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Maximal torus

In the theory of compact Lie groups, a maximal torus is defined as a compact, connected, abelian subgroup T of a compact, connected Lie group [G](/page/G) that is maximal with respect to inclusion among such subgroups, and is isomorphic to a product of circle groups (S^1)^k for some positive integer k.^[1] This dimension k is known as the rank of [G](/page/G), providing a fundamental invariant that classifies the group's structure.^[2] Maximal tori play a central role in the study of Lie groups, as all such tori in [G](/page/G) are conjugate under the action of [G](/page/G), meaning any two can be mapped to each other by an inner automorphism.^[1] A key property is that every element of G is conjugate to an element within any fixed maximal torus T, ensuring that T captures the "toroidal" essence of the group's conjugacy classes.^[2] The normalizer N_G(T) of T in G gives rise to the Weyl group W(G, T) = N_G(T)/T, a finite group that acts faithfully on T by conjugation and is independent of the choice of maximal torus up to isomorphism.^[3] This Weyl group is crucial for decomposing the Lie algebra of G into root spaces relative to the Lie algebra of T, facilitating the analysis of representations and symmetries.^[1] In representation theory, maximal tori enable the classification of irreducible representations of G through their restrictions to T, where characters decompose into weights under the Weyl group action.^[3] For example, in the special unitary group SU(2), the maximal torus consists of diagonal matrices \begin{pmatrix} e^{i\phi} & 0 \\ 0 & e^{-i\phi} \end{pmatrix}, with the Weyl group of order 2 reflecting the group's non-abelian nature.^[2] These structures also underpin integration formulas, such as the Weyl integration formula, which reduces integrals over G to integrals over T weighted by the Weyl group order.^[1]

Fundamentals

Definition

In a compact connected Lie group G, a maximal torus T is defined as a maximal connected compact abelian subgroup that is isomorphic to (S^1)^r, where r is the rank of G.^[1] This means T cannot be properly contained in any larger connected compact abelian subgroup of G.^[1] A torus in this context refers to any connected compact abelian Lie group, which is necessarily isomorphic to a product of circle groups. The maximality condition distinguishes it from smaller tori within G, ensuring it captures the full abelian structure at that level. Commonly, such a maximal torus is denoted as T \cong U(1)^r, where U(1) is the unitary group of degree 1, equivalent to the circle group S^1. For matrix Lie groups like the unitary group U(n), the Lie algebra \mathfrak{t} of the maximal torus T consists of diagonal skew-Hermitian matrices.^[4] The Lie algebra \mathfrak{t} of T corresponds to a Cartan subalgebra of the Lie algebra \mathfrak{g} of G. While the definition is standard for compact connected Lie groups, noncompact real Lie groups may lack compact maximal tori, though their compact real forms and complexifications possess analogous structures.

Rank and dimension

The rank r of a compact connected Lie group G is defined as the dimension of any maximal torus T in G, and this dimension is invariant for all maximal tori in G.^[5] This invariance follows from the fact that all maximal tori in G are conjugate under the action of G, ensuring they share the same topological and algebraic structure.^[6] Maximal tori in G correspond bijectively to Cartan subalgebras in the Lie algebra \mathfrak{g} of G, where a Cartan subalgebra \mathfrak{h} is a maximal abelian subalgebra consisting entirely of semisimple elements.^[5] Specifically, the Lie algebra \mathfrak{t} of a maximal torus T is itself a Cartan subalgebra of \mathfrak{g}, and the dimension of \mathfrak{h} equals the rank r.^[3] In this correspondence, every element of T is semisimple, reflecting the abelian and toral nature of \mathfrak{t}.^[5] For a semisimple Lie algebra \mathfrak{g}, the rank r coincides with the dimension of the Cartan subalgebra and equals the number of fundamental weights in the associated root system.^[6] Thus, \dim T = r provides a key invariant that links the geometric structure of G to the algebraic properties of \mathfrak{g}.^[5]

Examples

Classical Lie groups

In the unitary group U(n), a maximal torus is formed by the diagonal matrices with entries on the unit circle in the complex plane, explicitly given by

\diag(e^{i\theta_1}, \dots, e^{i\theta_n}),

where \theta_j \in \mathbb{R} for j = 1, \dots, n. This subgroup is isomorphic to the n-dimensional torus (S^1)^n and has dimension n. For the special unitary group SU(n), the maximal torus is the intersection of the above torus with SU(n), consisting of those diagonal matrices satisfying the determinant condition \det = 1, or equivalently \sum_{j=1}^n \theta_j \equiv 0 \pmod{2\pi}. This imposes one linear relation on the angles, yielding a connected abelian subgroup of dimension n-1, which is the rank of SU(n). In the special orthogonal group SO(2n), a maximal torus comprises block-diagonal matrices built from n orthogonal $2 \times 2 rotation blocks along the diagonal:

\diag \left( R(\theta_1), \dots, R(\theta_n) \right),

where

R(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}

and \theta_j \in \mathbb{R}. Each block contributes one parameter, so the torus has dimension n.^[7] The special orthogonal group SO(2n+1) admits a similar maximal torus, formed by block-diagonal matrices with the same n rotation blocks as in SO(2n), augmented by a terminal $1 \times 1 identity block

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to preserve the odd dimension. This construction again yields a dimension of n, matching the rank of the group. For the compact symplectic group Sp(n) (isomorphic to USp(2n)), the maximal torus is the subgroup of block-diagonal matrices with n blocks of the form

\begin{pmatrix} e^{i\theta_j} & 0 \\ 0 & e^{-i\theta_j} \end{pmatrix},

where \theta_j \in \mathbb{R}, ensuring compatibility with the symplectic structure. This parametrization gives an n-dimensional torus. A concrete low-dimensional illustration occurs in SO(2), which is itself a maximal torus, parametrized by rotations through angle \theta \in [0, 2\pi) via the matrix

\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.

This one-dimensional case exemplifies the general block structure for higher even orthogonal groups.^[7]

Exceptional and other groups

In the exceptional compact simple Lie groups, maximal tori take the form of products of circles, with the number of factors equal to the rank of the group. The group G_2 has rank 2 and thus a maximal torus isomorphic to (S^1)^2, while F_4 has rank 4 with maximal torus (S^1)^4. Similarly, E_6 has rank 6 and maximal torus (S^1)^6, E_7 rank 7 with (S^1)^7, and E_8 rank 8 with (S^1)^8.^[8] These structures underscore the universality of maximal tori across Lie group classifications, paralleling but distinct from the diagonal tori in classical groups like \mathrm{SU}(n).^[9] The spin groups provide another extension beyond classical orthogonal groups. The group \mathrm{Spin}(2n), a double cover of \mathrm{SO}(2n), has a maximal torus of dimension n, which lifts the standard maximal torus of \mathrm{SO}(2n) consisting of block-diagonal rotations in planes.^[10] This torus is again isomorphic to (S^1)^n, preserving the rank under the covering map. For abelian Lie groups, the concept simplifies directly. If G = T^r is itself a torus of rank r, then G serves as its own maximal torus. Although the theory centers on finite-dimensional compact Lie groups, maximal tori appear in infinite-dimensional analogs like loop groups and Kac-Moody groups, where they often involve extensions such as T \times S^1 \times T for a finite-dimensional torus T, but with more complex topology.^[11] The discussion here emphasizes finite-dimensional compact cases. Maximal tori in semisimple Lie groups can also be constructed via the Iwasawa decomposition G = K A N, where K is the maximal compact subgroup containing a maximal torus T of G, and A is a maximal abelian subspace in the orthogonal complement of the Lie algebra of K.^[6] In the fully compact semisimple case, K = G and T is central to the group's structure.

Properties

Conjugacy classes

In a connected compact Lie group G, all maximal tori are conjugate to each other. That is, given any two maximal tori T_1 and T_2 in G, there exists an element g \in G such that g T_1 g^{-1} = T_2.^[6] Moreover, every element of G lies in some maximal torus, which implies that the union of all conjugates of a fixed maximal torus T is the entire group G.^[6] In particular, since elements of compact Lie groups are semisimple, every semisimple element of G is conjugate to an element of T.^[12] This conjugacy property extends to more general settings. In a connected reductive algebraic group G over an algebraically closed field (such as the complex numbers), all maximal tori are likewise conjugate.^[13] Every semisimple element of such a G is contained in some maximal torus and thus conjugate to an element in a fixed maximal torus.^[14] For complex semisimple Lie groups, which are reductive algebraic groups over \mathbb{C}, maximal tori are conjugate and each is contained in a Borel subgroup (a maximal solvable connected subgroup); conversely, all Borel subgroups are conjugate and each contains a maximal torus.^[15] The proof of conjugacy relies on the corresponding result for Lie algebras: the Lie algebra \mathfrak{g} of G has all Cartan subalgebras conjugate under the adjoint action of the connected component of the automorphism group. A Cartan subalgebra \mathfrak{h} is the Lie algebra of a maximal torus T = \exp(\mathfrak{h}), and conjugacy in the algebra lifts to the group via the exponential map. For complex semisimple Lie algebras, this follows from the density of regular elements (those whose centralizer has minimal dimension, equal to the rank) and the fact that centralizers of regular elements are Cartan subalgebras; a sketch involves simultaneous triangularization of the adjoint action of commuting semisimple elements in faithful representations, ensuring maximality by dimension.

Centralizers and normalizers

In a compact connected Lie group G, the centralizer of a maximal torus T is the subgroup C_G(T) = \{ g \in G \mid g t g^{-1} = t \ \forall t \in T \}.^[16] Since T is maximal abelian, C_G(T) = T.^[16] The normalizer of T in G is the subgroup N_G(T) = \{ g \in G \mid g T g^{-1} = T \}, which is the largest subgroup of G in which T is normal.^[2] The quotient N_G(T)/T is finite and isomorphic to the Weyl group of G with respect to T.^[16] This quotient acts on T by conjugation, reflecting the discrete symmetries preserving the torus.^[2] In the unitary group U(n), the standard maximal torus T consists of diagonal unitary matrices. Here, C_{U(n)}(T) = T, while N_{U(n)}(T) comprises monomial matrices, which are products of permutation matrices and diagonal unitary matrices; the quotient N_{U(n)}(T)/T is thus the symmetric group S_n.^[17] The normalizer N_G(T) plays a key role in the structure of G, particularly in the Bruhat decomposition, where G decomposes into double cosets B w B for w \in N_G(T)/T and a Borel subgroup B containing T; T is normal in N_G(T), facilitating this cell decomposition.^[18]

Structure Theory

Root systems

In the context of a compact connected semisimple Lie group G with Lie algebra \mathfrak{g}, the Lie algebra \mathfrak{t} of a maximal torus T serves as a Cartan subalgebra.^[19] Considering the complexification \mathfrak{g}_\mathbb{C} and \mathfrak{t}_\mathbb{C}, the adjoint action of \mathfrak{t}_\mathbb{C} on \mathfrak{g}_\mathbb{C} induces a root space decomposition: \mathfrak{g}_\mathbb{C} = \mathfrak{t}_\mathbb{C} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, where \Phi \subset \mathfrak{t}_\mathbb{C}^* is the set of roots, consisting of nonzero linear functionals \alpha: \mathfrak{t}_\mathbb{C} \to \mathbb{C} such that the root space \mathfrak{g}_\alpha = \{ X \in \mathfrak{g}_\mathbb{C} \mid \mathrm{ad}(t) X = \alpha(t) X \ \forall t \in \mathfrak{t}_\mathbb{C} \} is nontrivial.^[20] Each root space \mathfrak{g}_\alpha is one-dimensional.^[19] The root system \Phi is a finite subset of \mathfrak{t}_\mathbb{C}^* that spans \mathfrak{t}_\mathbb{C}^* and is closed under negation, meaning if \alpha \in \Phi, then -\alpha \in \Phi.^[20] A choice of Borel subalgebra \mathfrak{b}_\mathbb{C} \subset \mathfrak{g}_\mathbb{C} containing \mathfrak{t}_\mathbb{C} determines a set of positive roots \Phi^+, consisting of those roots \alpha for which \mathfrak{g}_\alpha \subset \mathfrak{b}_\mathbb{C}, with the remaining roots \Phi^- = -\Phi^+.^[21] The simple roots form a basis \Pi \subset \Phi^+ for the real span of \Phi, such that every positive root is a nonnegative integer linear combination of elements of \Pi, and the Weyl chamber is the open cone in \mathfrak{t}_\mathbb{R} (identified via the Killing form) where \alpha(h) > 0 for all \alpha \in \Pi.^[21] Choose normalized root vectors e_\alpha \in \mathfrak{g}_\alpha for \alpha \in \Phi^+ and define f_\alpha = -e_{-\alpha} \in \mathfrak{g}_{-\alpha}. The Lie bracket satisfies [e_\alpha, e_{-\beta}] = \delta_{\alpha,\beta} h_\alpha for the coroot h_\alpha \in \mathfrak{t}_\mathbb{C} with \alpha(h_\alpha) = 2, and [e_\alpha, f_\beta] = N_{\alpha, \beta} e_{\alpha + \beta} whenever \alpha + \beta \in \Phi (a nonzero root), where N_{\alpha, \beta} is a nonzero structure constant, and is zero when \alpha + \beta \notin \Phi \cup \{0\}.^[22]^[23]

Weyl group

The Weyl group W of a maximal torus T in a connected compact Lie group G is defined as the quotient W = N_G(T)/C_G(T), where N_G(T) = \{ g \in G \mid g T g^{-1} = T \} is the normalizer of T in G, and C_G(T) = \{ g \in G \mid g t = t g \ \forall t \in T \} is the centralizer of T in G. Since T is abelian and maximal, C_G(T) = T, so W \cong N_G(T)/T. This group is finite and acts as a reflection group on the Lie algebra \mathfrak{t} of T.^[2]^[24] The Weyl group is generated by reflections s_\alpha associated to the simple roots \alpha in a choice of positive roots for the root system \Phi of G relative to T. Each reflection acts on elements t \in \mathfrak{t} by the formula

s_\alpha(t) = t - \alpha(t) \alpha^\vee,

where \alpha^\vee is the coroot of \alpha. These reflections have order 2 and satisfy the relations of a Coxeter group defined by the Dynkin diagram of the root system.^[24]^[25]^[26] The Weyl group acts faithfully on T by conjugation: for a representative n \in N_G(T) of w \in W, the action is t \mapsto n t n^{-1} for t \in T, which descends to a faithful action on \mathfrak{t} since the adjoint action of T on itself is trivial. On the root system \Phi, W acts by permuting the roots while preserving the set of positive roots up to choice, with the longest element w_0 \in W (of maximal length in the Coxeter presentation) satisfying w_0(\alpha) = -\alpha for all \alpha \in \Phi. The order of W is finite and given by the Coxeter group formula associated to the root system; for example, when G = \mathrm{SU}(n), W \cong S_n and |W| = n!.^[2]^[24]

Applications

Representation theory

In the representation theory of compact Lie groups, maximal tori play a central role in classifying finite-dimensional irreducible representations. For a compact connected Lie group G with maximal torus T, the irreducible representations of G are parametrized by dominant weights in the dual space \mathfrak{t}^* of the Lie algebra \mathfrak{t} of T. Specifically, these weights \Lambda are integral linear functionals on \mathfrak{t} that are dominant with respect to a choice of positive roots, meaning \langle \Lambda, \alpha^\vee \rangle \geq 0 for all positive coroots \alpha^\vee, where the root system is determined by the adjoint action of T on \mathfrak{g}/\mathfrak{t}.^[27]^[4] The highest weight theorem establishes that every finite-dimensional irreducible representation of G admits a highest weight vector, which is annihilated by the unipotent radical of a Borel subgroup containing T and has weight in the weight lattice that is dominant. This vector generates the entire representation under the action of G, and there is a bijection between the set of dominant integral weights and the set of equivalence classes of finite-dimensional irreducible representations of G. For example, in the case of SU(n), the dominant weights correspond to partitions, indexing the irreducible representations uniquely.^[27]^[4] The character of an irreducible representation V_\lambda of highest weight \lambda, restricted to the maximal torus T, is given by the Weyl character formula:

\chi_\lambda(t) = \frac{\sum_{w \in W} \varepsilon(w) \, e^{w(\lambda + \rho)}}{\sum_{w \in W} \varepsilon(w) \, e^{w(\rho)}},

where W is the Weyl group (the normalizer of T in G modulo T), \varepsilon(w) is the sign of the Weyl group element w, and \rho is the half-sum of the positive roots. This formula expresses the character as a ratio of alternating sums over the Weyl group orbits, providing an explicit way to compute traces on T. Alternatively, on T, the character can be viewed as a product over positive roots: \chi_\lambda(t) = \prod_{\alpha > 0} (1 - e^{-\alpha(t)})^{-1} times a numerator adjustment, but the full Weyl formula accounts for the highest weight structure.^[27]^[4] Restricting an irreducible representation of G to the maximal torus T decomposes it into a direct sum of one-dimensional weight spaces, with multiplicities given by the coefficients in the character expansion. These multiplicities are determined by the combinatorics of the root system and Weyl group action, reflecting the internal structure of the representation without altering its irreducibility over G.^[4]^[27]

Integration and Weyl formula

In the context of compact connected Lie groups, the Haar measure on G admits a decomposition involving the maximal torus T and the Weyl group W = N_G(T)/T. Specifically, the integration over G can be reduced to integration over T and the quotient G/T, accounting for the |W|-fold covering of conjugacy classes by the map (yT, t) \mapsto y t y^{-1}. This decomposition implies a volume relation where the normalized Haar measure satisfies \int_G dg = \frac{1}{|W|} \int_T \int_{G/T} | \Delta(t) |^2 d \, dt, with the Jacobian factor | \Delta(t) |^2 ensuring the measure is properly induced.^[28] For class functions f on G, which are constant on conjugacy classes (so f(y t y^{-1}) = f(t)), the Weyl integration formula simplifies to

\int_G f(g) \, dg = \frac{1}{|W|} \int_T f(t) \, |\Delta(t)|^2 \, dt,

where the Weyl denominator is

\Delta(t) = \prod_{\alpha > 0} \left( e^{\alpha(t)/2} - e^{-\alpha(t)/2} \right)

and the product runs over positive roots \alpha in the root system of G with respect to T. This formula, introduced by Hermann Weyl, leverages the bi-invariance of the Haar measure and the structure of maximal tori to compute integrals efficiently.^[28]^[29] In the general case for continuous functions f, the formula extends to

\int_G f(g) \, dg = \frac{1}{|W|} \int_T |\Delta(t)|^2 \left( \int_{G/T} f(y t y^{-1}) \, d \right) dt,

where the inner integral averages f over the G-orbit of t. This reflects the stratification of G into conjugacy classes intersecting T transversely, modulated by the Weyl group action.^[28]^[29] A concrete example arises for G = \mathrm{SU}(2), where T = \{ \diag(e^{i\theta}, e^{-i\theta}) \mid \theta \in [0, 2\pi) \} with normalized Haar measure d\theta / (2\pi) on T, and |W| = 2. For class functions, the formula becomes

\int_{\mathrm{SU}(2)} f(g) \, dg = \frac{1}{2} \int_0^{2\pi} f\left( \diag(e^{i\theta}, e^{-i\theta}) \right) \, 4 \sin^2(\theta) \, \frac{d\theta}{2\pi},

which simplifies to \frac{1}{\pi} \int_0^{2\pi} f\left( \diag(e^{i\theta}, e^{-i\theta}) \right) \sin^2(\theta) \, d\theta, as \Delta(t) = 2i \sin(\theta) up to normalization. This explicit form facilitates computations in low dimensions.^[30] These formulas find application in representation theory, particularly for computing dimensions of irreducible representations via character orthogonality: the dimension of a representation with character \chi is \dim = \int_G \chi(g) \, dg, evaluated using the Weyl formula over T.^[28]

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