Compact connected Lie groups are classified by root data

The center $Z(G)$ is closed, being the intersection of the closed centralizers of individual elements. By [L7] its identity component $Z(G{)}^0$ is a compact connected abelian Lie group. Its Lie algebra is $\mathfrak{z}$: differentiation gives one inclusion; conversely if $X$ is central in $\mathfrak{g}$, then $\mathrm{Ad}(\exp (tX))=I$ by [L7], and naturality shows these exponentials commute with every exponential of $G$. Exponential neighborhoods generate connected $G$ (the generated subgroup is open and so are all its cosets), so $\exp (tX)\in Z(G{)}^0$. Hence [L4] gives $Z(G{)}^0=\exp (\mathfrak{z})$. Only this identity component is asserted to be a torus; the whole center can be disconnected. For the adjoint module, the invariant inner product of [L8] gives every invariant subspace an invariant orthogonal complement; finite-dimensional induction makes the adjoint module completely reducible. Thus [L8] yields $\mathfrak{g}=\mathfrak{z}\oplus \mathfrak{s},\mathfrak{s}=[\mathfrak{g},\mathfrak{g}]\text{ semisimple}\mathrm{.}$ The product $Z(G{)}^{0}T$ is a compact connected abelian subgroup, hence a torus by [L4]; maximality of $T$ gives $Z(G{)}^0\subseteq T$ and therefore $\mathfrak{z}\subseteq \mathfrak{t}$. Decomposing each $H\in \mathfrak{t}$ under the displayed direct sum gives $H=Z+S$ with $Z\in \mathfrak{z}\subseteq \mathfrak{t}$ and hence $S=H-Z\in \mathfrak{t}\cap \mathfrak{s}$. Consequently $\mathfrak{t}=\mathfrak{z}\oplus {\mathfrak{t}}_s$ for ${\mathfrak{t}}_s=\mathfrak{t}\cap \mathfrak{s}$.

We establish the finite-kernel duality used below. If $M\subseteq X^{\ast }(U)$ is a full-rank finite-index sublattice of a torus character lattice, Smith form [L4] supplies bases in which $M=d_1\mathbb{Z}\oplus \cdots \oplus d_n\mathbb{Z}\subseteq {\mathbb{Z}}^n$, with $d_i>0$. The torus coordinate characters of this basis identify U with $(S^1{)}^n$ by the exponential-lattice description in [L4]. Its common kernel $C_M$ is ${\prod }_i{\mu }_{d_i}$ and is finite. A character vanishes on $C_M$ exactly when each coordinate exponent is divisible by $d_i$, so the pullback character lattice of $U\mathrm{/}{C}_M$ is exactly M. For rank zero both groups are trivial. This also shows that characters recover the full exponential lattice: in lattice coordinates the condition that all differentiated characters take values in $2\pi i\mathbb{Z}$ forces each coordinate to be integral.

For an abstract root datum, put $V={\mathrm{span}}_{\mathbb{R}}\mathrm{\Phi }\subseteq X_{\mathbb{R}}$. The paired reflections $s_{\alpha }$ preserve X, the root set, the dual lattice and the root–coroot correspondence by the defining axioms. On V the group they generate is finite, since it acts faithfully on its spanning finite root set. Average any positive inner product on V over that finite group. Each $s_{\alpha }{\mathrm{\mid }}_V$ is now an orthogonal reflection negating $\alpha$ and fixing $\ker ⟨\cdot ,{\alpha }^{\vee }⟩\cap V$, so $⟨v,{\alpha }^{\vee }⟩=2(v,\alpha )\mathrm{/}(\alpha ,\alpha )$ for $v\in V$. In particular $V\cap {\bigcap }_{\alpha }\ker ⟨\cdot ,{\alpha }^{\vee }⟩=0$. The full reflection group W on $X_{\mathbb{R}}$ is also finite: an element fixing all roots fixes their paired coroots, while $wx-x\in V$ and all its coroot pairings vanish, hence $wx=x$. Thus W injects into the finite permutation group of the roots. This argument includes empty roots with $V=0$ and $W=1$.

Average a rational positive inner product on $X_{\mathbb{Q}}$ over W. The common fixed subspace Z is the common annihilator of the coroots, and $X_{\mathbb{R}}=V\oplus Z$ orthogonally: orthogonal complements of the reflecting hyperplanes are precisely the root lines. The projection $x_s$ to V is rational, for it equals $x-\mathrm{\mid }W{\mathrm{\mid }}^{-1}{\sum }_{w\in W}wx$, and has the same coroot pairings as x. On V, the roots span, are finite and reflection-invariant, and have integral Cartan numbers by the datum. They are reduced in the ordinary real sense: if $c\alpha$ is another root, $2c$ and $2\mathrm{/}c$ are integers, so $\mathrm{\mid }c\mathrm{\mid }$ is $1\mathrm{/}2,1$ or 2; the first and last are excluded by the integer-multiple reducedness axiom, applied in the appropriate direction. Thus they form a reduced crystallographic root system with the prescribed coroot functionals.

Integrate $\mathfrak{s}\hookrightarrow \mathfrak{g}$ by [L3] from its simply connected compact group $G_{\mathrm{d}\mathrm{e}\mathrm{r}}^{\mathrm{s}\mathrm{c}}$ of [L2], writing the map as j. Multiplication $\mu (z,x)=zj(x)$ from $Z(G{)}^0\times G_{\mathrm{d}\mathrm{e}\mathrm{r}}^{\mathrm{s}\mathrm{c}}$ is a homomorphism because the first factor is central. Its differential is the direct-sum isomorphism established in step 1.1, hence it is a local diffeomorphism by exponential charts [L7]. Its image is open, so equals connected G. Its kernel is closed and discrete, hence finite in the compact domain. For a kernel element, conjugation by the connected domain is a continuous map into that discrete kernel and thus constant, proving centrality. Translates of a sufficiently small identity chart by kernel elements give disjoint sheets over the same target neighborhood, proving the covering assertion.

To prove uniqueness, let an isomorphism of root data be given contravariantly as $F:X^{\ast }(T_2)\to X^{\ast }(T_1)$. Its real dual, in differentiated-character coordinates, defines $L:{\mathfrak{t}}_1\to {\mathfrak{t}}_2$ by $d\chi (LH)=d(F\chi )(H)$. It maps exponential lattices bijectively by step 1.2. It maps the central annihilators of all roots to one another, and maps compact coroot tangents to one another by compatibility with the perfect pairing. Choose simple roots in the first system and the corresponding base in the second. In each semisimple complexification choose their triples with $\sigma (e_i)=-f_i$, $\sigma (h_i)=-h_i$ as in [L5]. The Cartan matrices agree, so [L6] gives a complex isomorphism sending the first triples to the second. It commutes with compact conjugation on every generator and hence on the generated algebra, so restricts to a real semisimple isomorphism. On the Cartan it agrees with L because the $i{h}_i$ span its semisimple part. Extend by L on the central summand from step 1.1; this is a real Lie-algebra isomorphism $\mathrm{\ell }:{\mathfrak{g}}_1\to {\mathfrak{g}}_2$ restricting to L. No unproved lifting of a diagram automorphism is used.

For the abstract datum put $X_V=X\cap V$. It is saturated: $nx\in V$ implies $x\in V$ for $n\mathrm{\ne }0$. Smith form [L4] then shows $Y=X\mathrm{/}{X}_V$ is free, with rank $\mathrm{rank}X-\dim V$. Projection from step 2.1 gives $x_s\in P$, since its coroot pairings are those of x and are integral. The map $j_X:X\to P\oplus Y$, $x\mapsto (x_s,x+X_V)$, is injective: a kernel element is in $X_V$ and has zero projection to V. The ranks are equal, so Smith form makes its image finite index. By [L2] take $G_{sc}$ with a maximal torus $T_{sc}$ whose character lattice is P and whose roots and coroots are identified with those in V. Let $A=\mathrm{Hom}(Y,S^1)$; a basis of Y makes this a torus with character lattice Y by [L4]. Thus $j_X$ embeds X in $X^{\ast }(T_{sc}\times A)$ and sends a root $\alpha$ to $(\alpha ,0)$.

Write $U_i={\mathfrak{z}}_i\times G_{\mathrm{d}\mathrm{e}\mathrm{r},i}^{\mathrm{s}\mathrm{c}}$, with additive vector-group first factor. The natural map $p_i:U_i\to G_i$ is the composite of $\exp :{\mathfrak{z}}_i\to Z(G_i{)}^0$ with the covering in step 2.2, and is a surjective local-diffeomorphism homomorphism. Its domain is simply connected: loops in the vector factor contract by scalar multiplication and loops in the second factor contract by its defining simple connectivity. Choose a maximal torus in the semisimple factor so that the Lie algebra of ${\widehat{T}}_i={\mathfrak{z}}_i\times T_{sc,i}$ maps to ${\mathfrak{t}}_i$. This is possible since the inverse image of ${\mathfrak{t}}_i$ in ${\mathfrak{s}}_i$ is maximal abelian, and its exponential closure is a maximal torus. Naturality and surjectivity of torus exponentials give $p_i({\widehat{T}}_i)=T_i$. Conversely if $p_i(u)\in T_i$, its adjoint action on ${\mathfrak{t}}_i$ is trivial, so the semisimple component of u centralizes $T_{sc,i}$ by naturality and lies in it by [L5]. Hence $p_i^{-1}(T_i)={\widehat{T}}_i$. If ${\mathrm{\Lambda }}_i=\ker ({\exp }_{T_i})$, exponential surjectivity on ${\widehat{T}}_i$ therefore gives $\ker p_i={\exp }_{U_i}((d{p}_i{)}^{-1}{\mathrm{\Lambda }}_i)$.

In $\tilde{T}=T_{sc}\times A$ let C be the common kernel of all characters in $j_X(X)$. Step 1.2 makes C finite. Each element of C has all its root characters equal to one, so it acts trivially on every root space and on the Cartan algebra of $G_{sc}$ by [L1]. Its adjoint action is therefore the identity; naturality and connected generation by exponentials show it is central in $G_{sc}\times A$. Consequently $G_X=(G_{sc}\times A)\mathrm{/}C$ is a compact connected Lie group by [L7]. Its torus $\tilde{T}\mathrm{/}C$ is maximal: its Lie algebra is maximal abelian, and a containing torus with the same Lie algebra is equal by exponential charts and connectedness. The quotient has the same Lie algebra, its characters on this torus are exactly $j_X(X)$ by step 1.2, and its roots are $j_X(\mathrm{\Phi })$. The coroot paired with $j_X(x)$ evaluates as $⟨x_s,{\alpha }^{\vee }⟩=⟨x,{\alpha }^{\vee }⟩$, so perfect duality [L4] identifies its cocharacters and paired coroots with the prescribed $X^{\vee },{\mathrm{\Phi }}^{\vee }$. This realizes the entire datum, including the torus case V=0.

Integrate $\mathrm{\ell }$ from step 2.3 by [L3] to an isomorphism $\tilde{\mathrm{\ell }}:U_1\to U_2$: integrate its inverse as well, and uniqueness makes both compositions the identity. Its restriction on torus Lie algebras is L, which carries ${\mathrm{\Lambda }}_1$ to ${\mathrm{\Lambda }}_2$. Naturality and the kernel formula of step 3.2 give $\tilde{\mathrm{\ell }}(\ker p_1)=\ker p_2$. It consequently descends to a group isomorphism $G_1\to G_2$. The descended maps in both directions are smooth locally by the covering charts, so this is a Lie-group isomorphism. Conversely a group isomorphism takes maximal tori to maximal tori; [L5] conjugates the image to any chosen target torus. Differentiation, conjugation of root spaces and the compact coroot construction preserve the paired root datum. Thus root-data isomorphism is equivalent to group isomorphism. Together with realization in step 4.1 and the finite cover in step 2.2 this proves the statement.