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In mathematics, a Lie algebra (pronounced /l/ "Lee") is a vector space together with a non-associative operation called the Lie bracket, an alternating bilinear map , satisfying the Jacobi 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 vectors with the bracket operation defined by the cross product This is skew-symmetric since , and instead of associativity it satisfies the Jacobi identity:

This is the Lie algebra of the Lie group of rotations of space, and each vector 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 .

Contents

HistoryEdit

Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the 1870s,[1] and independently discovered by Wilhelm Killing[2] in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group is used.

DefinitionsEdit

Definition of a Lie algebraEdit

A Lie algebra is a vector space   over some field  [nb 1] together with a binary operation   called the Lie bracket satisfying the following axioms:

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

Using bilinearity to expand the Lie bracket   and using alternativity shows that   for all elements x, y in  , showing that bilinearity and alternativity together imply

 
for all elements x, y in  . If the field's characteristic is not 2 then anticommutativity implies alternativity.[3]

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

Generators and dimensionEdit

Elements of a Lie algebra   are said to generate it if the smallest subalgebra containing these elements is   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 classification of low-dimensional real Lie algebras for other small examples.

Subalgebras, ideals and homomorphismsEdit

The Lie bracket is not associative, meaning that   need not equal  . (However, it is flexible.) Nonetheless, much of the terminology of associative rings and algebras is commonly applied to Lie algebras. A Lie subalgebra is a subspace   which is closed under the Lie bracket. An ideal   is a subalgebra satisfying the stronger condition:[4]

 

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 algebra   and an ideal   in it, one constructs the factor algebra or quotient algebra  , and the first isomorphism theorem holds for Lie algebras.

Since the Lie bracket is a kind of infinitesimal commutator of the corresponding Lie group, we say that two elements   commute if their bracket vanishes:  .

The centralizer subalgebra of a subset   is the set of elements commuting with S: that is,  . The centralizer of   itself is the center  . Similarly, for a subspace S, the normalizer subalgebra of S is  .[5] Equivalently, if S is a Lie subalgebra,   is the largest subalgebra such that   is an ideal of  .

Direct sum and semidirect productEdit

For two Lie algebras   and  , their direct sum Lie algebra is the vector space  consisting of all pairs  , with the operation

 

so that the copies of   commute with each other:   Let   be a Lie algebra and   an ideal of  . If the canonical map   splits (i.e., admits a section), then   is said to be a semidirect product of   and  ,  . See also semidirect 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).

DerivationsEdit

A derivation on the Lie algebra   (or on any non-associative algebra) is a linear map   that obeys the Leibniz law, that is,

 

for all  . The inner derivation associated to any   is the adjoint mapping   defined by  . (This is a derivation as a consequence of the Jacobi identity.) If   is semisimple, every derivation is inner.

The derivations form a vector space  , which is a Lie algebra under the commutator bracket  , where  denotes composition of mappings. The inner derivations form a Lie subalgebra of  .

Split Lie algebraEdit

Let V be a finite-dimensional vector space over a field F,   the Lie algebra of linear transformations and   a Lie subalgebra. Then   is said to be split if the roots of the characteristic polynomials of all linear transformations in   are in the base field F.[6] More generally, a finite-dimensional Lie algebra   is said to be split if it has a Cartan subalgebra   such that, for the adjoint representation  , the image   is split;[7] see split Lie algebra for further information.

ExamplesEdit

Vector spacesEdit

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

Associative algebra with commutator bracketEdit

  • 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.[8] 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  ,[9] with the bracket  , where   denotes matrix multiplication. This is the Lie algebra of the general linear group, consisting of invertible matrices.

Special matricesEdit

Two important subalgebras of   are:

Matrix Lie algebrasEdit

A complex matrix group is a Lie group consisting of matrices,  , where the multiplication of G is matrix multiplication. The corresponding Lie algebra   is 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 of   is given by the commutator of matrices,  . Given the Lie algebra, one can recover the Lie group as the image of the matrix exponential mapping   defined by  , which converges for every matrix  : that is,  .

The following are examples of Lie algebras of matrix Lie groups:[11]

  • 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 dimensionsEdit

  • 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, for   or  , the resulting group elements are upper triangular 2×2 matrices with unit lower diagonal:
 

Three dimensionsEdit

  • 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 the Heisenberg group is thus representable as a product of group generators, i.e., matrix exponentials of these Lie algebra generators,
 
  • The Lie algebra   of the group SO(3) is spanned by the three matrices[12]
 
The commutation relations among these generators are
 
 
 
The three-dimensional Euclidean space   with the Lie bracket given by the cross product of vectors has the same commutation relations as above: thus, it is isomorphic to  . This Lie algebra appears in quantum mechanics as the components of the angular momentum operator.

Infinite dimensionsEdit

  • 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 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:
 

RepresentationsEdit

DefinitionsEdit

Given a vector space V, let   denote the Lie algebra consisting of all linear endomorphisms of V, with bracket given by  . A representation of a Lie algebra   on V is a Lie algebra homomorphism

 

A representation is said to be faithful if its kernel is zero. Ado's theorem[13] states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space.

Adjoint representationEdit

For any Lie algebra  , we can define a representation

 

given by  ; it is a representation on the vector space   called the adjoint representation.

Goals of representation theoryEdit

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[14] 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 theorem of the highest weight.

Representation theory in physicsEdit

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 angular momentum operators, whose commutation relations are those of the Lie algebra   of the rotation 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 quantum hydrogen atom, for example, quantum mechanics textbooks give (without calling it that) a classification of the irreducible representations of the Lie algebra  .

Structure theory and classificationEdit

Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups.

Abelian, nilpotent, and solvableEdit

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   is abelian if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in  . Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups such as vector spaces   or tori  , and are all of the form   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   is nilpotent if the lower central series

 

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

 

is nilpotent.

More generally still, a Lie algebra   is 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 semisimpleEdit

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 algebra   is 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 criterionEdit

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   defined by the formula

 

where tr denotes the trace of a linear operator. A Lie algebra   is semisimple if and only if the Killing form is nondegenerate. A Lie algebra   is solvable if and only if  

ClassificationEdit

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[clarification needed] be accomplished in general.

Relation to Lie groupsEdit

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 group   with Lie algebra  . 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 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   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  , there is a unique simply connected Lie group   with 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 structuresEdit

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 ringEdit

A Lie ring arises as a generalisation of Lie algebras, or through the study of the lower central series of groups. A Lie ring is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity. More specifically we can define a Lie ring   to be an abelian group with an operation   that 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 be Lie groups under addition. Any Lie algebra is an example of a Lie ring. Any associative ring can be made into a Lie ring by defining a bracket operator  . Conversely to any Lie algebra there is a corresponding ring, called the universal 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.

ExamplesEdit

  • 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 commutator   gives the bracket operation the appropriate Lie theoretic properties.

See alsoEdit

RemarksEdit

  1. ^ Bourbaki (1989, Section 2.) allows more generally for a module over a commutative ring with unit element.

NotesEdit

  1. ^ O'Connor & Robertson 2000
  2. ^ O'Connor & Robertson 2005
  3. ^ Humphreys 1978, p. 1
  4. ^ Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.
  5. ^ Jacobson 1962, pg. 28
  6. ^ Jacobson 1962, pg. 42
  7. ^ Jacobson 1962, pg. 108
  8. ^ Bourbaki 1989, §1.2. Example 1.
  9. ^ Bourbaki 1989, §1.2. Example 2.
  10. ^ Humphreys p.2
  11. ^ Hall 2015 Section 3.4
  12. ^ Hall 2015 Example 3.27
  13. ^ Jacobson 1962, Ch. VI
  14. ^ Hall 2015, Theorem 10.9

ReferencesEdit

  • Beltiţă, Daniel (2006). Smooth Homogeneous Structures in Operator Theory. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. 137. Boca Raton, FL: Chapman & Hall/CRC Press. ISBN 978-1-4200-3480-6. MR 2188389.
  • Boza, Luis; Fedriani, Eugenio M.; Núñez, Juan (2001-06-01). "A new method for classifying complex filiform Lie algebras". Applied Mathematics and Computation. 121 (2–3): 169–175. doi:10.1016/s0096-3003(99)00270-2. ISSN 0096-3003.
  • Bourbaki, Nicolas (1989). Lie Groups and Lie Algebras: Chapters 1-3. Berlin·Heidelberg·New York: Springer. ISBN 978-3-540-64242-8.
  • Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0
  • Hall, 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.
  • Hofmann, Karl H.; Morris, Sidney A (2007). The Lie Theory of Connected Pro-Lie Groups. European Mathematical Society. ISBN 978-3-03719-032-6.
  • Humphreys, 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.
  • Jacobson, Nathan (1979) [1962]. Lie algebras. New York: Dover Publications, Inc. ISBN 978-0-486-63832-4.
  • Kac, Victor G.; et al. Course notes for MIT 18.745: Introduction to Lie Algebras. Archived from the original on 2010-04-20.CS1 maint: BOT: original-url status unknown (link)
  • Mubarakzyanov, G.M. (1963). "On solvable Lie algebras". Izv. Vys. Ucheb. Zaved. Matematika. 1 (32): 114–123.
  • O'Connor, J.J; Robertson, E.F. (2000). "Biography of Sophus Lie". MacTutor History of Mathematics Archive.
  • O'Connor, J.J; Robertson, E.F. (2005). "Biography of Wilhelm Killing". MacTutor History of Mathematics Archive.
  • Popovych, R.O.; Boyko, V.M.; Nesterenko, M.O.; Lutfullin, M.W.; et al. (2003). "Realizations of real low-dimensional Lie algebras". J. Phys. A: Math. Gen. 36 (26): 7337–7360. arXiv:math-ph/0301029. doi:10.1088/0305-4470/36/26/309.
  • Serre, Jean-Pierre (2006). Lie Algebras and Lie Groups (2nd ed.). Springer. ISBN 978-3-540-55008-2.
  • Steeb, Willi-Hans (2007). Continuous Symmetries, Lie Algebras, Differential Equations and Computer Algebra (second ed.). Hackensack, NJ: World Scientific. doi:10.1142/6515. ISBN 978-981-270-809-0. MR 2382250.
  • Varadarajan, Veeravalli S. (2004). Lie Groups, Lie Algebras, and Their Representations (1st ed.). Springer. ISBN 978-0-387-90969-1.

External linksEdit