Lie group–Lie algebra correspondence

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In mathematics, Lie group–Lie algebra correspondence allows one to study Lie groups, which are geometric objects, in terms of Lie algebras, which are linear objects. In this article, a Lie group refers to a real Lie group. For the complex and p-adic cases, see complex Lie group and p-adic Lie group.

In this article, manifolds (in particular Lie groups) are assumed to be second countable; in particular, they have at most countably many connected components.

Basics[edit]

The Lie algebra of a Lie group[edit]

There are various ways one can understand the construction of the Lie algebra of a Lie group G. One approach uses left-invariant vector fields. A vector field X on G is said to be invariant under left translations if, for any g, h in G,

where and is the differential of between tangent spaces. (In other words, it is -related to itself for any g in G.)

Let be the set of all left-translation-invariant vector fields on G. It is a real vector space. Moreover, it is closed under Lie bracket; i.e., is left-translation-invariant if X, Y are. Thus, is a Lie subalgebra of the Lie algebra of all vector fields on G and is called the Lie algebra of G. One can understand this more concretely by identifying the space of left-invariant vector fields with the tangent space at the identity, as follows: Given a left-invariant vector field, one can take its value at the identity, and given a tangent vector at the identity, one can extend it to a left-invariant vector field. Thus, the Lie algebra can be thought of as the tangent space at the identity and the bracket of X and Y in can be computed by extending them to left-invariant vector fields, taking the commutator of the vector fields, and then evaluating at the identity.

There is also another incarnation of as the Lie algebra of primitive elements of the Hopf algebra of distributions on G with support at the identity element; for this, see #Related constructions below.

Matrix Lie groups[edit]

Suppose G is a closed subgroup of GL(n;C), and thus a Lie group, by the closed subgroups theorem. Then the Lie algebra of G may be computed as

[1][2]

For example, one can use the criterion to establish the correspondence for classical compact groups (cf. the table in "compact Lie groups" below.)

Homomorphisms[edit]

If

is a Lie group homomorphism, then its differential at the identity element

is a Lie algebra homomorphism (brackets go to brackets), which has the following properties:

  • for all X in Lie(G), where "exp" is the exponential map
  • .[3]
  • If the image of f is closed,[4] then [5] and the first isomorphism theorem holds: f induces the isomorphism of Lie groups:
.
  • The chain rule holds: if and are Lie group homomorphisms, then

In particular, if H is a closed subgroup[6] of a Lie group G, then is a Lie subalgebra of . Also, if f is injective, then f is an immersion and so G is said to be an immersed (Lie) subgroup of H. For example, is an immersed subgroup of H. If f is surjective, then f is a submersion and if, in addition, G is compact, then f is a principal bundle with the structure group its kernel. (Ehresmann's lemma)

Other properties[edit]

Let be a direct product of Lie groups and projections. Then the differentials give the canonical identification:

.

If are Lie subgroups of a Lie group, then

Let G be a connected Lie group. If H is a Lie group, then any Lie group homomorphism is uniquely determined by its differential . Precisely, there is the exponential map (and one for H) such that and, since G is connected, this determines f uniquely.[7] In general, if U is a neighborhood of the identity element in a connected topological group G, then coincides with G, since the former is an open (hence closed) subgroup. Now, defines a local homeomorphism from a neighborhood of the zero vector to the neighborhood of the identity element. For example, if G is the Lie group of invertible real square matrices of size n (general linear group), then is the Lie algebra of real square matrices of size n and .

The correspondence[edit]

The correspondence between Lie groups and Lie algebras includes the following three main results.

  • Lie's third theorem: Every finite-dimensional real Lie algebra is the Lie algebra of some simply connected Lie group.[8]
  • The homomorphisms theorem: If is a Lie algebra homomorphism and if G is simply connected, then there exists a (unique) Lie group homomorphism such that .[9]
  • The subgroups–subalgebras theorem: If G is a Lie group and is a Lie subalgebra of , then there is a unique connected Lie subgroup (not necessarily closed) H of G with Lie algebra .[10]

In the second part of the correspondence, the assumption that G is simply connected cannot be omitted. For example, the Lie algebras of SO(3) and SU(2) are isomorphic,[11] but there is no corresponding homomorphism of SO(3) into SU(2).[12] Rather, the homomorphism goes from the simply connected group SU(2) to the non-simply connected group SO(3).[13] If G and H are both simply connected and have isomorphic Lie algebras, the above result allows one to show that G and H are isomorphic.[14] One method to construct f is to use the Baker–Campbell–Hausdorff formula.[15]

Proof of Lie's third theorem[edit]

Perhaps the most elegant proof of the first result above uses Ado's theorem, which says any finite-dimensional Lie algebra (over a field of any characteristic) is a Lie subalgebra of the Lie algebra of square matrices. The proof goes as follows: by Ado's theorem, we assume is a Lie subalgebra. Let G be the subgroup of generated by and let be a simply connected covering of G; it is not hard to show that is a Lie group and that the covering map is a Lie group homomorphism. Since , this completes the proof.

Example: Each element X in the Lie algebra gives rise to the Lie algebra homomorphism

By Lie's third theorem, as and exp for it is the identity, this homomorphism is the differential of the Lie group homomorphism for some immersed subgroup H of G. This Lie group homomorphism, called the one-parameter subgroup generated by X, is precisely the exponential map and H its image. The preceding can be summarized to saying that there is a canonical bijective correspondence between and the set of one-parameter subgroups of G.[16]

Proof of the homomorphisms theorem[edit]

One approach to proving the second part of the Lie group-Lie algebra correspondence (the homomorphisms theorem) is to use the Baker–Campbell–Hausdorff formula, as in Section 5.7 of Hall's book.[17] Specifically, given the Lie algebra homomorphism from to , we may define locally (i.e., in a neighborhood of the identity) by the formula

,

where is the exponential map for G, which has an inverse defined near the identity. We now argue that f is a local homomorphism. Thus, given two elements near the identity and (with X and Y small), we consider their product . According to the Baker–Campbell–Hausdorff formula, we have , where

,

with indicating other terms expressed as repeated commutators involving X and Y. Thus,

because is a Lie algebra homomorphism. Using the Baker–Campbell–Hausdorff formula again, this time for the group H, we see that this last expression becomes , and therefore we have

Thus, f has the homomorphism property, at least when X and Y are sufficiently small. It is important to emphasize that this argument is only local, since the exponential map is only invertible in a small neighborhood of the identity in G and since the Baker–Campbell–Hausdorff formula only holds if X and Y are small. Note also that we have not yet used the assumption that G is simply connected.

The next stage in the argument is to extend f from a local homomorphism to a global one. The extension is done by defining f it along a path and then using the simple connectedness of G to show that the definition is independent of the choice of path.

Lie group representations[edit]

A special case of Lie correspondence is a correspondence between finite-dimensional representations of a Lie group and representations of the associated Lie algebra.

The general linear group is a (real) Lie group and any Lie group homomorphism

is called a representation of the Lie group G. The differential

,

is then a Lie algebra homomorphism called a Lie algebra representation. (The differential is often simply denoted by .)

The homomorphisms theorem (mentioned above as part of the Lie group-Lie algebra correspondence) then says that if is the simply connected Lie group whose Lie algebra is , every representation of comes from a representation of G. The assumption that G be simply connected is essential. Consider, for example, the rotation group SO(3), which is not simply connected. There is one irreducible representation of the Lie algebra in each dimension, but only the odd-dimensional representations of the Lie algebra come from representations of the group.[18] (This observation is related to the distinction between integer spin and half-integer spin in quantum mechanics.) On the other hand, the group SU(2) is simply connected with Lie algebra isomorphic to that of SO(3), so every representation of the Lie algebra of SO(3) does give rise to a representation of SU(2).

The adjoint representation[edit]

An example of a Lie group representation is the adjoint representation of a Lie group G; each element g in a Lie group G defines an automorphism of G by conjugation: ; the differential is then an automorphism of the Lie algebra . This way, we get a representation , called the adjoint representation. The corresponding Lie algebra homomorphism is called the adjoint representation of and is denoted by . One can show , which in particular implies that the Lie bracket of is determined by the group law on G.

By Lie's third theorem, there exists a subgroup of whose Lie algebra is . ( is in general not a closed subgroup; only an immersed subgroup.) It is called the adjoint group of .[19] If G is connected, it fits into the exact sequence:

where is the center of G. If the center of G is discrete, then Ad here is a covering map.

Let G be a connected Lie group. Then G is unimodular if and only if for all g in G.[20]

Let G be a Lie group acting on a manifold X and Gx the stabilizer of a point x in X. Let . Then

  • .
  • If the orbit is locally closed, then the orbit is a submanifold of X and .[21]

For a subset A of or G, let

be the Lie algebra centralizer and the Lie group centralizer of A. Then .

If H is a closed connected subgroup of G, then H is normal if and only if is an ideal and in such a case .

Abelian Lie groups[edit]

Let G be a connected Lie group. Since the Lie algebra of the center of G is the center of the Lie algebra of G (cf. the previous §), G is abelian if and only if its Lie algebra is abelian.

If G is abelian, then the exponential map is a surjective group homomorphism.[22] The kernel of it is a discrete group (since the dimension is zero) called the integer lattice of G and is denoted by . By the first isomorphism theorem, induces the isomorphism .

By the rigidity argument, the fundamental group of a connected Lie group G is a central subgroup of a simply connected covering of G; in other words, G fits into the central extension

Equivalently, given a Lie algebra and a simply connected Lie group whose Lie algebra is , there is a one-to-one correspondence between quotients of by discrete central subgroups and connected Lie groups having Lie algebra .

For the complex case, complex tori are important; see complex Lie group for this topic.

Compact Lie groups[edit]

Let G be a connected Lie group with finite center. Then the following are equivalent.

  • G is compact.
  • (Weyl) The simply connected covering of G is compact.
  • The adjoint group is compact.
  • There exists an embedding as a closed subgroup.
  • The Killing form on is negative definite.
  • For each X in , is diagonalizable and has zero or purely imaginary eigenvalues.
  • There exists an invariant inner product on .

It is important to emphasize that the equivalence of the preceding conditions holds only under the assumption that G has finite center. Thus, for example, if G is compact with finite center, the universal cover is also compact. Clearly, this conclusion does not hold if G has infinite center, e.g., if . Note also that the last three conditions above are purely Lie algebraic in nature.

Compact Lie group Complexification of associated Lie algebra Root system
SU(n+1) An
SO(2n+1) Bn
Sp(n) Cn
SO(2n) Dn

If G is a compact Lie group, then

where the left-hand side is the Lie algebra cohomology of and the right-hand side is the de Rham cohomology of G. (Roughly, this is a consequence of the fact that any differential form on G can be made left invariant by the averaging argument.)

Related constructions[edit]

Let G be a Lie group. The associated Lie algebra of G may be alternatively defined as follows. Let be the algebra of distributions on G with support at the identity element with the multiplication given by convolution. is in fact a Hopf algebra. The Lie algebra of G is then , the Lie algebra of primitive elements in .[23] By the Milnor–Moore theorem, there is the canonical isomorphism between the universal enveloping algebra of and .

See also[edit]

Notes[edit]

  1. ^ Helgason 1978, Ch. II, § 2, Proposition 2.7.
  2. ^ Hall 2015 Section 3.3
  3. ^ More generally, if H' is a closed subgroup of H, then
  4. ^ This requirement cannot be omitted; see also https://math.stackexchange.com/q/329753
  5. ^ Bourbaki, Ch. III, § 3, no. 8, Proposition 28
  6. ^ Bourbaki, Ch. III, § 1, Proposition 5
  7. ^ Hall 2015 Corollary 3.49
  8. ^ Hall 2015 Theorem 5.25
  9. ^ Hall 2015 Theorem 5.6
  10. ^ Hall 2015 Theorem 5.20
  11. ^ Hall 2015 Example 3.27
  12. ^ Hall 2015 Proposition 4.35
  13. ^ Hall 2015 Section 1.4
  14. ^ Hall 2015 Corollary 5.7
  15. ^ Hall 2015 Section 5.7
  16. ^ Hall 2015 Theorem 2.14
  17. ^ Hall 2015
  18. ^ Hall, 2015 & Section 4.7
  19. ^ Helgason 1978, Ch II, § 5
  20. ^ Bourbaki, Ch. VII, § 6, no. 2, Corollary 4. to Proposition 1.
  21. ^ Bourbaki, Ch. III, § 1, no. 7, Proposition 14.
  22. ^ It's surjective because as is abelian.
  23. ^ Bourbaki, Ch. III, § 3. no. 7

References[edit]

  • Bourbaki, N. (1981), Groupes et Algèbres de Lie (Chapitre 3), Éléments de Mathématique, Hermann
  • Duistermaat, J.J.; Kolk, A. (2000), Lie groups, Universitext, Springer, doi:10.1007/978-3-642-56936-4, ISBN 3540152938
  • 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.
  • Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press, ISBN 0-12-338460-5

External links[edit]