx$$. With this bracket, $$A$$ is a Lie algebra.  The associative algebra A is called an enveloping algebra of the Lie algebra $$(A, [\,\cdot\,, \cdot \,])$$. Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see.
 * The associative algebra of s of an F-vector space $$V$$ with the above Lie bracket is denoted $$\mathfrak{gl}(V)$$.
 * For a finite dimensional vector space $$V = F^n$$, the previous example becomes the Lie algebra of n × n matrices, denoted $$\mathfrak{gl}(n, F)$$ or $$\mathfrak{gl}_n(F)$$, with the bracket $$[X,Y]=X\cdot Y-Y\cdot X$$, where $$\cdot$$ denotes matrix multiplication. This is the Lie algebra of the, consisting of invertible matrices.

Special matrices
Two important subalgebras of $$\mathfrak{gl}_n(F)$$ are:


 * The matrices of zero form the  $$\mathfrak{sl}_n(F)$$, the Lie algebra of the  $$\mathrm{SL}_n(F)$$.
 * The matrices form the unitary Lie algebra $$\mathfrak u(n)$$, the Lie algebra of the  U(n).

Matrix Lie algebras
A complex is a Lie group consisting of matrices, $$G\subset M_n(\mathbb{C})$$, where the multiplication of G is matrix multiplication. The corresponding Lie algebra $$\mathfrak g$$ is the space of matrices which are tangent vectors to G inside the linear space $$M_n(\mathbb{C})$$: this consists of derivatives of smooth curves in G at the identity: "$\mathfrak{g} = \{ X = c'(0) \in M_n(\mathbb{C}) \ \mid\ \text{ smooth } c : \mathbb{R}\to G, \ c(0) = I \}.$"The Lie bracket of $$\mathfrak{g}$$ is given by the commutator of matrices, $$[X,Y]=XY-YX$$. Given the Lie algebra, one can recover the Lie group as the image of the mapping $$\exp: M_n(\mathbb{C})\to M_n(\mathbb{C})$$ defined by $$\exp(X) = I + X + \tfrac{1}{2!}X^2+\cdots$$, which converges for every matrix $$X$$: that is, $$G=\exp(\mathfrak g)$$. The following are examples of Lie algebras of matrix Lie groups:


 * The $${\rm SL}_n(\mathbb{C})$$, consisting of all n × n matrices with determinant 1. Its Lie algebra $$\mathfrak{sl}_n(\mathbb{C})$$consists of all n × n matrices with complex entries and trace 0. Similarly, one can define the corresponding real Lie group $${\rm SL}_n(\mathbb{R})$$ and its Lie algebra $$\mathfrak{sl}_n(\mathbb{R})$$.
 * The $$U(n)$$ consists of n × n unitary matrices (satisfying $$U^*=U^{-1}$$). Its Lie algebra $$\mathfrak{u}(n)$$ consists of skew-self-adjoint matrices ($$X^*=-X$$).
 * The special group $$\mathrm{SO}(n)$$, consisting of real determinant-one orthogonal matrices ($$A^{\mathrm{T}}=A^{-1}$$). Its Lie algebra $$\mathfrak{so}(n)$$ consists of real skew-symmetric matrices ($$X^{\rm T}=-X$$). The full orthogonal group $$\mathrm{O}(n)$$, without the determinant-one condition, consists of $$\mathrm{SO}(n)$$ and a separate connected component, so it has the same Lie algebra as $$\mathrm{SO}(n)$$. Similarly, one can define a complex version of this group and algebra, simply by allowing complex  matrix entries.

Two dimensions

 * On any field $$F$$ there is, up to isomorphism, a single two-dimensional nonabelian Lie algebra. With generators x, y, its bracket is defined as $$ \left [x, y\right ] = y$$. It generates the.


 * This can be realized by the matrices:
 * $$ x= \left( \begin{array}{cc} 1 & 0\\ 0 & 0 \end{array}\right), \qquad y=  \left( \begin{array}{cc} 0 & 1\\ 0 & 0 \end{array}\right).   $$

Since
 * $$ \left( \begin{array}{cc} 1 & c\\ 0 & 0 \end{array}\right)^{n+1} = \left( \begin{array}{cc} 1 & c\\ 0 & 0 \end{array}\right)$$

for any natural number $$n$$ and any $$c$$, one sees that the resulting Lie group elements are upper triangular 2×2 matrices with unit lower diagonal:
 * $$ \exp(a\cdot{}x+b\cdot{}y)= \left( \begin{array}{cc} e^a & \tfrac{b}{a}(e^a-1)\\ 0 & 1 \end{array}\right) = 1 + \tfrac{e^a-1}{a}\left(a\cdot{}x+b\cdot{}y\right). $$

Three dimensions

 * The $${\rm H}_3(\mathbb{R})$$ is a three-dimensional Lie algebra generated by elements $x$,  $y$  and  $z$ with Lie brackets


 * $$[x,y]=z,\quad [x,z]=0, \quad [y,z]=0$$.
 * It is realized as the space of 3&times;3 strictly upper-triangular matrices, with the commutator Lie bracket:

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 $$


 * Any element of the is thus representable as a product of group generators, i.e., s of these Lie algebra generators,
 * $$\left( \begin{array}{ccc}

1&a&c\\ 0&1&b\\ 0&0&1 \end{array}\right)= e^{by} e^{cz} e^{ax}~. $$


 * The Lie algebra $$\mathfrak{so}(3)$$ of the group SO(3) is spanned by the three matrices

F_1 = \left( \begin{array}{ccc} 0&0&0\\ 0&0&-1\\ 0&1&0 \end{array}\right),\quad F_2 = \left( \begin{array}{ccc} 0&0&1\\ 0&0&0\\ -1&0&0 \end{array}\right),\quad F_3 = \left( \begin{array}{ccc} 0&-1&0\\ 1&0&0\\ 0&0&0 \end{array}\right)~.\quad $$
 * The commutation relations among these generators are
 * $$[F_1, F_2] = F_3,$$
 * $$[F_2, F_3] = F_1,$$
 * $$[F_3, F_1] = F_2.$$
 * The three-dimensional $$\mathbb{R}^3$$ with the Lie bracket given by the  of  has the same commutation relations as above: thus, it is isomorphic to $$\mathfrak{so}(3)$$. This Lie algebra is unitarily equivalent to the usual  angular-momentum component operators for spin-1 particles in.
 * The three-dimensional $$\mathbb{R}^3$$ with the Lie bracket given by the  of  has the same commutation relations as above: thus, it is isomorphic to $$\mathfrak{so}(3)$$. This Lie algebra is unitarily equivalent to the usual  angular-momentum component operators for spin-1 particles in.

Infinite dimensions

 * An important class of infinite-dimensional real Lie algebras arises in . The space of smooth s on a M forms a Lie algebra, where the Lie bracket is defined to be the . One way of expressing the Lie bracket is through the formalism of s, which identifies a vector field X with a first order partial differential operator LX acting on smooth functions by letting LX(f) be the directional derivative of the function f in the direction of X. The Lie bracket [X,Y] of two vector fields is the vector field defined through its action on functions by the formula:
 * $$ L_{[X,Y]}f=L_X(L_Y f)-L_Y(L_X f).\,$$


 * are a large class of infinite-dimensional Lie algebras whose structure is very similar to the finite-dimensional cases above.
 * The is an infinite-dimensional Lie algebra that contains all s as subalgebras.
 * The is of paramount importance in.

Definitions
Given a vector space V, let $$\mathfrak{gl}(V)$$ denote the Lie algebra consisting of all linear s of V, with bracket given by $$[X,Y]=XY-YX$$. A representation of a Lie algebra $$\mathfrak{g}$$ on V is a Lie algebra homomorphism
 * $$\pi: \mathfrak g \to \mathfrak{gl}(V).$$

A representation is said to be faithful if its kernel is zero. states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space.

Adjoint representation
For any Lie algebra $$\mathfrak{g}$$, we can define a representation
 * $$\operatorname{ad}\colon\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})$$

given by $$\operatorname{ad}(x)(y) = [x, y]$$; it is a representation on the vector space $$\mathfrak{g}$$ called the.

Goals of representation theory
One important aspect of the study of Lie algebras (especially semisimple Lie algebras) is the study of their representations. (Indeed, most of the books listed in the references section devote a substantial fraction of their pages to representation theory.) Although Ado's theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra $$\mathfrak{g}$$. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather the goal is to understand all possible representation of $$\mathfrak{g}$$, up to the natural notion of equivalence. In the semisimple case, says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The irreducible representations, in turn, are classified by a.

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 would be the s, whose commutation relations are those of the Lie algebra $$\mathfrak{so}(3)$$ of the. Typically, the space of states is very 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 quantum, for example, quantum mechanics textbooks give (without calling it that) a classification of the irreducible representations of the Lie algebra $$\mathfrak{so}(3)$$.

Structure theory and classification
Lie algebras can be classified to some extent. In particular, this has an application to 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 ) connected Lie groups such as vector spaces $$\mathbb{K}^n$$ or $$\mathbb{T}^n$$, 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  if the


 * $$ \mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > {{Wikipedia link|\mathfrak{g},\mathfrak{g}],\mathfrak{g}] > {{Wikipedia link|{Wikipedia link|\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] > \cdots$$

becomes zero eventually. By {{Wikipedia link|Engel's theorem}}, a Lie algebra is nilpotent if and only if for every u in $$\mathfrak{g}$$ the {{Wikipedia link|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 {{Wikipedia link|solvable Lie algebra|solvable}} if the {{Wikipedia link|derived series}}:


 * $$ \mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > {{Wikipedia link|\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}}} > {{Wikipedia link|{Wikipedia link|\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}}},{{Wikipedia link|\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}}}} > \cdots$$

becomes zero eventually.

Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its {{Wikipedia link|radical of a Lie algebra|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 "" if it has no non-trivial ideals and is not abelian. (That is to say, a one-dimensional—necessarily abelian—Lie algebra is by definition not simple, even though it has no nontrivial ideals.) A Lie algebra $$\mathfrak{g}$$ is called  if it is isomorphic to a direct sum of simple algebras. There are several equivalent characterizations of semisimple algebras, such as having no nonzero solvable ideals.

The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground field F has zero, any finite-dimensional representation of a semisimple Lie algebra is  (i.e., direct sum of irreducible representations.) In general, a Lie algebra is called  if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.

Cartan's criterion
gives conditions for a Lie algebra to be nilpotent, solvable, or semisimple. It is based on the notion of the, a on $$\mathfrak{g}$$ defined by the formula
 * $$K(u,v)=\operatorname{tr}(\operatorname{ad}(u)\operatorname{ad}(v)),$$

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

Classification
The expresses an arbitrary Lie algebra as a  of its solvable radical and a semisimple Lie algebra, almost in a canonical way. Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their s.

Relation to Lie groups
Although Lie algebras are often studied in their own right, historically they arose as a means to study s.

We now briefly outline the relationship between Lie groups and Lie algebras. Any Lie group gives rise to a canonically determined Lie algebra (concretely, the tangent space at the identity). Conversely, for any finite-dimensional Lie algebra $$\mathfrak g$$, there exists a corresponding connected Lie group $$G$$ with Lie algebra $$\mathfrak g$$. This is ; see the. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are locally isomorphic, and in particular, have the same. For instance, the  and the   give rise to the same Lie algebra, which is isomorphic to $$\mathbb{R}^3$$ with the cross-product, but SU(2) is a simply-connected twofold cover of SO(3).

If we consider simply connected Lie groups, however, we have a one-to-one correspondence: For each (finite-dimensional real) Lie algebra $$\mathfrak g$$, there is a unique simply connected Lie group $$G$$ with Lie algebra $$\mathfrak g$$.

The correspondence between Lie algebras and Lie groups is used in several ways, including in the and the related matter of the  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, once the classification of Lie algebras is known (solved by et al. in the  case).

If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a (for example, in Diff(S1), one may find diffeomorphisms arbitrarily close to the identity that are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.

Real form and complexification
Given a $$\mathfrak g$$, a real Lie algebra $$\mathfrak{g}_0$$ is said to be a  of $$\mathfrak g$$ if the  $$\mathfrak{g}_0 \otimes_{\mathbb R} \mathbb{C} \simeq \mathfrak{g}$$ is isomorphic to $$\mathfrak{g}$$. A real form need not be unique; for example, $$\mathfrak{sl}_2 \mathbb{C}$$ has two real forms $$\mathfrak{sl}_2 \mathbb{R}$$ and $$\mathfrak{su}_2$$.

Given a semisimple finite-dimensional complex Lie algebra $$\mathfrak g$$, a  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 isomorphisms). A  is a real form that is the Lie algebra of a compact Lie group. A compact form exists and is also unique.

Lie algebra with additional structures
A Lie algebra can be equipped with some additional structures that are assumed to be compatible with the bracket. For example, a is a Lie algebra with a graded vector space structure. If it also comes with differential (so that the underlying graded vector space is a ), then it is called a.

A is a  in the category of Lie algebras; in other words, it is obtained by replacing the underlying set with a  (so it might be better thought of as a family of Lie algebras).

Lie ring
A Lie ring arises as a generalisation of Lie algebras, or through the study of the of. A Lie ring is defined as a with multiplication that is  and satisfies the. More specifically we can define a Lie ring $$L$$ to be an with an operation $$[\cdot,\cdot]$$ that has the following properties:


 * Bilinearity:


 * $$ [x + y, z] = [x, z] + [y, z], \quad [z, x + y] = [z, x] + [z, y] $$


 * for all x, y, z &isin; L.


 * The Jacobi identity:


 * $$ [x,[y,z}} + [y,[z,x}} + [z,[x,y}} = 0 \quad $$


 * for all x, y, z in L.


 * For all x in L:


 * $$ [x,x]=0 \quad $$

Lie rings need not be s under addition. Any Lie algebra is an example of a Lie ring. Any can be made into a Lie ring by defining a bracket operator $$[x,y] = xy - yx$$. Conversely to any Lie algebra there is a corresponding ring, called the.

Lie rings are used in the study of finite s through the Lazard correspondence'. The lower central factors of a p-group are finite abelian p-groups, so modules over Z/pZ. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the of two coset representatives. The Lie ring structure is enriched with another module homomorphism, the pth power map, making the associated Lie ring a so-called restricted Lie ring.

Lie rings are also useful in the definition of a s and their endomorphisms by studying Lie algebras over rings of integers such as the. The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and the reducing modulo p to get a Lie algebra over a finite field.

Examples

 * Any Lie algebra over a general instead of a  is an example of a Lie ring.  Lie rings are not s under addition, despite the name.
 * Any associative ring can be made into a Lie ring by defining a bracket operator
 * $$[x,y] = xy - yx.$$


 * For an example of a Lie ring arising from the study of, let $$G$$ be a group with $$(x,y) = x^{-1}y^{-1}xy$$ the commutator operation, and let $$G = G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots \supseteq G_n \supseteq \cdots$$ be a  in $$G$$ &mdash; that is the commutator subgroup $$(G_i,G_j)$$ is contained in $$G_{i+j}$$ for any $$i,j$$. Then


 * $$L = \bigoplus G_i/G_{i+1}$$


 * is a Lie ring with addition supplied by the group operation (which will be × in each homogeneous part), and the bracket operation given by


 * $$[xG_i, yG_j] = (x,y)G_{i+j}\ $$


 * extended linearly. The centrality of the series ensures the commutator $$(x,y)$$ gives the bracket operation the appropriate Lie theoretic properties.