In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Lie group–Lie algebra correspondence allows one to correspond a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
to a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
or vice versa, and study the conditions for such a relationship. Lie groups that are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
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
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
and the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
respectively) which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other. However, by restricting our attention to the
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
Lie groups, the Lie group-Lie algebra correspondence will be
one-to-one
One-to-one or one to one may refer to:
Mathematics and communication
*One-to-one function, also called an injective function
*One-to-one correspondence, also called a bijective function
*One-to-one (communication), the act of an individual comm ...
.
In this article, a Lie group refers to a real Lie group. For the complex and ''p''-adic cases, see
complex Lie group
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mat ...
and
''p''-adic Lie group. In this article, manifolds (in particular Lie groups) are assumed to be
second countable
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T is second-countable if there exists some countable collection \mat ...
; in particular, they have at most countably many connected components.
Basics
The Lie algebra of a Lie group
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.,
, Y
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
/math> 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
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
:
For example, one can use the criterion to establish the correspondence for
classical compact groups (cf. the table in "compact Lie groups" below.)
Homomorphisms
If
:
is a
Lie group homomorphism
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
, then its differential at the identity element
:
is a
Lie algebra homomorphism
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
(brackets go to brackets), which has the following properties:
*
for all ''X'' in Lie(''G''), where "exp" is the
exponential map
*
.
*If the image of ''f'' is closed, then
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 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
In mathematics, or specifically, in differential topology, Ehresmann's lemma or Ehresmann's fibration theorem states that if a smooth mapping f\colon M \rightarrow N, where M and N are smooth manifolds, is
# a surjective submersion, and
# a ...
)
Other properties
Let
be a
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
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. 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
The correspondence between Lie groups and Lie algebras includes the following three main results.
*
Lie's third theorem In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra \mathfrak over the real numbers is associated to a Lie group ''G''. The theorem is part of the Lie group–Lie algebra correspondence.
Histori ...
: Every finite-dimensional real Lie algebra is the Lie algebra of some
simply connected Lie group.
*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
.
*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
.
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, but there is no corresponding homomorphism of SO(3) into SU(2). Rather, the homomorphism goes from the simply connected group SU(2) to the non-simply connected group SO(3). 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. One method to construct ''f'' is to use the
Baker–Campbell–Hausdorff formula.
For readers familiar with
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
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
with Lie group
.
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
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
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
:\varphi : \mathbb \rightarrow G
from the real line \mathbb (as an additive group) to some other topological group G.
If \varphi is ...
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''.
Proof of the homomorphisms theorem
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. 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. 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
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
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
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
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is g ...
. (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)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
, 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. (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
An example of a Lie group representation is the
adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...
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
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...
of
and is denoted by
. One can show
, Y
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
/math>, which in particular implies that the Lie bracket of
is determined by the
group law
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. The ...
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
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n ...
of
. 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''.
Let ''G'' be a Lie group acting on a manifold ''X'' and ''G''
''x'' 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
.
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
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. 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
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
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
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', whe ...
are important; see
complex Lie group
In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\mat ...
for this topic.
Compact Lie groups
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
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...
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.
If ''G'' is a compact Lie group, then
:
where the left-hand side is the
Lie algebra cohomology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to prope ...
of
and the right-hand side is the
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapte ...
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
Let ''G'' be a Lie group. The associated Lie algebra
of ''G'' may be alternatively defined as follows. Let
be the algebra of
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
s on ''G'' with support at the identity element with the multiplication given by
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
.
is in fact a
Hopf algebra Hopf is a German surname. Notable people with the surname include:
*Eberhard Hopf (1902–1983), Austrian mathematician
*Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
*Heinz Hopf (actor) (1934–2001), Swedis ...
. The Lie algebra of ''G'' is then
, the Lie algebra of
primitive elements in
.
By the
Milnor–Moore theorem
In algebra, the Milnor–Moore theorem, introduced by classifies an important class of Hopf algebras, of the sort that often show up as cohomology rings in algebraic topology.
The theorem states: given a connected, graded, cocommutative Hopf al ...
, there is the canonical isomorphism
between the
universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra.
Universal enveloping algebras are used in the representati ...
of
and
.
See also
*
Compact Lie algebra
In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algeb ...
*
Milnor–Moore theorem
In algebra, the Milnor–Moore theorem, introduced by classifies an important class of Hopf algebras, of the sort that often show up as cohomology rings in algebraic topology.
The theorem states: given a connected, graded, cocommutative Hopf al ...
*
Formal group
*
Malcev Lie algebra In mathematics, a Malcev Lie algebra, or Mal'tsev Lie algebra, is a generalization of a rational nilpotent Lie algebra, and Malcev groups are similar. Both were introduced by , based on the work of .
Definition
According to a Malcev Lie algebra ...
*
Distribution on a linear algebraic group
Citations
References
*
*
*
*
*
External links
Notes for Math 261A Lie groups and Lie algebras*
Formal Lie theory in characteristic zero a blog post by Akhil Mathew
{{DEFAULTSORT:Lie group-Lie algebra correspondence
Differential geometry
Lie algebras
Lie groups
Manifolds