Cartan SubalgebraEdit
Cartan subalgebras occupy a foundational place in the theory of Lie algebras, providing a fixed, abelian core around which the entire structure of the algebra can be understood. In the standard setting of finite-dimensional complex Lie algebras, a Cartan subalgebra h is a maximal toral subalgebra—equivalently, a nilpotent subalgebra that is self-normalizing. For complex semisimple Lie algebras, Cartan subalgebras are abelian and consist of semisimple elements; they give rise to a highly organized decomposition of the ambient algebra via roots. This decomposition is not just a bookkeeping tool: it determines the principal invariants of the algebra, guides the representation theory, and leads to the celebrated classification by root systems and Dynkin diagrams.
A typical way to view Cartan subalgebras is through the root space decomposition. If g is a complex Lie algebra and h ⊂ g is a Cartan subalgebra, then the adjoint action of h on g splits g into simultaneous eigenspaces g = h ⊕ ⊕α∈Δ gα, where each gα is the α-root subspace and Δ ⊂ h* is the set of roots. The action of h is diagonalizable on g, and the nonzero linear functionals α for which gα ≠ 0 are precisely the roots. The collection Δ, together with the pairing induced by the Lie bracket, forms a root system, a highly structured object that encodes far-reaching information about g. The roots also determine a Dynkin diagram, whose connected components classify g up to isomorphism. See root system and Dynkin diagram for more.
In concrete terms, Cartan subalgebras can be constructed and studied via familiar matrix realizations. For the classic example g = sl(n, C), the Cartan subalgebra h consists of all diagonal traceless matrices. In this case dim h = n − 1, and the corresponding root system is of type A_{n−1}. More generally, for many classical families such as so(n, C) and sp(2n, C), one can describe h explicitly as a family of diagonal or block-diagonal matrices that commute with a chosen set of elements, and then analyze the induced root spaces gα. These explicit descriptions illuminate how the abstract theory plays out in familiar algebraic settings.
Existence and conjugacy of Cartan subalgebras are central results in the structure theory. In a finite-dimensional complex semisimple Lie algebra g, any two Cartan subalgebras are conjugate under inner automorphisms of g; in particular, the dimension of h—the rank of g—is an invariant of g. This conjugacy principle ensures that the root system Δ attached to h is intrinsic to g, not an artifact of a particular choice of h. In the real setting, one encounters a richer landscape. Real semisimple Lie algebras admit Cartan subalgebras that can be split (maximally noncompact, consisting of semisimple elements that are diagonalizable over the reals) or compact (maximal among compact subalgebras). The choice of real form and the position of the Cartan subalgebra within it influence representation theory and harmonic analysis on the corresponding Lie group.
Cartan subalgebras play a crucial role in representation theory. Weights are linear functionals on h, and their associated weight spaces decompose representations of g into a direct sum of weight spaces. The celebrated Weyl character formula and the highest weight theory hinge on this weight-space framework and on the precise structure of the root system. The interplay among h, Δ, and the Weyl group W = N_G(h)/Z_G(h) governs many essential features, including branching rules, tensor product decompositions, and the geometry of orbits. See Weyl group and highest weight representation for related topics.
Beyond their role in the classification and representation theory of Lie algebras, Cartan subalgebras anchor many constructions in adjacent areas of mathematics and mathematical physics. They appear in the study of complex semisimple Lie groups via maximal tori, in the theory of differential operators and flag varieties through Borel and parabolic subalgebras, and in the formulation of invariant theory and symmetric spaces. The structure constants and root data derived from a Cartan subalgebra feed into the Cartan–Weyl framework that underpins much of the modern algebraic and geometric approach to symmetry.
Historical notes and development. The concept of a Cartan subalgebra crystallized from Elie Cartan’s work on the structure and classification of Lie algebras in the early 20th century. Over time, the Cartan subalgebra emerged as a unifying device that translates the noncommutative geometry of a Lie algebra into the more tractable language of linear functionals and eigenvalue patterns. The resulting framework, including the root system and Dynkin diagram classifications, remains a cornerstone of semisimple Lie theory and its applications to representation theory and mathematical physics. See Cartan subalgebra for the topic’s primary entry and Lie algebra for broader context.
Definition and basic properties
- A Cartan subalgebra h of a Lie algebra g is a nilpotent subalgebra that is self-normalizing: N_g(h) = h. In complex finite-dimensional semisimple g, such subalgebras are automatically abelian and consist of semisimple elements.
- For complex semisimple g, all Cartan subalgebras are conjugate under the inner automorphism group of g, and dim(h) equals the rank of g.
- The adjoint action of h on g yields a root space decomposition g = h ⊕ ⊕α∈Δ gα with α(h) ≠ 0 for gα ≠ 0, and Δ ⊂ h* is the root system.
Real forms and variants
- In real semisimple Lie algebras, Cartan subalgebras can be split (maximally noncompact) or compact (maximally compact), among other possibilities. Different choices reflect distinct geometric and spectral properties of the associated Lie groups.
- The real forms and their Cartan subalgebras influence harmonic analysis, representation theory, and the geometry of corresponding symmetric spaces.
Applications and connections
- Root data: The root system Δ and the corresponding Dynkin diagram encode the entire isomorphism class of the complex semisimple g. See root system and Dynkin diagram.
- Representation theory: Weights, highest weight modules, and character formulas (e.g., the Weyl character formula) are formulated with respect to a Cartan subalgebra. See highest weight representation and Weyl group.
- Connections to geometry: Cartan subalgebras relate to maximal tori in Lie groups, to flag varieties, and to Borel subalgebras, driving geometric interpretations of representation theory. See Borel subalgebra.