Lie Groups, Lie Algebras, and Exponential and Adjoint Maps

For a finite-dimensional Lie group $G$, the tangent space $\mathfrak g=T_eG$ at the identity becomes a Lie algebra. Its vectors are the initial velocities of one-parameter subgroups, its bracket is the first noncommutative correction to local multiplication, and the exponential map sends $X\in\mathfrak g$ to the finite transformation reached at unit parameter. Conjugation by $g\in G$ differentiates to $\operatorname{Ad}_g$ on $\mathfrak g$, while differentiating $\operatorname{Ad}$ gives $\operatorname{ad}_X(Y)=[X,Y]$.

This data determines the group law locally near the identity and controls smooth homomorphisms from a connected group once they exist. It does not recover disconnected components, periods, covering kernels, or which infinitesimal actions descend to a chosen global group. The exponential map is therefore a local bridge, not generally a global coordinate system or a global homomorphism from the additive Lie algebra.

Required background. Groups, Actions, Quotients, and Covers supplies actions, covering homomorphisms, kernels, and descent.

Lie groups and their tangent algebras

The page treats finite-dimensional real Lie groups. Standard complex matrix groups may also be complex Lie groups, but here their tangent algebras are real vector spaces unless stated otherwise. We use only the differential geometry needed to define tangent maps, invariant vector fields, and local coordinates.

The page does not classify Lie groups, representations, roots and weights, Lorentz multiplets, bundles, moment maps, infinite-dimensional symmetry groups, Lie superalgebras, or central extensions. It also does not construct Noether currents or quantum charge operators.

From a smooth group to its tangent algebra

A finite-dimensional Lie group is a finite-dimensional smooth manifold $G$ whose multiplication and inversion maps,

$$\begin{aligned} m:G\times G&\longrightarrow G, & m(g,h)&=gh, \\ \iota:G&\longrightarrow G, & \iota(g)&=g^{-1}, \end{aligned}$$

are smooth. Thus $U(1)$, $SO(n)$, $U(n)$, $SU(n)$, and $GL(n,\mathbb R)$ are Lie groups, whereas the group axioms alone do not supply the smooth structure. These definitions and the invariant-field construction below are developed in Etingof 2020, §§2.1, 5.4, and 8.1–8.4, PDF.

Let $e$ be the identity and $L_g(h)=gh$ be left translation. Every $X\in T_eG$ extends uniquely to a left-invariant vector field

$$X^L(g)=(\mathrm dL_g)_eX.$$

It is left-invariant because $(\mathrm dL_h)_gX^L(g)=X^L(hg)$, and uniqueness follows by evaluating any left-invariant field at $e$. Diffeomorphisms preserve vector-field brackets, so $[X^L,Y^L]$ is again left-invariant. This defines

$$\mathfrak g:=T_eG, \qquad [X,Y]_{\mathfrak g}:=[X^L,Y^L]_e.$$

Bilinearity, antisymmetry, and the Jacobi identity are inherited from the vector-field bracket. The smooth group has therefore produced an algebraic object, but the bracket contains differential information from multiplication near $e$; it is not an extra arbitrary operation placed on $T_eG$.

For a matrix Lie group, tangent vectors are matrices and

$$X^L(g)=gX, \qquad [X,Y]_{\mathfrak g}=XY-YX.$$

Differentiating the defining matrix equations gives, for example,

$$\begin{aligned} \mathfrak u(n) &=\{X\in M_n(\mathbb C):X^\dagger=-X\}, \\ \mathfrak{su}(n) &=\{X\in\mathfrak u(n):\operatorname{tr}X=0\}, \\ \mathfrak{so}(n) &=\{X\in M_n(\mathbb R):X^{\mathsf T}=-X\}. \end{aligned}$$

In particular, elements of $\mathfrak u(n)$ and $\mathfrak{su}(n)$ are intrinsically anti-Hermitian matrices. Hermitian physics generators are a basis convention related to these tangent matrices below.

Active actions carry a sign choice

The preceding bracket used left-invariant vector fields on $G$. A left action of $G$ on another manifold $M$ produces a related but sign-sensitive construction. With the direct convention

$$X_M(x) = \left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0} \exp(tX)\cdot x,$$

the map $X\mapsto X_M$ is an anti-homomorphism:

$$[X_M,Y_M]=-[X,Y]_M.$$

The left action of $G$ on itself is the quickest check: its field at $g$ is $(\mathrm dR_g)_eX$, which is right-invariant, and right-invariant fields carry the opposite bracket sign.

The groups-and-actions page fixed the induced active action on scalar-valued functions as

$$(g\cdot f)(x)=f(g^{-1}\cdot x).$$

Differentiating this action gives

$$\begin{aligned} (D_Xf)(x) &= \left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0} f\bigl(\exp(-tX)\cdot x\bigr) \\ &=-X_Mf(x), \qquad [D_X,D_Y]=D_{[X,Y]}. \end{aligned}$$

The inverse in the finite action is exactly what turns the induced function-space generators into a Lie-algebra homomorphism. One may instead define fundamental vector fields with $\exp(-tX)$ from the outset. Either choice is valid, but combining formulas from the two choices silently reverses commutator signs.

One-parameter subgroups and the exponential map

A one-parameter subgroup is a smooth homomorphism $\gamma:(\mathbb R,+)\to G$. For every $X\in\mathfrak g$, the left-invariant vector field $X^L$ has a unique integral curve through $e$, and left invariance plus uniqueness gives

$$\gamma_X(s+t)=\gamma_X(s)\gamma_X(t), \qquad \dot\gamma_X(0)=X.$$

Conversely, the initial velocity of a one-parameter subgroup determines it uniquely. The exponential map is

$$\exp_G:\mathfrak g\longrightarrow G, \qquad \exp_G(X)=\gamma_X(1),$$

so that

$$\gamma_X(t)=\exp_G(tX), \qquad \exp_G((s+t)X)=\exp_G(sX)\exp_G(tX).$$

For a matrix Lie group this is the usual convergent matrix series

$$\exp X = \sum_{k=0}^{\infty}\frac{X^k}{k!}.$$

Since $\left.\frac{\mathrm d}{\mathrm dt}\right|_0\exp(tX)=X$, one has

$$(\mathrm d\exp_G)_0=\operatorname{id}_{\mathfrak g}.$$

The inverse-function theorem then supplies neighborhoods $0\in V\subset\mathfrak g$ and $e\in U\subset G$ for which $\exp:V\to U$ is a diffeomorphism. These results, including naturality under homomorphisms, are proved in Etingof 2020, §7.1, Proposition 7.1 through Theorem 7.5, PDF and cross-checked in Kirillov 2008, §3.1, Theorem 3.7 and Remark 3.8, PDF.

An identity neighborhood generates the identity component $G_0$: the subgroup generated by an open identity neighborhood is open, and its cosets show that it is also closed in $G_0$. Thus products of sufficiently small exponentials reach every element of a connected group. This does not say that every element is one exponential.

Two small examples mark the global limits.

First,

$$\exp_{U(1)}(i\theta)=e^{i\theta}$$

is surjective but not injective; its kernel is $2\pi i\mathbb Z$. Second, the exponential map of $SL(2,\mathbb R)$ is not surjective. Consider

$$A= \begin{pmatrix} -2&0\\ 0&-\tfrac12 \end{pmatrix}, \qquad \det A=1, \qquad \operatorname{tr}A=-\frac52.$$

If $A=e^X$ for a real trace-zero $2\times2$ matrix $X$, Cayley–Hamilton would give $X^2=-\det(X)\mathbf1$. When $\det X\leq0$, $\operatorname{tr}(e^X)=2\cosh\sqrt{-\det X}\geq2$; when $\det X>0$, $\operatorname{tr}(e^X)=2\cos\sqrt{\det X}\geq-2$. Both contradict $\operatorname{tr}A=-5/2$.

Nor is $\exp$ generally a homomorphism from the additive vector space $\mathfrak g$. If $[X,Y]=0$, then

$$\exp(X+Y)=\exp X\,\exp Y.$$

Without commutation there are bracket corrections. More sharply, if $\exp(t(X+Y))=\exp(tX)\exp(tY)$ for every sufficiently small $t$, the local formula in the next section forces $[X,Y]=0$. Accidental equality at one isolated parameter value is a weaker statement because the exponential can have periods.

The bracket is visible in local multiplication

On a sufficiently small neighborhood of $0$, where one logarithm branch is fixed, the Baker–Campbell–Hausdorff formula begins

$$\begin{aligned} \log\bigl(\exp X\,\exp Y\bigr) ={}&X+Y+\frac12[X,Y] \\ &+\frac1{12}[X,[X,Y]] +\frac1{12}[Y,[Y,X]] +\cdots. \end{aligned}$$

It reconstructs the local group product from iterated brackets. The series may also be treated formally, but it must not be used as a global logarithm identity. Its local convergence and reconstruction role are stated in Kirillov 2008, §3.7, Theorem 3.37 and Corollary 3.38, PDF; the universal Lie-polynomial expansion is treated in Etingof 2020, §14.3, PDF.

Applying the formula successively gives the leading group-commutator test

$$\begin{aligned} &\exp(tX)\exp(sY)\exp(-tX)\exp(-sY) \\ &\qquad = \exp\left( ts[X,Y] +O\bigl(t^2s+ts^2\bigr) \right). \end{aligned}$$

Thus the Lie bracket is the first obstruction to commuting infinitesimal motions. This also supplies a convention check: reversing the order of the group commutator reverses the sign of its leading bracket.

Differentiating and integrating homomorphisms

Let $F:G\to H$ be a smooth group homomorphism and write $\mathrm dF_e:\mathfrak g\to\mathfrak h$ for its differential at the identity. The composite $t\mapsto F(\exp_G(tX))$ is the one-parameter subgroup of $H$ with initial velocity $\mathrm dF_eX$. Uniqueness therefore proves

$$F(\exp_GX)=\exp_H(\mathrm dF_eX).$$

Differentiating the local commutator, or equivalently using invariant fields, gives

$$\mathrm dF_e[X,Y] = [\mathrm dF_eX,\mathrm dF_eY].$$

Thus differentiation is functorial and sends group homomorphisms to Lie-algebra homomorphisms. If $G$ is connected, a smooth homomorphism from $G$ is determined by its differential: the exponential formula fixes it on an identity neighborhood, and that neighborhood generates $G$.

The reverse direction has a global hypothesis. A Lie-algebra homomorphism integrates uniquely from a connected, simply connected source group (Etingof 2020, §9.3, Theorem 9.12, PDF). If the desired source is a quotient of its simply connected cover, the integrated homomorphism descends only when it kills the covering kernel. This is the descent criterion from the groups-and-actions page, now applied after integration.

For example, every real $s$ defines a Lie-algebra homomorphism

$$\varphi_s:\mathfrak u(1)\longrightarrow\mathfrak u(1), \qquad \varphi_s(i\theta)=is\theta.$$

On the universal cover it integrates to $\theta\mapsto e^{is\theta}$. It descends to the declared $2\pi$-periodic group $U(1)$ exactly when $s\in\mathbb Z$. The local algebra allows every real $s$; the global period selects the integer weights.

Adjoint maps transport infinitesimal directions

For $g\in G$, conjugation

$$C_g(h)=ghg^{-1}$$

is a Lie-group automorphism fixing $e$. Its differential defines the uppercase adjoint map

$$\operatorname{Ad}_g:=(\mathrm dC_g)_e :\mathfrak g\longrightarrow\mathfrak g.$$

Because $C_{gh}=C_g\circ C_h$ and differentials preserve brackets,

$$\begin{aligned} \operatorname{Ad}_{gh} &=\operatorname{Ad}_g\operatorname{Ad}_h, \\ \operatorname{Ad}_g[X,Y] &=[\operatorname{Ad}_gX,\operatorname{Ad}_gY]. \end{aligned}$$

For a matrix group these formulas reduce to

$$\operatorname{Ad}_gX=gXg^{-1}.$$

Applying exponential naturality to $C_g$ gives the finite–infinitesimal compatibility relation

$$g\exp(X)g^{-1} = \exp(\operatorname{Ad}_gX).$$

Now differentiate $\operatorname{Ad}_{\exp(tX)}$ at $t=0$. The lowercase adjoint map is

$$\operatorname{ad}_X(Y) := \left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0} \operatorname{Ad}_{\exp(tX)}Y =[X,Y].$$

The same one-parameter-subgroup argument, now in $GL(\mathfrak g)$, yields

$$\operatorname{Ad}_{\exp X} = \exp(\operatorname{ad}_X).$$

Jacobi is equivalently the operator identity

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

These identities are proved in Etingof 2020, §§4.5 and 8.1–8.2, PDF and Kirillov 2008, §§2.6 and 3.3, PDF. The terms must remain typed: $\operatorname{Ad}_g$ is a linear map on $\mathfrak g$, $\operatorname{ad}_X$ is the bracket operator, and $A^\dagger$ is a Hermitian adjoint.

The center gives one more local–global check. Always $Z(G)\subseteq\ker\operatorname{Ad}$. If $G$ is connected, equality holds (Etingof 2020, §9.2, Proposition 9.8, PDF): an element in the kernel induces a conjugation automorphism with identity differential, so connectedness makes that automorphism the identity on all of $G$. Without connectedness, the claim fails; for a nonabelian discrete Lie group the Lie algebra is zero and every element lies in $\ker\operatorname{Ad}$, although not every element is central.

Hermitian and anti-Hermitian generator conventions

The site’s gauge convention uses Hermitian generators $T_a$:

$$[T_a,T_b]=if_{ab}{}^cT_c.$$

The actual tangent matrices in a unitary group are instead

$$t_a=-iT_a.$$

Substitution and a round trip give

$$\begin{aligned} [t_a,t_b] &=(-i)^2[T_a,T_b] =f_{ab}{}^ct_c, \\ \exp(\theta^at_a) &=\exp(-i\theta^aT_a), \\ [T_a,T_b] &=[it_a,it_b] =if_{ab}{}^cT_c. \end{aligned}$$

No total antisymmetry of $f_{ab}{}^c$ is assumed without an invariant inner product and a compatible basis.

For $SU(2)$, take

$$T_a=\frac{\sigma_a}{2}, \qquad t_a=-\frac{i\sigma_a}{2}.$$

Then

$$[t_a,t_b]=\epsilon_{ab}{}^ct_c, \qquad U(\theta,\mathbf n) = \exp(\theta n^at_a) = \exp\left( -\frac{i\theta}{2}\, \mathbf n\cdot\boldsymbol\sigma \right),$$

which is the convention used on the groups-and-actions page.

The groups-and-actions page wrote the circle coordinate as $e^{+i\alpha}$. The Hermitian choice $T=1$ has tangent basis $t=-i$, so the same oriented point is $\exp(st)=e^{-is}$ with $s=-\alpha$. This parameter reversal is not a physical sign change; it is the translation between the declared circle coordinate and the chosen Hermitian basis.

Same local algebra, different global groups

The matrix-group exponential, adjoint action, and BCH calculations can be cross-checked in Hall 2015, Chapters 1–3 and 5. For the recurring $SU(2)$–$SO(3)$ global-form comparison, see Kosmann-Schwarzbach 2022, pp. 89–102.

The covering homomorphism $SU(2)\to SO(3)$ from the groups-and-actions page is locally a diffeomorphism, so its differential is a Lie-algebra isomorphism. Define

$$(J_a)_{bc}=-\epsilon_{abc}.$$

Then

$$[J_a,J_b]=\epsilon_{ab}{}^cJ_c, \qquad [t_a,t_b]=\epsilon_{ab}{}^ct_c,$$

and the differential of the cover sends $t_a\mapsto J_a$. Hence

$$\mathfrak{su}(2)\cong\mathfrak{so}(3).$$

The isomorphism does not see the group kernel $\{\pm\mathbf1\}$. It therefore cannot decide whether a particular $SU(2)$ action descends to $SO(3)$; that still requires testing the kernel on the object being transformed. Likewise, $\mathbb R$ and $U(1)$ have isomorphic one-dimensional abelian Lie algebras but different periods.

These examples answer the principal question sharply: the Lie algebra encodes infinitesimal transformations, local multiplication, and finite motions generated near the identity. It does not by itself encode global form or the allowed representation spectrum.

Controlled QFT bridge: phase and translation of one scalar field

Let

$$G=\mathbb R^{1,3}\times U(1)$$

act actively on a smooth complex scalar field by

$$\bigl((a,e^{i\alpha})\cdot\phi\bigr)(x) = e^{iq\alpha}\phi(x-a), \qquad q\in\mathbb Z.$$

This is one configuration-space action containing an internal continuous symmetry and a spacetime translation. The inverse in $x-a$ is the induced function-action convention fixed on the groups-and-actions page, while $q\in\mathbb Z$ is the global $U(1)$ descent condition proved there.

Differentiate the one-parameter subgroup $(\varepsilon v,e^{i\varepsilon\beta})$. Its infinitesimal variation is

$$\delta_{(v,\beta)}\phi(x) = -v^\mu\partial_\mu\phi(x) +iq\beta\,\phi(x).$$

Exponentiating the two commuting terms recovers the finite action. The minus sign in the translation term comes from pullback by the inverse spacetime transformation; the phase sign comes from the declared $e^{+i\alpha}$ circle coordinate.

For the complex scalar action

$$S[\phi] = \int\mathrm d^4x\, \left( \partial_\mu\phi^*\partial^\mu\phi -m^2\phi^*\phi \right)$$

with the site’s $(+---)$ metric, the constant phase cancels and a translation changes only the integration variable, subject to the usual domain and boundary assumptions. The phase example is Tong 2006, §1.3.4; the translation check is his §1.3.2, with the parameter sign translated to the active-action convention declared here.

The calculation does not identify a Noether current, define a quantum charge operator, establish a commutator with fields, or test anomalies, spontaneous breaking, or gauge redundancy. Those claims need additional physical hypotheses and belong to the linked continuation.

Common pitfalls

Treating $\mathfrak g$ as the whole group. The algebra sees local data at the identity. Periods, disconnected components, cover kernels, and descent remain global questions.

Using the exponential as global coordinates. It is a local diffeomorphism near $0$, but it can be noninjective and nonsurjective. Products of exponentials are not the same as one global logarithm.

Adding exponents as if the algebra were abelian. The identity $\exp(X+Y)=\exp X\exp Y$ is guaranteed when $[X,Y]=0$. Otherwise BCH begins with the correction $\tfrac12[X,Y]$.

Mixing left- and right-invariant signs. The direct infinitesimal field of an active left action is an anti-homomorphism. The inverse in the induced action on functions restores the homomorphism sign.

Calling Hermitian matrices elements of $\mathfrak{su}(n)$. The tangent matrices are anti-Hermitian. Hermitian $T_a$ are related by $t_a=-iT_a$, including the corresponding sign in the finite exponential.

Conflating three adjoints. $\operatorname{Ad}_g$ acts on the Lie algebra, $\operatorname{ad}_X$ brackets with $X$, and $A^\dagger$ is a Hermitian adjoint. They have different domains and definitions.

Turning an infinitesimal action into a conserved charge. Exponentiation only supplies a finite transformation when the global descent conditions hold. Conservation and quantum implementation require dynamics, equations of motion, boundary conditions, domains, and anomaly checks.

Exercises

1. Recover the circle locally and globally

Identify $\mathfrak u(1)$, compute its bracket and exponential map, and use the result to explain both the local equivalence and the global difference between $\mathbb R$ and $U(1)$.

Solution

Differentiating $z^*z=1$ at $z=1$ gives $\mathfrak u(1)=i\mathbb R$. Matrix multiplication is commutative here, so the bracket vanishes. The exponential is $\exp(i\theta)=e^{i\theta}$ and has kernel $2\pi i\mathbb Z$. The map $r\mapsto ir$ identifies the zero-bracket Lie algebras of $\mathbb R$ and $U(1)$, but only $U(1)$ identifies parameters differing by $2\pi$.

2. Detect the first noncommutative correction

Suppose

$$\exp(t(X+Y))=\exp(tX)\exp(tY)$$

for every sufficiently small $t$. Use BCH to show that $[X,Y]=0$. Give two $2\times2$ matrices for which the equality therefore fails locally.

Solution

In the logarithm neighborhood, the left side has logarithm $t(X+Y)$, whereas BCH gives

$$tX+tY+\frac{t^2}{2}[X,Y]+O(t^3)$$

on the right. Equality for all small $t$ forces the coefficient of $t^2$ to vanish. For example,

$$X= \begin{pmatrix}0&1\\0&0\end{pmatrix}, \qquad Y= \begin{pmatrix}0&0\\1&0\end{pmatrix}$$

satisfy $[X,Y]=\operatorname{diag}(1,-1)\neq0$, so the proposed equality fails for sufficiently small nonzero $t$.

3. Round-trip the generator convention

Starting from Hermitian generators with $[T_a,T_b]=if_{ab}{}^cT_c$, set $t_a=-iT_a$. Derive the bracket and finite exponential in the $t_a$ basis, then translate the bracket back.

Solution

Direct substitution gives

$$[t_a,t_b] =(-i)^2[T_a,T_b] =-if_{ab}{}^cT_c =f_{ab}{}^ct_c.$$

The finite element is $\exp(\theta^at_a)=\exp(-i\theta^aT_a)$. Since $T_a=it_a$,

$$[T_a,T_b] =-[t_a,t_b] =-f_{ab}{}^ct_c =if_{ab}{}^cT_c,$$

which recovers the starting convention.

4. Separate the rotation algebra from the rotation group

Verify that $t_a=-i\sigma_a/2$ and $(J_a)_{bc}=-\epsilon_{abc}$ obey the same commutator. Why does this not prove that every $SU(2)$ representation is an $SO(3)$ representation?

Solution

The Pauli identity $[\sigma_a,\sigma_b]=2i\epsilon_{ab}{}^c\sigma_c$ gives

$$[t_a,t_b]=\epsilon_{ab}{}^ct_c.$$

A direct index calculation, or the identity $J_a\mathbf v=\mathbf e_a\times\mathbf v$, gives $[J_a,J_b]=\epsilon_{ab}{}^cJ_c$. Thus $t_a\mapsto J_a$ is a Lie-algebra isomorphism. The group cover still has kernel $\{\pm\mathbf1\}$, however, and an $SU(2)$ representation descends only if $-\mathbf1$ acts trivially. The algebra does not record that test.

5. Transfer the finite action to QFT notation

For

$$\bigl((a,e^{i\alpha})\cdot\phi\bigr)(x) =e^{iq\alpha}\phi(x-a),$$

differentiate the curve $(a,e^{i\alpha})=(\varepsilon v,e^{i\varepsilon\beta})$. Then name two claims that the resulting infinitesimal formula does not prove.

Solution

Expanding to first order,

$$\begin{aligned} e^{iq\varepsilon\beta}\phi(x-\varepsilon v) &= \bigl(1+iq\varepsilon\beta\bigr) \bigl(\phi(x)-\varepsilon v^\mu\partial_\mu\phi(x)\bigr) +O(\varepsilon^2), \end{aligned}$$

so

$$\delta\phi =-v^\mu\partial_\mu\phi+iq\beta\phi.$$

The formula alone does not prove, for example, that a chosen action is invariant, that a conserved current exists with suitable boundary behavior, that a quantum charge operator is well defined, or that the symmetry is nonanomalous. Any two of these distinctions answer the second part.

Where to continue

• Pages that build directly on this material include Compact Lie Groups, Roots, Weights, and Weyl Structure and Lorentz Field Representations and Poincaré Particle Representations. Each states its additional prerequisites and hypotheses.
• The same Lie-group input is hard-required by Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities and Hamiltonian Group Actions and Moment Maps. This page does not teach either destination’s additional geometry.
• For the first physical application to continuous internal and spacetime symmetries, continue to Continuous Symmetries, Generators, and Charges. That physical page has additional prerequisites; this page alone does not satisfy them.

References

• Pavel Etingof, Lie Groups and Lie Algebras I, PDF, MIT OpenCourseWare 18.745, Fall 2020, §§2.1, 4.5, 5.4, 7.1–7.2, 8.1–8.4, 9.2–9.3, and 14.3. These open notes develop Lie groups, invariant fields, the tangent bracket, exponential naturality, $\operatorname{Ad}/\operatorname{ad}$, local multiplication, and the local–global integration boundary.
• Brian C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, second edition, Graduate Texts in Mathematics 222, Springer, 2015, Chapters 1–3 and 5. This cross-checks the matrix-group calculations, exponentials, adjoint action, and BCH formulas.
• Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras, Cambridge University Press, 2008, §§2.6 and 3.1–3.7; an author-posted preliminary text is available as an Open PDF and supplies exact theorem locators. These sections establish invariant fields, the exponential map, commutators, adjoint maps, and the local convergence qualification on BCH.
• Yvette Kosmann-Schwarzbach, Groups and Symmetries: From Finite Groups to Lie Groups, second edition, Springer, 2022, “Lie Groups $SU(2)$ and $SO(3)$,” pp. 89–102. This supports continuity with the chapter’s recurring rotation-cover comparison.
• David Tong (2006), Quantum Field Theory, §§1.3.2 and 1.3.4, “An Example: Translations and the Energy-Momentum Tensor” and “Internal Symmetries,” Cambridge Part III lecture notes, supports the bounded translation and complex-scalar phase example; its translation parameter is translated to the active convention used here, and currents and charges remain at the physical continuation.