Lie group–Lie algebra correspondence

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In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvious counterexample is and (see real coordinate space and the circle group respectively) which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other. However, for simply connected Lie groups, the Lie group-Lie algebra correspondence is one-to-one.[1]

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

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The Lie algebra of a Lie group

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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   is defined by   and   is the differential of   between tangent spaces.

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. This correspondence is one-to-one in both directions, so is bijective. 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 bracket of the vector fields, and then evaluating the result 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

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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[2][3]

 

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

Homomorphisms

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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
  •  .[4]
  • If the image of f is closed,[5] then  [6] 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[7] 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

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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.[8] 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

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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.[9]
  • 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  .[10]
  • 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  .[11]

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,[12] but there is no corresponding homomorphism of SO(3) into SU(2).[13] Rather, the homomorphism goes from the simply connected group SU(2) to the non-simply connected group SO(3).[14] 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.[15] One method to construct f is to use the Baker–Campbell–Hausdorff formula.[16]

For readers familiar with category theory the correspondence can be summarised as follows: First, the operation of associating to each connected Lie group   its Lie algebra  , and to each homomorphism   of Lie groups the corresponding differential   at the neutral element, is a (covariant) functor   from the category of connected (real) Lie groups to the category of finite-dimensional (real) Lie-algebras. This functor has a left adjoint functor   from (finite dimensional) Lie algebras to Lie groups (which is necessarily unique up to canonical isomorphism). In other words there is a natural isomorphism of bifunctors

 

  is the (up to isomorphism unique) simply-connected Lie group with Lie algebra  . The associated natural unit morphisms   of the adjunction are isomorphisms, which corresponds to   being fully faithful (part of the second statement above). The corresponding counit   is the canonical projection   from the simply connected covering; its surjectivity corresponds to   being a faithful functor.

Proof of Lie's third theorem

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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 closed (without taking the closure one can get pathological dense example as in the case of the irrational winding of the torus) 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.[17]

Proof of the homomorphisms theorem

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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.[18] 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. 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. The assumption that G is simply connected has not yet been used.

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 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

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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.[19] (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

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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  .[20] 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.[21]

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  .[22]

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

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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.[23] 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

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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  . 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.)

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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  .[24] By the Milnor–Moore theorem, there is the canonical isomorphism   between the universal enveloping algebra of   and  .

See also

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Citations

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  1. ^ Lee 2012, p. 530.
  2. ^ Helgason 1978, Ch. II, § 2, Proposition 2.7.
  3. ^ Hall 2015 Section 3.3
  4. '^ More generally, if H is a closed subgroup of H, then  
  5. ^ This requirement cannot be omitted; see also https://math.stackexchange.com/q/329753
  6. ^ Bourbaki 1981, Ch. III, § 3, no. 8, Proposition 28
  7. ^ Bourbaki 1981, Ch. III, § 1, Proposition 5
  8. ^ Hall 2015 Corollary 3.49
  9. ^ Hall 2015 Theorem 5.25
  10. ^ Hall 2015 Theorem 5.6
  11. ^ Hall 2015 Theorem 5.20
  12. ^ Hall 2015 Example 3.27
  13. ^ Hall 2015 Proposition 4.35
  14. ^ Hall 2015 Section 1.4
  15. ^ Hall 2015 Corollary 5.7
  16. ^ Hall 2015 Section 5.7
  17. ^ Hall 2015 Theorem 2.14
  18. ^ Hall 2015
  19. ^ Hall 2015, Section 4.7
  20. ^ Helgason 1978, Ch II, § 5
  21. ^ Bourbaki 1981, Ch. III, § 3, no. 16, Corollary to Proposition 55.
  22. ^ Bourbaki 1981, Ch. III, § 1, no. 7, Proposition 14.
  23. ^ It's surjective because   as   is abelian.
  24. ^ Bourbaki 1981, Ch. III, § 3. no. 7

References

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  • 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, vol. 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
  • Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771.
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