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Lie Groups

Field: Group Theory

Groups that are also smooth manifolds, with group operations being smooth.

Sequence of Expressions

A real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication μ:G×G→Gμ(x,y)=xy\mu :G\times G\to G\quad \mu (x,y)=xy means that μ is a smooth mapping of the product manifoldG × G into G. The two requirements can be combined to the single requirement that the mapping (x,y)↦x−1y(x,y)\mapsto x^{-1}y be a smooth mapping of the product manifold into G. - The 2×2 realinvertible matrices form a group under multiplication, called general linear group of degree 2 and denoted by GL⁡(2,R)\operatorname {GL} (2,\mathbb {R} ) or by ⁠ GL⁡2(R)\operatorname {GL} _{2}(\mathbb {R} ) ⁠: GL⁡(2,R)={A=(abcd):det⁡A=ad−bc≠0}.\operatorname {GL} (2,\mathbb {R} )=\left\{A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\det A=ad-bc\neq 0\right\}. This is a four-dimensional noncompact real Lie group; it is an open subset of ⁠ R4\mathbb {R} ^{4} ⁠. This group is disconnected; it has two connected components corresponding to the positive and negative values of the determinant. - The rotation matrices form a subgroup of ⁠ GL⁡(2,R)\operatorname {GL} (2,\mathbb {R} ) ⁠, denoted by ⁠ SO⁡(2,R)\operatorname {SO} (2,\mathbb {R} ) ⁠. It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is diffeomorphic to the circle. Using the rotation angle φ\varphi as a parameter, this group can be parametrized as follows: SO⁡(2,R)={(cos⁡φ−sin⁡φsin⁡φcos⁡φ):φ∈R / 2πZ}.\operatorname {SO} (2,\mathbb {R} )=\left\{{\begin{pmatrix}\cos \varphi &-\sin \varphi \\\sin \varphi &\cos \varphi \end{pmatrix}}:\varphi \in \mathbb {R} \ /\ 2\pi \mathbb {Z} \right\}. Addition of the angles corresponds to multiplication of the elements of ⁠ SO⁡(2,R)\operatorname {SO} (2,\mathbb {R} ) ⁠, and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps. - The affine group of one dimension is a two-dimensional matrix Lie group, consisting of 2×22\times 2 real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1. Thus, the group consists of matrices of the form A=(ab01),a>0, b∈R.A=\left({\begin{array}{cc}a&b\\0&1\end{array}}\right),\quad a>0,\,b\in \mathbb {R} . We now present an example of a group with an uncountable number of elements that is not a Lie group under a certain topology. The group given by H={(e2πiθ00e2πiaθ): θ∈R}⊂T2={(e2πiθ00e2πiϕ): θ,ϕ∈R},H=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi ia\theta }\end{matrix}}\right):\,\theta \in \mathbb {R} \right\}\subset \mathbb {T} ^{2}=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\phi }\end{matrix}}\right):\,\theta ,\phi \in \mathbb {R} \right\}, with a∈R∖Qa\in \mathbb {R} \setminus \mathbb {Q} a fixedirrational number, is a subgroup of the torus T2\mathbb {T} ^{2} that is not a Lie group when given the subspace topology. If we take any small neighborhood UU of a point hh in ⁠ HH ⁠, for example, the portion of HH in UU is disconnected. The group HH winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of ⁠ T2\mathbb {T} ^{2} ⁠. A portion of the group HH inside ⁠ T2\mathbb {T} ^{2} ⁠. Small neighborhoods of the element h∈Hh\in H are disconnected in the subset topology on ⁠ HH ⁠ The group HH can, however, be given a different topology, in which the distance between two points h1,h2∈Hh_{1},h_{2}\in H is defined as the length of the shortest path in the group HH joining h1h_{1} to ⁠ h2h_{2} ⁠. In this topology, HH is identified homeomorphically with the real line by identifying each element with the number θ\theta in the definition of ⁠ HH ⁠. With this topology, HH is just the group of real numbers under addition and is therefore a Lie group. The group HH is an example of a "Lie subgroup" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts. Let GL⁡(n,C)\operatorname {GL} (n,\mathbb {C} ) denote the group of n×nn\times n invertible matrices with entries in ⁠ C\mathbb {C} ⁠. Any closed subgroup of GL⁡(n,C)\operatorname {GL} (n,\mathbb {C} ) is a Lie group; Lie groups of this sort are called matrix Lie groups. Since most of the interesting examples of Lie groups can be realized as matrix Lie groups, some textbooks restrict attention to this class, including those of Hall, Rossmann, and Stillwell. Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map. The following are standard examples of matrix Lie groups. - The special linear groups over R\mathbb {R} and ⁠ C\mathbb {C} ⁠, SL⁡(n,R)\operatorname {SL} (n,\mathbb {R} ) and ⁠ SL⁡(n,C)\operatorname {SL} (n,\mathbb {C} ) ⁠, consisting of n×nn\times n matrices with determinant 11 and entries in R\mathbb {R} or C\mathbb {C} - The unitary groups and special unitary groups, U⁡(n,C)\operatorname {U} (n,\mathbb {C} ) and ⁠ SU⁡(n,C)\operatorname {SU} (n,\mathbb {C} ) ⁠, consisting of n×nn\times n complex matrices satisfying U∗=U−1U^{*}=U^{-1} (and also det⁡(U)=1\det(U)=1 in the case of SU⁡(n)\operatorname {SU} (n) ), where U∗U^{*} is the conjugate transpose of UU - The orthogonal groups and special orthogonal groups, O⁡(n,R)\operatorname {O} (n,\mathbb {R} ) and ⁠ SO⁡(n,R)\operatorname {SO} (n,\mathbb {R} ) ⁠, consisting of n×nn\times n real matrices satisfying RT=R−1R^{\mathrm {T} }=R^{-1} (and also det⁡(R)=1\det(R)=1 in the case of SO⁡(n,R)\operatorname {SO} (n,\mathbb {R} ) ), where RTR^{\mathrm {T} } is the transpose of RR All of the preceding examples fall under the heading of the classical groups. A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: SL⁡(2,C)\operatorname {SL} (2,\mathbb {C} ) ), and holomorphic maps. Similarly, using an alternate metric completion of ⁠ Q\mathbb {Q} ⁠, one can define a p-adic Lie group over the p-adic numbers, a topological group which is also an analytic p-adic manifold, such that the group operations are analytic. In particular, each point has a p-adic neighborhood. Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, Gleason, Montgomery and Zippin showed that if G is a topological manifold with continuous group operations, then there exists exactly one analytic structure on G which turns it into a Lie group (see also Hilbert–Smith conjecture). If the underlying manifold is allowed to be infinite-dimensional (for example, a Hilbert manifold), then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple groups. The language of category theory provides a concise definition for Lie groups: a Lie group is a group object in the category of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to Lie supergroups. This categorical point of view leads also to a different generalization of Lie groups, namely Lie groupoids, which are groupoid objects in the category of smooth manifolds with a further requirement. A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds nor topological manifolds. Precisely, a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to a matrix Lie group, a closed subgroup of GL⁡(n,C)\operatorname {GL} (n,\mathbb {C} ) and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows: - Given a Lie group G in the usual manifold sense, the Lie group–Lie algebra correspondence (or a version of Lie's third theorem) constructs a closed Lie subgroup G′⊂GL⁡(n,C)G'\subset \operatorname {GL} (n,\mathbb {C} ) such that G,G′G,G' share the same Lie algebra; thus, they are locally isomorphic. Hence, GG satisfies the above topological definition. - Conversely, let GG be a topological group that is a Lie group in the above topological sense and choose a matrix Lie group G′G' that is locally isomorphic to ⁠ GG ⁠ around the respective identities. Then, by a version of the closed subgroup theorem, G′G' is a real-analytic manifold and then, through the local isomorphism, G acquires a structure of a manifold near the identity element. One then shows that the group law on G can be given by formal power series; so the group operations are real-analytic and GG itself is a real-analytic manifold. The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, the topology of a Lie group together with the group law determines the geometry of the group. - ^Rossmann 2001, Chapter 2 - ^Hall 2015 Corollary 3.45 - ^Hall 2015 - ^Rossmann 2001 - ^Stillwell 2008 - ^Kobayashi & Oshima 2005, Definition 5.3 - ^Bruhat, F. (1958). "Lectures on Lie Groups and Representations of Locally Compact Groups"(PDF). Tata Institute of Fundamental Research, Bombay. Cite error: There are tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).
A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds nor topological manifolds. Precisely, a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to a matrix Lie group, a closed subgroup of GL⁡(n,C)\operatorname {GL} (n,\mathbb {C} ) and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows: - Given a Lie group G in the usual manifold sense, the Lie group–Lie algebra correspondence (or a version of Lie's third theorem) constructs a closed Lie subgroup G′⊂GL⁡(n,C)G'\subset \operatorname {GL} (n,\mathbb {C} ) such that G,G′G,G' share the same Lie algebra; thus, they are locally isomorphic. Hence, GG satisfies the above topological definition. - Conversely, let GG be a topological group that is a Lie group in the above topological sense and choose a matrix Lie group G′G' that is locally isomorphic to ⁠ GG ⁠ around the respective identities. Then, by a version of the closed subgroup theorem, G′G' is a real-analytic manifold and then, through the local isomorphism, G acquires a structure of a manifold near the identity element. One then shows that the group law on G can be given by formal power series; so the group operations are real-analytic and GG itself is a real-analytic manifold. The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, the topology of a Lie group together with the group law determines the geometry of the group. - ^Kobayashi & Oshima 2005, Definition 5.3 - ^Bruhat, F. (1958). "Lectures on Lie Groups and Representations of Locally Compact Groups"(PDF). Tata Institute of Fundamental Research, Bombay. Cite error: There are tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).
To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are "infinitesimally close" to the identity, and the Lie bracket of the Lie algebra is related to the commutator of two such infinitesimal elements. Before giving the abstract definition we give a few examples: - The Lie algebra of the vector space R^{n} is just R^{n} with the Lie bracket given by     [A, B] = 0. (In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.) - The Lie algebra of the general linear group GL(n, C) of invertible matrices is the vector space M(n, C) of square matrices with the Lie bracket given by     [A, B] = AB − BA. - If G is a closed subgroup of GL(n, C) then the Lie algebra of G can be thought of informally as the matrices m of M(n, C) such that 1 + εm is in G, where ε is an infinitesimal positive number with ε^{2} = 0 (of course, no such real number ε exists). For example, the orthogonal group O(n, R) consists of matrices A with AA^{T} = 1, so the Lie algebra consists of the matrices m with (1 + εm)(1 + εm)^{T} = 1, which is equivalent to m + m^{T} = 0 because ε^{2} = 0. - The preceding description can be made more rigorous as follows. The Lie algebra of a closed subgroup G of GL(n, C), may be computed as Lie⁡(G)={X∈M(n;C)∣exp⁡(tX)∈G for all t in R},\operatorname {Lie} (G)=\{X\in M(n;\mathbb {C} )|\operatorname {exp} (tX)\in G{\text{ for all }}t{\text{ in }}\mathbb {\mathbb {R} } \}, where exp(tX) is defined using the matrix exponential. It can then be shown that the Lie algebra of G is a real vector space that is closed under the bracket operation, ⁠ [X,Y]=XY−YX[X,Y]=XY-YX ⁠. The concrete definition given above for matrix groups is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not even obvious that the Lie algebra is independent of the representation we use. To get around these problems we give the general definition of the Lie algebra of a Lie group (in 4 steps): - Vector fields on any smooth manifold M can be thought of as derivationsX of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket [X, Y] = XY − YX, because the Lie bracket of any two derivations is a derivation. - If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra. - We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations L_{g}(h) = gh. This shows that the space of left invariant vector fields (vector fields satisfying L_{g}_{*}X_{h} = X_{gh} for every h in G, where L_{g}_{*} denotes the differential of L_{g}) on a Lie group is a Lie algebra under the Lie bracket of vector fields. - Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v^_{g} = L_{g}_{*}v. This identifies the tangent spaceT_{e}G at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur g.{\mathfrak {g}}. Thus the Lie bracket on g{\mathfrak {g}} is given explicitly by [v, w] = [v^, w^]_{e}. This Lie algebra g{\mathfrak {g}} is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element. Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily. For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras. We could also define a Lie algebra structure on T_{e} using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space T_{e}. The Lie algebra structure on T_{e} can also be described as follows: the commutator operation (x, y) → xyx^{−1}y^{−1} on G × G sends (e, e) to e, so its derivative yields a bilinear operation on T_{e}G. This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket, and it is equal to twice the one defined through left-invariant vector fields. If G and H are Lie groups, then a Lie group homomorphism f : G → H is a smooth group homomorphism. In the case of complex Lie groups, such a homomorphism is required to be a holomorphic map. However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real) analytic. The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let ϕ:G→H\phi :G\to H be a Lie group homomorphism and let ϕ∗\phi _{*} be its derivative at the identity. If we identify the Lie algebras of G and H with their tangent spaces at the identity elements, then ϕ∗\phi _{*} is a map between the corresponding Lie algebras: ϕ∗:g→h,\phi _{*}:{\mathfrak {g}}\to {\mathfrak {h}}, which turns out to be a Lie algebra homomorphism (meaning that it is a linear map which preserves the Lie bracket). In the language of category theory, we then have a covariant functor from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity. Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Observe that, by the above, a continuous homomorphism from a Lie group GG to a Lie group HH is an isomorphism of Lie groups if and only if it is bijective. Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras. The first result in this direction is Lie's third theorem, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected. To put it differently, the global structure of a Lie group is not determined by its Lie algebra; for example, if Z is any discrete subgroup of the center of G then G and G/Z have the same Lie algebra (see the table of Lie groups for examples). An example of importance in physics are the groups SU(2) and SO(3). These two groups have isomorphic Lie algebras, but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not. On the other hand, if we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic. (See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups. A Lie group GG is said to be simply connected if every loop in GG can be shrunk continuously to a point in ⁠ GG ⁠. This notion is important because of the following result that has simple connectedness as a hypothesis: Theorem: Suppose GG and HH are Lie groups with Lie algebras g{\mathfrak {g}} and h{\mathfrak {h}} and that f:g→hf:{\mathfrak {g}}\rightarrow {\mathfrak {h}} is a Lie algebra homomorphism. If GG is simply connected, then there is a unique Lie group homomorphism ϕ:G→H\phi :G\rightarrow H such that ⁠ ϕ∗=f\phi _{*}=f ⁠, where ϕ∗\phi _{*} is the differential of ϕ\phi at the identity. Lie's third theorem says that every finite-dimensional real Lie algebra is the Lie algebra of a Lie group. It follows from Lie's third theorem and the preceding result that every finite-dimensional real Lie algebra is the Lie algebra of a unique simply connected Lie group. An example of a simply connected group is the special unitary group SU(2), which as a manifold is the 3-sphere. The rotation group SO(3), on the other hand, is not simply connected. (See Topology of SO(3).) The failure of SO(3) to be simply connected is intimately connected to the distinction between integer spin and half-integer spin in quantum mechanics. Other examples of simply connected Lie groups include the special unitary group SU(n), the spin group (double cover of rotation group) Spin(n) for ⁠ n≥3n\geq 3 ⁠, and the compact symplectic group Sp(n). Methods for determining whether a Lie group is simply connected or not are discussed in the article on fundamental groups of Lie groups. The exponential map from the Lie algebra M(n;C)\mathrm {M} (n;\mathbb {C} ) of the general linear group GL(n;C)\mathrm {GL} (n;\mathbb {C} ) to GL(n;C)\mathrm {GL} (n;\mathbb {C} ) is defined by the matrix exponential, given by the usual power series: exp⁡(X)=1+X+X22!+X33!+⋯\exp(X)=1+X+{\frac {X^{2}}{2!}}+{\frac {X^{3}}{3!}}+\cdots for matrices ⁠ XX ⁠. If GG is a closed subgroup of ⁠ GL(n;C)\mathrm {GL} (n;\mathbb {C} ) ⁠, then the exponential map takes the Lie algebra of GG into ⁠ GG ⁠; thus, we have an exponential map for all matrix groups. Every element of GG that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra. The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows. For each vector XX in the Lie algebra g{\mathfrak {g}} of GG (i.e., the tangent space to GG at the identity), one proves that there is a unique one-parameter subgroup c:R→Gc:\mathbb {R} \rightarrow G such that ⁠ c′(0)=Xc'(0)=X ⁠. Saying that cc is a one-parameter subgroup means simply that cc is a smooth map into GG and that c(s+t)=c(s)c(t)\$$ for all $s$ and ⁠ $t$ ⁠. The operation on the right hand side is the group multiplication in ⁠ $G$ ⁠. The formal similarity of this formula with the one valid for the exponential function justifies the definition \exp(X)=c(1). This is called the exponential map, and it maps the Lie algebra ${\mathfrak {g}}$ into the Lie group ⁠ $G$ ⁠. It provides a diffeomorphism between a neighborhood of 0 in ${\mathfrak {g}}$ and a neighborhood of $e$ in ⁠ $G$ ⁠. This exponential map is a generalization of the exponential function for real numbers (because $\mathbb {R}$ is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because $\mathbb {C}$ is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because $M(n,\mathbb {R} )$ with the regular commutator is the Lie algebra of the Lie group $\mathrm {GL} (n,\mathbb {R} )$ of all invertible matrices). Because the exponential map is surjective on some neighbourhood $N$ of ⁠ $e$ ⁠, it is common to call elements of the Lie algebra infinitesimal generators of the group ⁠ $G$ ⁠. The subgroup of $G$ generated by $N$ is the identity component of ⁠ $G$ ⁠. The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker–Campbell–Hausdorff formula: there exists a neighborhood $U$ of the zero element of ⁠ ${\mathfrak {g}}$ ⁠, such that for $X,Y\in U$ we have \exp(X)\,\exp(Y)=\exp \left(X+Y+{\tfrac {1}{2}}[X,Y]+{\tfrac {1}{12}}[\,[X,Y],Y]-{\tfrac {1}{12}}[\,[X,Y],X]-\cdots \right), where the omitted terms are known and involve Lie brackets of four or more elements. In case $X$ and $Y$ commute, this formula reduces to the familiar exponential law ⁠ $\exp(X)\exp(Y)=\exp(X+Y)$ ⁠. The exponential map relates Lie group homomorphisms. That is, if $\phi :G\to H$ is a Lie group homomorphism and $\phi _{*}:{\mathfrak {g}}\to {\mathfrak {h}}$ the induced map on the corresponding Lie algebras, then for all $x\in {\mathfrak {g}}$ we have \phi (\exp(x))=\exp(\phi _{*}(x)).$$ In other words, the following diagram commutes, (In short, exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups.) The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL(2, R) is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled on C^{∞}Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1. A Lie subgroup $H$ of a Lie group $G$ is a Lie group that is a subset of $G$ and such that the inclusion map from $H$ to $G$ is an injectiveimmersion and group homomorphism. According to Cartan's theorem, a closed subgroup of $G$ admits a unique smooth structure which makes it an embedded Lie subgroup of $G$ —i.e. a Lie subgroup such that the inclusion map is a smooth embedding. Examples of non-closed subgroups are plentiful; for example take $G$ to be a torus of dimension 2 or greater, and let $H$ be a one-parameter subgroup of irrational slope, i.e. one that winds around in G. Then there is a Lie group homomorphism $\varphi :\mathbb {R} \to G$ with ⁠ $\mathrm {im} (\varphi )=H$ ⁠. The closure of $H$ will be a sub-torus in ⁠ $G$ ⁠. The exponential map gives a one-to-one correspondence between the connected Lie subgroups of a connected Lie group $G$ and the subalgebras of the Lie algebra of ⁠ $G$ ⁠. Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of $G$ which determines which subalgebras correspond to closed subgroups. - ^Helgason 1978, Ch. II, § 2, Proposition 2.7 - ^Hall 2015 - ^Hall 2015 Theorem 3.20 - ^But see Hall 2015, Proposition 3.30 and Exercise 8 in Chapter 3 - ^Hall 2015 Corollary 3.50 - ^Hall 2015 Theorem 5.20 - ^Hall 2015 Example 3.27 - ^Hall 2015 Section 1.3.4 - ^Hall 2015 Corollary 5.7 - ^Hall 2015 Theorem 5.6 - ^Hall 2015 Section 13.2 - ^Hall 2015 Theorem 3.42 - ^"Introduction to Lie groups and algebras : Definitions, examples and problems"(PDF). State University of New York at Stony Brook. 2006. Archived from the original(PDF) on 28 September 2011. Retrieved 11 October 2014. - ^Hall 2015 Theorem 5.20 Cite error: There are tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).
Let GG be a set equipped with a smooth manifold structure such that the underlying topological space is Hausdorff and second-countable. GG is a Lie Group if it is also a group (G,m,i)(G, \text{m}, \text{i}) such that the group multiplication map m:G×G→G\text{m}: G \times G \to G and the inversion map i:G→G\text{i}: G \to G are smooth maps. Specifically, if \phi: U \to \bb{R}^n are local coordinate charts for GG, the maps m\text{m} and i\text{i} must induce smooth functions on the coordinate representations. Formally, for any point g∈Gg \in G, there exist charts ϕ1,ϕ2,ϕ3\phi_1, \phi_2, \phi_3 around g,g1,g2g, g_1, g_2 respectively, such that the coordinate representations of the operations are smooth: \n\nm∗:C∞(U1)×C∞(U2)→C∞(U3)\text{m}_{*}: \text{C}^{\infty}(U_1) \times \text{C}^{\infty}(U_2) \to \text{C}^{\infty}(U_3) \ni∗:C∞(U)→C∞(U)\text{i}_{*}: \text{C}^{\infty}(U) \to \text{C}^{\infty}(U) \nwhere C∞\text{C}^{\infty} denotes the space of smooth functions, and the maps m∗\text{m}_{*} and i∗\text{i}_{*} are defined by the local coordinates of the group operations. Equivalently, GG is a Lie Group if and only if it is a smooth manifold such that the group operations m:G×G→G\text{m}: G \times G \to G and i:G→G\text{i}: G \to G are smooth maps in the manifold topology.