Lie algebra

In mathematics, a Lie algebra 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, pronounced /liː/ ("lee"), not /laɪ/ ("lie") ) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used.

Definition and first properties

A Lie algebra is a type of algebra over a field; it is a vector space over some field F together with a binary operation [·, ·]


called the commutator or the Lie bracket, which satisfies the following axioms:
:


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


for all elements x, y in When F is a field of characteristic two, one has to impose the stronger condition


:


for all x in
:


for all x, y, z in


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:



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 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 of a Lie algebra that is closed under the Lie bracket is called a Lie subalgebra. If a subspace satisfies a stronger condition that



then I is called an ideal in the Lie algebra [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:



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

Categorical approach

A composition of two homomorphisms and is a homomorphism of the Lie algebras If a homomorphism is bijective, then it is invertible and is called an isomorphism, and these Lie algebras are called isomorphic. For many purposes, isomorphic Lie algebras are indistinguishable. The identity map on any Lie algebra is an isomorphism of the Lie algebra with itself.

Examples

  • Any vector space V 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 antisymmetry of the Lie bracket.
  • The three-dimensional Euclidean space R3 with the Lie bracket given by the cross product of vectors becomes a three-dimensional Lie algebra.
  • The Heisenberg algebra is a three-dimensional Lie algebra with generators x,y,z, whose commutation relations have the form
:
  • Any Lie group G defines an associated real Lie algebra The definition in general is somewhat technical, but in the case of real matrix groups, it can be formulated via the exponential map, or the matrix exponent. The Lie algebra consists of those matrices X for which
:
for all real numbers t. The Lie bracket of 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.
  • The real vector space of all n × n skew-hermitian matrices is closed under the commutator and forms a real Lie algebra denoted u(n). This is the Lie algebra of the unitary group U(n).
  • 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 the 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:
:
This Lie algebra is related to the pseudogroup of diffeomorphisms of M.
:
:
:

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, 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) both 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.

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. 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 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.

A Lie algebra is "simple" if it has no non-trivial ideals and is not abelian. A Lie algebra is called semisimple if its radical is zero. Equivalently, 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.

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 product 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 agebra 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 nilpotent if and only if the Killing form is identically zero, and semisimple if and only if the Killing form is nondegenerate. A Lie algebra is solvable if and only if

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 over F is equivalent to the complete reducibility of all finite-dimensional representations of An early proof of this statement proceeded via connection with compact groups (Weyl's unitary trick), but later entirely algebraic proofs were found.

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 the category of vector spaces together with a morphism [.,.]: AAA such that
where τ (ab) := ba and σ is the cyclic permutation braiding (id ⊗ τA,A) ° (τA,A ⊗ id). In diagrammatic form:

See also

Notes

1. ^ Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.

References

  • Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0
  • Brian C. Hall Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer, 2003. ISBN 0-387-40122-9
  • Humphreys, James E. Introduction to Lie Algebras and Representation Theory, Second printing, revised. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1978. ISBN 0-387-90053-5
  • Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4
  • Kac, Victor G. et al. Course notes for MIT 18.745: Introduction to Lie Algebras, http://www-math.mit.edu/~lesha/745lec/
  • Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations, 1st edition, Springer, 2004. ISBN 0-387-90969-9
  • O'Connor, J. J. & Robertson, E.F. Biography of Sophus Lie, MacTutor History of Mathematics Archive, http://www-history.mcs.st-and.ac.uk/Biographies/Lie.html
  • O'Connor, J. J. & Robertson, E.F. Biography of Wilhelm Killing, MacTutor History of Mathematics Archive, http://www-history.mcs.st-and.ac.uk/Biographies/Killing.html
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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In mathematics, a Lie group (IPA pronunciation: [liː], sounds like "Lee"), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important.
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In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space.
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Sophus Lie

Born November 17 1842(1842--)
Nordfjordeid, Norway
Died January 18 1899 (aged 58)
Christiania, Norway
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Hermann Klaus Hugo Weyl (November 9 1885 – December 9 1955) was a German mathematician. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is closely identified with the University of Göttingen tradition of mathematics, represented by
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Centuries: 19th century - 20th century - 21st century

1900s 1910s 1920s - 1930s - 1940s 1950s 1960s
1930 1931 1932 1933 1934
1935 1936 1937 1938 1939

- -
- The 1930s
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algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring.
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In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are
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field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
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In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. Binary operations can be accomplished using either a binary function or binary operator.
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In mathematics, a bilinear map is a function which is linear in both of its arguments. An example of such a map is multiplication of integers.

Definition

Let V, W and X be three vector spaces over the same base field F.
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In mathematics, anticommutativity refers to the property of a operation being anticommutative, i.e. being non commutative in a precise way. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence in physics: they
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In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must add the ring's multiplicative identity element (1) to itself to get the additive identity element (0); the ring is said to have
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In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity.
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associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. They are thus special algebras.
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In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.
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In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U(L). This construction passes from the non-associative structure L
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associativity is a property that a binary operation can have. It means that, within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed.
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In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. The branch of abstract algebra which studies rings is called ring theory.
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associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. They are thus special algebras.
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In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms.

History

The isomorphism theorems were originally formulated by Emmy Noether in her paper
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direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as subspaces. This is an example of a coproduct.
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In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that
f(x) = y.
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In mathematics, an isomorphism (Greek: isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings.
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identity map is a database access design pattern used to improve performance by providing a context-specific in-memory cache to prevent duplicate retrieval of the same object data from the database.
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Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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cross product is a binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which is perpendicular to the two input vectors. By contrast, the dot product produces a scalar result.
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spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
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