|Group theory → Lie groups|
Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds, with the property that the group operations of multiplication and inversion are smooth maps. Any Lie group gives rise to a Lie algebra. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to covering (Lie's third theorem). This correspondence between Lie groups and Lie algebras allows one to study Lie groups in terms of Lie algebras.
Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.
- 1 History
- 2 Definitions
- 3 Examples
- 4 Representations
- 5 Structure theory and classification
- 6 Relation to Lie groups
- 7 Lie algebra with additional structures
- 8 Lie ring
- 9 See also
- 10 Remarks
- 11 Notes
- 12 References
- 13 External links
Definition of a Lie algebra
- for all scalars a, b in F and all elements x, y, z in .
- for all x in .
- The Jacobi identity,
- 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.
It is customary to express a Lie algebra in lower-case fraktur, like . If a Lie algebra is associated with a Lie group, then the spelling of the Lie algebra is the same as that Lie group. For example, the Lie algebra of SU(n) is written as .
Consider , with the bracket defined by
where is the cross product. The bilinearity, skew-symmetry, and Jacobi identity are all known properties of the cross product. Concretely, if is the standard basis, then the bracket operation is completely determined by the relations:
(E.g., the relation follows from the above by the skew-symmetry of the bracket.)
Generators and dimension
Elements of a Lie algebra are said to be generators of the Lie algebra if the smallest subalgebra of containing them 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 also the classification of low-dimensional real Lie algebras for the low-dimensional case.
Subalgebras, ideals and homomorphisms
The Lie bracket is not associative in general, meaning that need not equal . (However, it is flexible.) 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 that is closed under the Lie bracket is called a Lie subalgebra. If a subspace satisfies a stronger condition that
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 in it, one constructs the factor algebra or quotient algebra , and the first isomorphism theorem holds for Lie algebras.
Let S be a subset of . The set of elements x such that for all s in S forms a subalgebra called the centralizer of S. The centralizer of itself is called the center of . Similar to centralizers, if S is a subspace, then the set of x such that is in S for all s in S forms a subalgebra called the normalizer of S.
Direct sum and semidirect product
Given two Lie algebras and , their direct sum is the Lie algebra consisting of the vector space , of the pairs , with the operation
for all x and y in the algebra. For any x, ad(x) (defined in section 4.2 below) is a derivation; a consequence of the Jacobi identity. Thus, the image of ad lies in the subalgebra of consisting of derivations on . A derivation that happens to be in the image of ad is called an inner derivation. If 𝔤 is semisimple, every derivation on 𝔤 is inner.
Split Lie algebra
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. 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; see split Lie algebra for further information.
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 antisymmetry of the Lie bracket.
- The real vector space of all n × n skew-hermitian matrices is closed under the commutator and forms a real Lie algebra denoted . This is the Lie algebra of the unitary group U(n).
- 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. 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 a -vector space with the above Lie bracket is denoted . If , the notation is or .
Every subalgebra (subspace closed under the Lie bracket) of a Lie algebra is a Lie algebra in its own right.
- The subspace of the general linear Lie algebra consisting of matrices of trace zero is a subalgebra, the special linear Lie algebra, denoted
Matrix Lie groups
Any Lie group G defines an associated real Lie algebra . The definition in general is somewhat technical, but in the case of a real matrix group G, it can be formulated via the exponential map, or the matrix exponential. The Lie algebra of G may be computed as
The Lie bracket of is given by the commutator of matrices, . The following are examples of Lie algebras of matrix Lie groups:
- The special linear group , consisting of all n × n matrices with real entries and determinant 1. Its Lie algebra consists of all n × n matrices with real entries and trace 0.
- The unitary group U(n) consists of n × n unitary matrices (those satisfying ). Its Lie algebra consists of skew-self-adjoint matrices (those satisfying ).
- The orthogonal and special orthogonal groups O(n) and SO(n) have the same Lie algebra, consisting of real, skew-symmetric matrices (those satisfying ).
- 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.
- So, for
- the resulting group elements are upper triangular 2×2 matrices with unit lower diagonal,
- The three-dimensional Euclidean space 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 generated by elements x, y and z with Lie brackets
- It is explicitly realized as the space of 3×3 strictly upper-triangular matrices, with the Lie bracket given by the matrix commutator,
- 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
- The commutation relations among these generators are
- These commutation relations are essentially the same as those among the x, y, and z components of the angular momentum operator in quantum mechanics.
- 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:
- A Kac–Moody algebra is an example of an infinite-dimensional Lie algebra.
- 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.
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
For any Lie algebra , we can define a representation
given by is a representation of on the vector space called the adjoint 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 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 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 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 what the irreducible representations of the given Lie algebra are. 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 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 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.
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 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 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.
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 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 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 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).
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:
- for all x, y, z ∈ L.
- 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.
- 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 commutative in each homogeneous part), and the bracket operation given by
- extended linearly. Note that the centrality of the series ensures the commutator gives the bracket operation the appropriate Lie theoretic properties.
- O'Connor & Robertson 2000
- O'Connor & Robertson 2005
- Humphreys 1978, p. 1
- Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.
- Jacobson 1962, pg. 28
- Jacobson 1962, pg. 42
- Jacobson 1962, pg. 108
- Bourbaki 1989, §1.2. Example 1.
- Bourbaki 1989, §1.2. Example 2.
- Humphreys p.2
- Helgason 1978, Ch. II, § 2, Proposition 2.7.
- Hall 2015 Section 3.3
- Hall 2015 Section 3.4
- Hall 2015 Example 3.27
- Jacobson 1962, Ch. VI
- Hall 2015, Theorem 10.9
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- 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.