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# Lie algebra

Inmathematics, a Lie algebra (pronounced/l/"Lee") is avector spacetogether with anon-associativeoperationcalled the Lie bracket, analternating bilinear map, satisfying theJacobi identity.

Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: 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 connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification 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 quantum mechanics and particle physics.

An elementary example is the space of three dimensional vectorswith the bracket operation defined by thecross productThis is skew-symmetric since, and instead of associativity it satisfies the Jacobi identity:
This is the Lie algebra of the Lie group ofrotations of space, and each vectormay 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.

## History

Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the 1870s,[2] and independently discovered by Wilhelm Killing[3] in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group is used.

## Definitions

### Definition of a Lie algebra

A Lie algebra is avector spaceover somefield[1] together with abinary operationcalled the Lie bracket satisfying the following axioms:
• Bilinearity,

for all scalars a, b in F and all elements x, y, z in.
• Alternativity,

for all x in.
• The Jacobi identity,

for all x, y, z in.
Using bilinearity to expand the Lie bracketand using alternativity shows thatfor all elements x, y in, showing that bilinearity and alternativity together imply
• Anticommutativity,

for all elements x, y in. If the field'scharacteristicis not 2 then anticommutativity implies alternativity.[4]
It is customary to denote a Lie algebra by a lower-casefrakturletter such as. If a Lie algebra is associated with aLie group, then the algebra is denoted by the fraktur version of the group: for example the Lie algebra ofSU(n)is.

### Generators and dimension

Elements of a Lie algebraare said togenerateit if the smallest subalgebra containing these elements isitself. 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 classification of low-dimensional real Lie algebras for other small examples.

### Subalgebras, ideals and homomorphisms

The Lie bracket is notassociative, meaning thatneed not equal. (However, it is *flexible*.) Nonetheless, much of the terminology of associativeringsandalgebrasis commonly applied to Lie algebras. A Lie subalgebra is a subspacewhich is closed under the Lie bracket. An idealis a subalgebra satisfying the stronger condition:[5]

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

As for associative rings, ideals are precisely the kernels of homomorphisms; given a Lie algebraand an idealin it, one constructs the factor algebra or quotient algebra, and thefirst isomorphism theoremholds for Lie algebras.
Since the Lie bracket is a kind of infinitesimalcommutatorof the corresponding Lie group, we say that two elementscommute if their bracket vanishes:.
Thecentralizersubalgebra of a subsetis the set of elements commuting with S: that is,. The centralizer ofitself is the center. Similarly, for a subspace S, thenormalizersubalgebra of S is.[6] Equivalently, if S is a Lie subalgebra,is the largest subalgebra such thatis an ideal of.

### Direct sum and semidirect product

For two Lie algebrasand, theirdirect sumLie algebra is the vector spaceconsisting of all pairs, with the operation
so that the copies ofcommute with each other:Letbe a Lie algebra andan ideal of. If the canonical mapsplits (i.e., admits a section), thenis said to be asemidirect productofand,. See alsosemidirect sum of Lie algebras.

Levi's theorem says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra (Levi subalgebra).

### Derivations

Aderivationon the Lie algebra(or on anynon-associative algebra) is alinear mapthat obeys theLeibniz law, that is,
for all. The inner derivation associated to anyis the adjoint mappingdefined by. (This is a derivation as a consequence of the Jacobi identity.) Ifissemisimple, every derivation is inner.
The derivations form a vector space, which is a Lie algebra under the commutator bracket, wheredenotes composition of mappings. The inner derivations form a Lie subalgebra of.

### Split Lie algebra

Let V be a finite-dimensional vector space over a field F,the Lie algebra of linear transformations anda Lie subalgebra. Thenis said to be split if the roots of the characteristic polynomials of all linear transformations inare in the base field F.[7] More generally, a finite-dimensional Lie algebrais said to be split if it has a Cartan subalgebrasuch that, for theadjoint representation, the imageis split;[8] seesplit Lie algebrafor further information.

## Examples

### Vector spaces

Any vector spaceendowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are calledabelian, 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 associative algebra over a field with multiplication , a Lie bracket may be defined by the commutator . With this bracket, is a Lie algebra.[9] The associative algebra A is called an enveloping algebra of the Lie algebra . Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see universal enveloping algebra.

• The associative algebra of endomorphisms of an F-vector space with the above Lie bracket is denoted .

• For a finite dimensional vector space , the previous example becomes the Lie algebra of n × n matrices, denoted or ,[10] with the bracket , where denotes matrix multiplication. This is the Lie algebra of the general linear group, consisting of invertible matrices.

### Special matrices

Two important subalgebras ofare:
• The matrices of trace zero form the special linear Lie algebra , the Lie algebra of the special linear group .[11]

• The skew-hermitian matrices form the unitary Lie algebra , the Lie algebra of the unitary group U(n).

### Matrix Lie algebras

A complexmatrix groupis a Lie group consisting of matrices,, where the multiplication of G is matrix multiplication. The corresponding Lie algebrais the space of matrices which are tangent vectors to G inside the linear space: this consists of derivatives of smooth curves in G at the identity:
The Lie bracket ofis given by the commutator of matrices,. Given the Lie algebra, one can recover the Lie group as the image of thematrix exponentialmappingdefined by, which converges for every matrix: that is,.

The following are examples of Lie algebras of matrix Lie groups:[12]

• The special linear group , consisting of all n × n matrices with determinant 1. Its Lie algebra consists of all n × n matrices with real entries and trace 0. Similarly, one can define the corresponding real Lie group and its Lie algebra .

• The unitary group consists of n × n unitary matrices (satisfying ). Its Lie algebra consists of skew-self-adjoint matrices ().

• The special orthogonal group , consisting of real determinant-one orthogonal matrices (). Its Lie algebra consists of real skew-symmetric matrices (). The full orthogonal group , without the determinant-one condition, consists of and a separate connected component, so it has the same Lie algebra as . Similarly, one can define a complex version of this group and algebra, simply by allowing complex matrix entries.

### Two dimensions

• On any field there is, up to isomorphism, a single two-dimensional nonabelian Lie algebra with generators x, y, and bracket defined as . It generates the affine group in one dimension.

This can be realized by the matrices:
and, foror, the resulting group elements are upper triangular 2×2 matrices with unit lower diagonal:

### Three dimensions

• The Heisenberg algebra is a three-dimensional Lie algebra generated by elements x, y and z with Lie brackets

.
It is realized as the space of 3×3 strictly upper-triangular matrices, with the commutator Lie bracket:
Any element of theHeisenberg groupis thus representable as a product of group generators, i.e.,matrix exponentialsof these Lie algebra generators,
• The Lie algebra of the group SO(3) is spanned by the three matrices[13]

The commutation relations among these generators are
The three-dimensionalEuclidean spacewith the Lie bracket given by thecross productofvectorshas the same commutation relations as above: thus, it is isomorphic to. This Lie algebra appears inquantum mechanicsas the components of theangular momentumoperator.

### Infinite dimensions

• An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields on a differentiable manifold M forms a Lie algebra, where the Lie bracket is defined to be the commutator of vector fields. One way of expressing the Lie bracket is through the formalism of Lie derivatives, which identifies a vector field X with a first order partial differential operator L**X acting on smooth functions by letting L**X(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:

• Kac–Moody algebras are a large class of infinite-dimensional Lie algebras whose structure is very similar to the finite-dimensional cases above.

• The Moyal algebra is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.

• The Virasoro algebra is of paramount importance in string theory.

## Representations

### Definitions

Given a vector space V, letdenote the Lie algebra consisting of all linearendomorphismsof V, with bracket given by. A representation of a Lie algebraon V is a Lie algebra homomorphism

A representation is said to be faithful if its kernel is zero. Ado's theorem[14] states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space.

For any Lie algebra, we can define a representation
given by; it is a representation on the vector spacecalled theadjoint representation.

### 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. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather the goal is to understand all possible representation of, up to the natural notion of equivalence. In the semisimple case,Weyl's theorem[15] 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 atheorem of the highest weight.

### 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 theangular momentum operators, whose commutation relations are those of the Lie algebraof therotation group SO(3). 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 quantumhydrogen atom, for example, quantum mechanics textbooks give (without calling it that) a classification of the irreducible representations of the Lie algebra.

## 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 algebrais *abelian
• if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in
. Abelian Lie algebras correspond to commutative (orabelian) connected Lie groups such as vector spacesortori, and are all of the formmeaning 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 algebrais *nilpotent
• if the
lower central series
becomes zero eventually. ByEngel's theorem, a Lie algebra is nilpotent if and only if for every u intheadjoint endomorphism

is nilpotent.

More generally still, a Lie algebrais said to be *solvable
• if the
derived series:

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. (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 algebrais called *semisimple
• 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 characteristic zero, any finite-dimensional representation of a semisimple Lie algebra is semisimple (i.e., direct sum of irreducible representations.) In general, a Lie algebra is called reductive if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.

### Cartan's criterion

Cartan's criteriongives conditions for a Lie algebra to be nilpotent, solvable, or semisimple. It is based on the notion of theKilling form, asymmetric bilinear formondefined by the formula
where tr denotes thetrace of a linear operator. A Lie algebrais semisimple if and only if the Killing form isnondegenerate. A Lie algebrais solvable if and only if

### Classification

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. Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems. However, the classification of solvable Lie algebras is a 'wild' problem, and cannot be accomplished in general.

## Relation to Lie groups

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

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, there exists a corresponding connected Lie groupwith Lie algebra. This isLie's third theorem; see theBaker–Campbell–Hausdorff formula. 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 sameuniversal cover. For instance, thespecial orthogonal groupSO(3)and thespecial unitary groupSU(2)give rise to the same Lie algebra, which is isomorphic towith 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, there is a unique simply connected Lie groupwith Lie algebra.

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).

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 that are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.

## 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 graded Lie algebra 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 chain complex), then it is called a differential graded Lie algebra.

A simplicial Lie algebra is a simplicial object in the category of Lie algebras; in other words, it is obtained by replacing the underlying set with a simplicial set (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 thelower central seriesofgroups. A Lie ring is defined as anonassociative ringwith multiplication that isanticommutativeand satisfies theJacobi identity. More specifically we can define a Lie ringto be anabelian groupwith an operationthat has the following properties:
• Bilinearity:

for all x, y, zL.
• The Jacobi identity:

for all x, y, z in L.
• For all x in L:

Lie rings need not beLie groupsunder addition. Any Lie algebra is an example of a Lie ring. Anyassociative ringcan be made into a Lie ring by defining a bracket operator. Conversely to any Lie algebra there is a corresponding ring, called theuniversal enveloping algebra.

Lie rings are used in the study of finite p-groups 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 commutator 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 p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic integers. 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 ring instead of a field is an example of a Lie ring. Lie rings are not Lie groups under addition, despite the name.

• Any associative ring can be made into a Lie ring by defining a bracket operator

• For an example of a Lie ring arising from the study of groups, let be a group with the commutator operation, and let be a central series in — that is the commutator subgroup is contained in for any . Then

is a Lie ring with addition supplied by the group operation (which will be × in each homogeneous part), and the bracket operation given by
extended linearly. The centrality of the series ensures the commutatorgives the bracket operation the appropriate Lie theoretic properties.

## References

[1]
Citation Linkopenlibrary.orgBourbaki, Nicolas (1989). Lie Groups and Lie Algebras: Chapters 1-3. Berlin·Heidelberg·New York: Springer. ISBN 978-3-540-64242-8., Section 2.) allows more generally for a module over a commutative ring with unit element.
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Citation Linkopenlibrary.orgO'Connor, J.J; Robertson, E.F. (2000). "Biography of Sophus Lie". MacTutor History of Mathematics Archive.
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[3]
Citation Linkopenlibrary.orgO'Connor, J.J; Robertson, E.F. (2005). "Biography of Wilhelm Killing". MacTutor History of Mathematics Archive.
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[4]
Citation Linkopenlibrary.orgHumphreys, James E. (1978). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. 9 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90053-7., p. 1
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Citation Linkopenlibrary.orgDue to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.
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Citation Linkopenlibrary.orgHall, Brian C. (2015). Lie groups, Lie algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. 222 (2nd ed.). Springer. doi:10.1007/978-3-319-13467-3. ISBN 978-3319134666. ISSN 0072-5285. Section 3.4
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