Lie Groups

Introduction

A Lie group is a group that is also a smooth manifold, with smooth group operations. Lie groups formalize continuous symmetry: the rotations of the plane and of space, the Lorentz group, the unitary group, and the invertible linear maps of a vector space are all Lie groups.

The treatment is introductory and purely mathematical. We assume familiarity with groups from Groups and Lie algebras from Lie Algebras: A General Introduction, together with the elementary notions of smooth manifolds (charts, smooth maps, tangent spaces, vector fields). Only a little differential geometry is needed, and it is recalled where used.

To a Lie group $G$ one associates its tangent space at the identity, $\mathrm{G} = T_eG$, equipped with a bracket built from the commutator of left-invariant vector fields. This linear object is the Lie algebra of $G$; it records the local structure of $G$ and discards its global topology. A connected, simply connected Lie group is determined up to isomorphism by its Lie algebra. We treat real and complex Lie groups of finite dimension.

Part I: Lie Groups and the Classical Matrix Groups

1. Smooth Manifolds (Recalled)

A smooth manifold of dimension $n$ is a second-countable Hausdorff space $M$ with a maximal atlas of charts $\phi : U \to \mathbb{R}^n$ whose transition maps are smooth; smooth maps and their derivatives $df_p : T_pM \to T_{f(p)}N$ are defined in charts. The pushforward of a vector field $X$ by a diffeomorphism $F$ is written $F_*X$.

2. Definition of a Lie Group

A Lie group is a group $G$ that is also a smooth manifold, such that the maps

$$ m : G \times G \to G, \quad m(g,h) = gh, \qquad \iota : G \to G, \quad \iota(g) = g^{-1} $$

are smooth. Its dimension is the dimension of the underlying manifold; the identity is $e$. It is enough to require that $m$ is smooth, for then $\iota$ is smooth as well (inverse function theorem applied to $(g,h) \mapsto (g,gh)$ at $(e,e)$); the sources in Further Reading require both operations, and the definitions agree.

For each $g$, left and right translations $L_g(h) = gh$ and $R_g(h) = hg$ are diffeomorphisms, with inverses $L_{g^{-1}}$ and $R_{g^{-1}}$. Conjugation

$$ \Psi_g = L_g \circ R_{g^{-1}} : h \mapsto ghg^{-1} $$

is an automorphism of $G$ and a diffeomorphism; since it fixes $e$, its derivative at $e$ is an invertible linear map of $T_eG$, the source of the adjoint representation (§15).

3. The Classical Matrix Groups

Let $\mathbb{K}$ be $\mathbb{R}$ or $\mathbb{C}$, and write $A^* = \overline{A}^{\,T}$ in the complex case. The classical groups are

$$ GL_n(\mathbb{K}) = \{A \in M_n(\mathbb{K}) : \det A \neq 0\}, $$

$$ SL_n(\mathbb{K}) = \{A \in GL_n(\mathbb{K}) : \det A = 1\}, $$

$$ O(n) = \{A \in GL_n(\mathbb{R}) : A^TA = I\}, \qquad SO(n) = \{A \in O(n) : \det A = 1\}, $$

$$ U(n) = \{A \in GL_n(\mathbb{C}) : A^*A = I\}, \qquad SU(n) = \{A \in U(n) : \det A = 1\}. $$

Two groups share the name symplectic: the compact symplectic (quaternionic unitary) group $Sp(n) = \{A \in GL_n(\mathbb{H}) : A^*A = I\}$, and the real symplectic group $Sp(2n,\mathbb{R}) = \{A \in GL_{2n}(\mathbb{R}) : A^TJA = J\}$ for a fixed invertible antisymmetric $J$. Note $Sp(1) \cong SU(2)$ and $Sp(2,\mathbb{R}) \cong SL_2(\mathbb{R})$.

The real dimensions are:

Group	Field	Real dimension
$GL_n(\mathbb{R})$	$\mathbb{R}$	$n^2$
$GL_n(\mathbb{C})$	$\mathbb{C}$	$2n^2$
$SL_n(\mathbb{R})$	$\mathbb{R}$	$n^2 - 1$
$SL_n(\mathbb{C})$	$\mathbb{C}$	$2n^2 - 2$
$O(n)$	$\mathbb{R}$	$n(n-1)/2$
$SO(n)$	$\mathbb{R}$	$n(n-1)/2$
$U(n)$	$\mathbb{C}$	$n^2$
$SU(n)$	$\mathbb{C}$	$n^2 - 1$
$Sp(n)$	$\mathbb{H}$	$n(2n+1)$
$Sp(2n,\mathbb{R})$	$\mathbb{R}$	$n(2n+1)$

Each is a closed subgroup of a general linear group, itself a Lie group with polynomial multiplication and rational inversion. The defining conditions are level sets of smooth maps with surjective derivative; for instance $A \mapsto A^TA$ has derivative $H \mapsto A^TH + H^TA$, so the regular value theorem makes $O(n)$ an embedded submanifold of dimension $n^2 - n(n+1)/2 = n(n-1)/2$, and the same argument gives the table (Cartan's closed subgroup theorem, §12, is an alternative). The group operations restrict from the ambient group, so each group is a Lie group. Here $O(n)$ is compact with identity component $SO(n)$, of index $2$, while $U(n)$ and $SU(n)$ are compact connected with $U(n)/SU(n) \cong U(1)$.

4. Further Examples

The additive group $\mathbb{R}^n$ is an abelian Lie group of dimension $n$, and $\mathbb{C}^n$ is a real Lie group of dimension $2n$. A torus

$$ T^n = \mathbb{R}^n/\mathbb{Z}^n = U(1)^n $$

is a compact connected abelian Lie group of dimension $n$; the circle $S^1 = U(1)$ is the case $n=1$.

The unit sphere $S^3 = \{q \in \mathbb{H} : |q| = 1\}$ is closed under quaternion multiplication and inversion, and

$$ S^3 \cong SU(2) \cong Sp(1). $$

It is compact, connected, and simply connected, and $S^3/\{\pm1\} \cong SO(3)$ with a two-sheeted covering $S^3 \to SO(3)$. This double cover of the rotation group by the unit quaternions is the group-theoretic reason quaternions describe rotations. Among spheres, only $S^0$, $S^1$, $S^3$ admit a Lie group structure, so $S^7$ does not.

The Heisenberg group

$$ H_3(\mathbb{R}) = \left\{ \begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix} : a,b,c \in \mathbb{R} \right\} $$

is a simply connected nilpotent Lie group of dimension $3$, with the Heisenberg Lie algebra. Semidirect products $G \rtimes H$ with a smooth action, such as the Euclidean group $\mathbb{R}^n \rtimes SO(n)$, are Lie groups as well.

Part II: The Lie Algebra of a Lie Group

5. Left-Invariant Vector Fields

A vector field $X$ on $G$ assigns to each $g$ a tangent vector $X_g \in T_gG$, smoothly in $g$, and is equivalently a derivation of $C^\infty(G)$. It is left-invariant if $(L_g)_*X = X$ for all $g$, that is, if

$$ X_{gh} = (dL_g)_h X_h \quad \text{for all } g,h \in G. $$

Taking $h = e$ gives $X_g = (dL_g)_eX_e$, so a left-invariant field is determined by its value at $e$; conversely $X_g = (dL_g)_eX$ defines a left-invariant field for any $X \in T_eG$. Evaluation at $e$ is a linear isomorphism

$$ \{\text{left-invariant vector fields}\} \xrightarrow{\ \sim\ } T_eG. $$

6. The Lie Bracket

Vector fields carry the commutator bracket

$$ [X,Y]f = X(Yf) - Y(Xf), \qquad f \in C^\infty(G), $$

which is bilinear, antisymmetric, and satisfies the Jacobi identity. It is natural under diffeomorphisms: $F_*[X,Y] = [F_*X, F_*Y]$. Since left translations are diffeomorphisms, $(L_g)_*[X,Y] = [X,Y]$ whenever $X, Y$ are left-invariant, so the left-invariant fields form a Lie subalgebra. Transporting the bracket across the isomorphism of §5 gives a bracket on $T_eG$.

Definition. The Lie algebra of $G$ is $\mathrm{G} = T_eG$ with this bracket, written $\mathrm{G} = \operatorname{Lie}(G)$.

In group terms the bracket is the infinitesimal commutator: for $X, Y \in \mathrm{G}$,

$$ \exp(tX)\exp(sY)\exp(-tX)\exp(-sY) = \exp\!\left(ts[X,Y] + \text{higher order}\right), $$

$$ [X,Y] = \frac{\partial^2}{\partial s\,\partial t}\Big|_{s=t=0} \exp(tX)\exp(sY)\exp(-tX)\exp(-sY). $$

The left-invariant convention is the standard one and is used throughout this series.

7. The Lie Algebra of a Matrix Group

If $G \subseteq GL_n(\mathbb{R})$ is a closed subgroup, its Lie algebra is

$$ \mathrm{G} = T_IG = \{X \in M_n(\mathbb{R}) : \exp(tX) \in G \text{ for all } t \in \mathbb{R}\}, $$

with bracket the matrix commutator

$$ [X,Y] = XY - YX. $$

Indeed the left-invariant fields determined by $X, Y$ are $A \mapsto AX$ and $A \mapsto AY$, whose commutator at $I$ is $XY - YX$. The tangent space is the kernel of the derivative of the defining equations: $d(\det)_I = \operatorname{tr}$ gives $\mathrm{SL}_n = \{\operatorname{tr}H = 0\}$; $A^TA = I$ gives $\mathrm{SO}(n) = \{H^T = -H\}$; and $A^*A = I$ gives $\mathrm{U}(n) = \{H^* = -H\}$. Similarly $\mathrm{SU}(n) = \{H^* = -H,\ \operatorname{tr}H = 0\}$, and $\mathrm{Sp}(n)$, $\mathrm{Sp}(2n,\mathbb{R})$ satisfy $H^* = -H$ and $H^TJ + JH = 0$ respectively. Each has the real dimension of the corresponding group in §3, and these algebras coincide with those classified in Lie Algebras: Categorization; in particular $\mathrm{SO}(3)$ is $\mathbb{R}^3$ with the cross product, and $\mathrm{GL}_2(\mathbb{C})$ is the biquaternions with the commutator bracket.

For an abelian group all brackets vanish, and different groups can share a Lie algebra: $S^1$ and $\mathbb{R}$ both have Lie algebra $\mathbb{R}$ but are not isomorphic. The Lie algebra records local data only.

Part III: The Exponential Map

8. One-Parameter Subgroups and the Exponential Map

A one-parameter subgroup of $G$ is a smooth homomorphism $\gamma : \mathbb{R} \to G$, so $\gamma(0) = e$ and $\gamma(s+t) = \gamma(s)\gamma(t)$. Left-invariant vector fields are complete (their flows extend by translation), so for each $X \in \mathrm{G}$ there is a unique one-parameter subgroup $\gamma_X$ with $\gamma_X(0) = e$ and $\gamma_X'(0) = X$. The exponential map is

$$ \exp : \mathrm{G} \to G, \qquad \exp(X) = \gamma_X(1). $$

9. Properties of the Exponential Map

(a) $\exp(tX) = \gamma_X(t)$ for all $t$, hence $\exp(0) = e$.

(b) $\exp(sX)\exp(tX) = \exp((s+t)X)$ and $\exp(-X) = \exp(X)^{-1}$.

(c) The map $\exp$ is smooth with $d\exp_0 = \operatorname{id}_{\mathrm{G}}$, so by the inverse function theorem some neighborhoods $U$ of $0$ and $V$ of $e$ satisfy $\exp(U) = V$ diffeomorphically; thus every element near $e$ has a unique small logarithm.

(d) For a Lie group homomorphism $\varphi : G \to H$, $$ \varphi(\exp_G X) = \exp_H(d\varphi_e X). $$

(e) For a matrix group, $\exp(X) = \sum_{k \geq 0} X^k/k!$, the matrix exponential.

(f) For $g \in G$, $\ g\exp(X)g^{-1} = \exp(\operatorname{Ad}_gX)$, with $\operatorname{Ad}_g$ as in §15.

(g) Baker–Campbell–Hausdorff: for sufficiently small $X, Y$ there is $Z \in \mathrm{G}$ with $\exp(X)\exp(Y) = \exp(Z)$ and $$ Z = X + Y + \frac{1}{2}[X,Y] + \frac{1}{12}[X,[X,Y]] - \frac{1}{12}[Y,[X,Y]] + \cdots, $$ a convergent series of iterated brackets. Thus the group law near $e$ is determined by the bracket, and $\exp(X)\exp(Y) = \exp(X+Y)$ whenever $[X,Y] = 0$.

(h) If $G$ is connected, $\exp(\mathrm{G})$ generates $G$.

10. Limits of the Exponential Map

The map $\exp$ is a local diffeomorphism at $0$, but in general it is neither injective nor surjective.

Not injective. For $G = S^1 = U(1)$ and $\mathrm{G} = i\mathbb{R}$, one has $\exp(2\pi i) = 1 = \exp(0)$.
Not surjective. For $G = SL_2(\mathbb{R})$, the matrix $$ A = \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix} $$ lies in $SL_2(\mathbb{R})$ but equals no $\exp(X)$ with $X$ real. A real matrix is the exponential of a real matrix exactly when it is invertible and each Jordan block belonging to a negative real eigenvalue occurs an even number of times; the eigenvalue $-1$ of $A$ has a single Jordan block of size $2$, so $A$ has no real logarithm.
Surjective cases. The map $\exp$ is surjective when $G$ is connected and compact (for example $U(n)$, $SU(n)$, $SO(n)$, $Sp(n)$, $T^n$), when $G = GL_n(\mathbb{C})$ (every invertible complex matrix has a complex logarithm), and when $G$ is connected and nilpotent; in the last case, if $G$ is simply connected then $\exp$ is a diffeomorphism, the standard example being the Heisenberg group $H_3(\mathbb{R})$. If $G$ is connected and abelian, then $\exp : \mathrm{G} \to G$ is a surjective homomorphism with discrete kernel, so $G \cong \mathrm{G}/\Gamma$ for a discrete subgroup $\Gamma$, as for $\mathbb{R}/\mathbb{Z} \cong S^1$ and $\mathbb{R}^n/\mathbb{Z}^n \cong T^n$.
Exponential coordinates. Since $\exp$ is a local diffeomorphism at $0$, the pair $(U,\exp^{-1})$ is a chart near $e$, called a system of exponential coordinates.

Part IV: Homomorphisms and the Lie Correspondence

11. Lie Group Homomorphisms and Their Differentials

A Lie group homomorphism is a map $\varphi : G \to H$ that is both a group homomorphism and smooth; it is an isomorphism if it is bijective with smooth inverse. Every continuous group homomorphism between Lie groups is automatically smooth.

Its derivative at the identity,

$$ d\varphi_e : \mathrm{G} \to \mathrm{H}, $$

is a Lie algebra homomorphism:

$$ d\varphi_e([X,Y]) = [d\varphi_eX, d\varphi_eY]. $$

The assignment $G \mapsto \mathrm{G}$ is functorial: $d(\psi \circ \varphi)_e = d\psi_e \circ d\varphi_e$ and $d(\operatorname{id}_G)_e = \operatorname{id}_{\mathrm{G}}$. For example, $\det$ differentiates to $\operatorname{tr}$, and the covering $SU(2) \to SO(3)$ has differential an isomorphism $\mathrm{SU}(2) \to \mathrm{SO}(3)$.

Integration theorem. If $G$ is simply connected and $\phi : \mathrm{G} \to \mathrm{H}$ is a Lie algebra homomorphism, then there is a unique Lie group homomorphism $\varphi : G \to H$ with $d\varphi_e = \phi$. (This is sometimes called Lie's second theorem; the numbering of Lie's theorems varies among sources.)

For a homomorphism $\varphi : G \to H$: $\operatorname{Lie}(\ker\varphi) = \ker d\varphi_e$; the image is an immersed Lie subgroup with $\operatorname{Lie}(\varphi(G)) = \operatorname{im}d\varphi_e$; if $G, H$ are connected of the same dimension and $d\varphi_e$ is an isomorphism, then $\varphi$ is a covering map, and an isomorphism if $G$ is simply connected; if $\varphi$ is surjective with $H$ connected, then $G/\ker\varphi \cong H$.

12. Lie Subgroups and Lie Subalgebras

A Lie subalgebra of $\mathrm{G}$ is a subspace $\mathrm{H}$ with $[\mathrm{H},\mathrm{H}] \subseteq \mathrm{H}$. An embedded Lie subgroup of $G$ is a subgroup that is an embedded submanifold; the group operations then restrict to smooth maps, and its Lie algebra is $\mathrm{H} = T_eH \subseteq T_eG$. An immersed Lie subgroup (analytic subgroup) is a subgroup with a smooth structure making the inclusion an injective immersion; it need not be embedded.

Cartan's closed subgroup theorem. Every closed subgroup of a Lie group is an embedded Lie subgroup. In particular the classical matrix groups are Lie groups, being closed in $GL_n(\mathbb{K})$.

Lie correspondence. Let $G$ have Lie algebra $\mathrm{G}$. Every Lie subalgebra $\mathrm{H} \subseteq \mathrm{G}$ is the Lie algebra of a unique connected immersed Lie subgroup $H \subseteq G$, and conversely the Lie algebra of such an $H$ is a subalgebra; these assignments are inverse, giving a bijection

$$ \{\text{Lie subalgebras of } \mathrm{G}\} \longleftrightarrow \{\text{connected immersed Lie subgroups of } G\}. $$

Under it $\dim H = \dim\mathrm{H}$, and inclusions correspond to inclusions. For subgroups, being embedded is equivalent to being closed, so an immersed subgroup is embedded exactly when it is closed.

Example. In the torus $T^2 = \mathbb{R}^2/\mathbb{Z}^2$, the subgroup $H = \{(e^{2\pi it}, e^{2\pi i\alpha t}) : t \in \mathbb{R}\}$ with $\alpha$ irrational is a connected immersed subgroup of dimension $1$, with Lie algebra the line $\mathbb{R}\cdot(1,\alpha)$. It is dense, hence neither closed nor embedded, and its closure is all of $T^2$; thus a one-dimensional subalgebra can generate a dense subgroup.

13. The Lie Correspondence for Simply Connected Groups

Lie's third theorem. Every finite-dimensional real Lie algebra $\mathrm{G}$ is the Lie algebra of some simply connected Lie group $G$.

Equivalence theorem. The assignment $G \mapsto \operatorname{Lie}(G)$ is an equivalence of categories between simply connected Lie groups and finite-dimensional real Lie algebras: essentially surjective by Lie's third theorem, full by the integration theorem, and faithful because a homomorphism is determined by its differential on a connected domain. Hence a connected simply connected Lie group is determined up to isomorphism by its Lie algebra.

Universal covering group. Every connected Lie group $G$ has a simply connected universal covering group $\tilde{G}$ with a covering homomorphism $\pi : \tilde{G} \to G$ and $\operatorname{Lie}(\tilde{G}) \cong \mathrm{G}$. The kernel $\pi_1(G)$ is a discrete central subgroup, and $G \cong \tilde{G}/\pi_1(G)$. Examples: $\mathbb{R} \to S^1$ with $\pi_1(S^1) = \mathbb{Z}$; $SU(2) \to SO(3)$ with $\pi_1(SO(3)) = \mathbb{Z}/2$; and $\operatorname{Spin}(n) \to SO(n)$ for $n \geq 3$, which is simply connected.

14. Connected, Simply Connected, and Compact Examples

Group	Connected	Simply connected	Compact
$\mathbb{R}^n$	yes	yes	no
$T^n$, $n \geq 1$	yes	no	yes
$S^1 = U(1)$	yes	no	yes
$S^3 = SU(2) = Sp(1)$	yes	yes	yes
$GL_n(\mathbb{R})$, $n \geq 1$	no	no	no
$GL_n(\mathbb{C})$, $n \geq 1$	yes	no	no
$SL_n(\mathbb{R})$, $n \geq 2$	yes	no	no
$SL_n(\mathbb{C})$, $n \geq 2$	yes	yes	no
$O(n)$	no	no	yes
$SO(n)$, $n \geq 2$	yes	no	yes
$U(n)$	yes	no	yes
$SU(n)$	yes	yes	yes
$Sp(n)$	yes	yes	yes

A compact connected Lie group has finite fundamental group exactly when its universal cover is compact; the circle shows that compactness alone does not force this, since $\pi_1(S^1) = \mathbb{Z}$. Also $SO(3) \cong \mathbb{RP}^3$, compact and connected but not simply connected.

Part V: The Adjoint Representation and Structure

15. The Adjoint Representation

For $g \in G$, conjugation $\Psi_g(h) = ghg^{-1}$ fixes $e$, so its derivative at $e$ is an invertible linear map

$$ \operatorname{Ad}_g = d(\Psi_g)_e \in GL(\mathrm{G}). $$

The map $\operatorname{Ad} : G \to GL(\mathrm{G})$, $g \mapsto \operatorname{Ad}_g$, is a smooth Lie group homomorphism, the adjoint representation of $G$; each $\operatorname{Ad}_g$ preserves the bracket, $\operatorname{Ad}_g[X,Y] = [\operatorname{Ad}_gX, \operatorname{Ad}_gY]$, and for a matrix group $\operatorname{Ad}_gX = gXg^{-1}$. Its derivative at the identity is the adjoint representation of the Lie algebra

$$ \operatorname{ad} = d\operatorname{Ad}_e : \mathrm{G} \to \mathrm{GL}(\mathrm{G}), \qquad \operatorname{ad}_XY = [X,Y]. $$

The Jacobi identity is exactly the statement that $\operatorname{ad}$ is a Lie algebra homomorphism:

$$ \operatorname{ad}_{[X,Y]} = [\operatorname{ad}_X, \operatorname{ad}_Y]. $$

For $g \in G$ and $X \in \mathrm{G}$ one has

$$ g\exp(X)g^{-1} = \exp(\operatorname{Ad}_gX), \qquad \operatorname{Ad}_{\exp X} = e^{\operatorname{ad}_X} = \sum_{k \geq 0} \frac{\operatorname{ad}_X^k}{k!}. $$

If $G$ is connected, then $\ker\operatorname{Ad} = Z(G)$, so the adjoint group satisfies $\operatorname{Ad}(G) \cong G/Z(G)$, with Lie algebra $\operatorname{ad}(\mathrm{G}) \cong \mathrm{G}/Z(\mathrm{G})$. Connectedness is needed here: for $O(2)$, whose Lie algebra is abelian of dimension $1$, the rotations in $SO(2)$ act trivially on $\mathrm{SO}(2)$ while the reflections act by $-1$, so $\ker\operatorname{Ad} = SO(2) \neq Z(O(2)) = \{\pm I\}$.

16. The Derived Subgroup; Solvable and Nilpotent Groups

The commutator of $g, h \in G$ is $[g,h] = ghg^{-1}h^{-1}$, and the derived subgroup $[G,G]$ is generated by all commutators. It is normal, and $G/[G,G]$ is the largest abelian quotient of $G$. Define the derived series by

$$ G^{(0)} = G, \qquad G^{(k+1)} = [G^{(k)},G^{(k)}], $$

and the lower central series by

$$ G_0 = G, \qquad G_{k+1} = [G, G_k]. $$

The group $G$ is solvable if $G^{(k)} = \{e\}$ for some $k$, and nilpotent if $G_k = \{e\}$ for some $k$. Every nilpotent group is solvable, but not conversely. These definitions parallel those for Lie algebras, and for connected $G$ the two theories agree: $\operatorname{Lie}([G,G]) = [\mathrm{G},\mathrm{G}]$, and $G$ is solvable (respectively nilpotent) if and only if $\mathrm{G}$ is.

Examples: abelian groups are nilpotent of step $1$; the Heisenberg group $H_3(\mathbb{R})$ is nilpotent of step $2$, with derived subgroup equal to its center; the invertible upper triangular matrices form a solvable group, whose unipotent subgroup (all diagonal entries $1$) is nilpotent; and for $n \geq 2$, $[\mathrm{GL}_n,\mathrm{GL}_n] = \mathrm{SL}_n$ and $[\mathrm{SL}_n,\mathrm{SL}_n] = \mathrm{SL}_n \neq 0$, so $\mathrm{GL}_n$, hence $GL_n$, is neither solvable nor nilpotent. The underlying Lie algebra facts are Engel's theorem, that a finite-dimensional $\mathrm{G}$ is nilpotent if and only if every $\operatorname{ad}_X$ is nilpotent, and Lie's theorem, that a solvable subalgebra of $\mathrm{GL}(V)$ over an algebraically closed field of characteristic $0$ is simultaneously triangularizable.

17. Quotients by Normal Closed Subgroups

Let $N \subseteq G$ be a closed normal subgroup. Then $G/N$ carries a unique smooth manifold structure making the quotient map $\pi : G \to G/N$ a smooth surjective submersion; with it, $G/N$ is a Lie group and

$$ \operatorname{Lie}(G/N) \cong \mathrm{G}/\mathrm{N}, \qquad \mathrm{N} = \operatorname{Lie}(N). $$

The map $\pi$ is a principal $N$-bundle, and $G/N$ is connected, respectively compact, when $G$ is. More generally, for any closed subgroup $H$ the homogeneous space $G/H$ has a unique smooth structure making $\pi : G \to G/H$ a submersion, with $\dim G/H = \dim G - \dim H$; when $H$ is normal this is a Lie group. Closedness is essential: the quotient by a dense subgroup such as the irrational line of §12 is not a manifold.

First isomorphism theorem. For a Lie group homomorphism $\varphi : G \to H$, the kernel is a closed normal subgroup, the image is an immersed Lie subgroup, and $\varphi$ induces an isomorphism of Lie groups $G/\ker\varphi \cong \varphi(G)$.

Examples: $\mathbb{R}/\mathbb{Z} \cong S^1$, $\mathbb{R}^n/\mathbb{Z}^n \cong T^n$, $SU(2)/\{\pm1\} \cong SO(3) \cong \mathbb{RP}^3$, $U(n)/SU(n) \cong U(1)$, $GL_n(\mathbb{R})/SL_n(\mathbb{R}) \cong \mathbb{R}^\times$, and $G/[G,G]$ is the maximal abelian quotient, equal to $\mathbb{R}^2$ for the Heisenberg group.

Part VI: Summary

19. Lie Groups in the Wider Corpus

The group of units of the biquaternion algebra is $\mathbb{B}^\times \cong GL_2(\mathbb{C})$, a real Lie group of dimension $8$ whose Lie algebra is $\mathrm{GL}_2(\mathbb{C})$ with the commutator bracket, in agreement with Lie Algebras: Categorization; the biquaternion exponential is treated in a separate article of the biquaternion series.
The unit quaternions form $S^3 \cong SU(2) \cong Sp(1)$, the double cover of $SO(3)$, and $SL_2(\mathbb{C})$ is the double cover of the identity component $SO^+(1,3)$ of the Lorentz group.
The spin groups $\operatorname{Spin}(n)$ for $n \geq 3$, the universal covers of $SO(n)$, link this article to the Clifford algebra and spinor articles of the series.

Summary

A Lie group is a group and a smooth manifold with smooth group operations; the classical matrix groups $GL_n$, $SL_n$, $O(n)$, $SO(n)$, $U(n)$, $SU(n)$, $Sp(n)$, and $Sp(2n,\mathbb{R})$ are the basic examples, with real dimensions as in §3.
Its Lie algebra $\mathrm{G} = T_eG$ is the space of left-invariant vector fields with the commutator bracket; for matrix groups $[X,Y] = XY - YX$.
The exponential map is a local diffeomorphism at $0$, natural in homomorphisms, and surjective for connected compact groups, connected nilpotent groups, and $GL_n(\mathbb{C})$, but not in general.
Lie group homomorphisms differentiate to Lie algebra homomorphisms, and the differential functor is an equivalence between simply connected Lie groups and finite-dimensional real Lie algebras; Lie subalgebras correspond to connected immersed Lie subgroups, closed subgroups being embedded.
The adjoint representation $\operatorname{Ad}$ differentiates to $\operatorname{ad}$, with $\ker\operatorname{Ad} = Z(G)$ for connected $G$; for such $G$, solvability and nilpotency of $G$ and $\mathrm{G}$ coincide, and quotients by closed normal subgroups have Lie algebra $\mathrm{G}/\mathrm{N}$.