Cartan MatrixEdit

The Cartan matrix is a square integer matrix that plays a central role in the study of semisimple Lie algebra. It encodes the geometric data of the simple roots chosen inside a Cartan subalgebra, and from it one can recover the structure of the root system that governs representation theory and symmetry. Although it arises from an abstract algebraic setting, the Cartan matrix has concrete consequences for classifying and working with finite-dimensional symmetry algebras that appear in mathematics and theoretical physics. See also root system and Cartan subalgebra for the foundational objects from which the Cartan matrix is built.

By construction, the Cartan matrix A = (a_ij) is defined relative to a choice of simple roots {alpha_1, ..., alpha_n} in a finite root system. The entries are a_ij = 2 (alpha_i, alpha_j) / (alpha_i, alpha_i), where (·,·) denotes the invariant inner product on the ambient space of the root system. This yields a matrix with a_ii = 2 for all i, and a_ij ≤ 0 for i ≠ j. In many cases the product a_ij a_ji takes only a few small values, namely 0, 1, 2, or 3, reflecting the angles between simple roots. The Cartan matrix is typically not symmetric, but it is always symmetrizable: there exists a diagonal matrix D with positive entries such that DA is symmetric. See also Dynkin diagram as the graphical companion that records the same data in a combinatorial form.

Definition and construction

Prerequisites and objects

Lie algebra: the algebraic object whose representations and symmetries are being organized.
Cartan subalgebra: a maximal toral subalgebra that provides a natural setting for decomposing the Lie algebra into root spaces.
Root system: the collection of vectors that encode how the Lie algebra decomposes into weight spaces.
Simple root: a basis of the root system from which every root is a integer combination with coefficients all of one sign.

The Cartan matrix

Given a basis of simple roots {alpha_i}, the Cartan matrix A is the n×n matrix with entries a_ij = 2 (alpha_i, alpha_j) / (alpha_i, alpha_i). The diagonal entries are a_ii = 2, and off-diagonal entries are non-positive integers. The matrix is not arbitrary; it satisfies integrality conditions and reflects the angles between simple roots. A is often described as symmetrizable, since one can find a positive diagonal D with DA symmetric: this property is crucial for connecting the matrix to representation theory and to the corresponding Weyl group.

Examples

Type A_n (the special linear case): the Cartan matrix is symmetric and tridiagonal with 2’s on the diagonal and -1’s on the off-diagonal adjacent entries. For example, the A_2 matrix is [[2, -1], [-1, 2]], corresponding to the Lie algebra A_2 Lie algebra in classical language.
Type G_2: a 2×2 matrix with entries [[2, -3], [-1, 2]], illustrating a non-symmetric case that is still symmetrizable.
Other classical families (B_n, C_n, D_n) and the exceptional types (E_6, E_7, E_8, F_4) each have their own standard Cartan matrices, and these matrices directly determine the corresponding Dynkin diagrams, see Dynkin diagram.

Relationship to Dynkin diagrams

From the Cartan matrix one can construct a Dynkin diagram by placing a node for each simple root and drawing bonds whose multiplicities are determined by the values of a_ij a_ji. The diagram encodes the same information as the Cartan matrix in a more geometric and combinatorial form, and the diagrammatic classification is what underpins the famous finite-type classification. See also Dynkin diagram for the details of this correspondence.

Properties and consequences

Symmetrizability: Every Cartan matrix A is symmetrizable, meaning there exists a diagonal D > 0 such that DA is symmetric. This facilitates the transfer of many results from linear algebra to the representation theory of the associated Lie algebra.
Weyl group: The simple roots and the Cartan matrix give rise to reflections that generate the Weyl group, a finite group of symmetries of the root system and its representations. See Weyl group for the larger symmetry picture.
Classification: Finite-type Cartan matrices correspond to finite root systems, which in turn are classified by connected Dynkin diagrams of types A, B, C, D, E, F, and G. This is the backbone of the classification of finite-dimensional semisimple Lie algebra over the complex numbers.
Representations: The Cartan matrix encodes data that govern weight spaces, highest weight modules, and the structure of irreducible representations. The data it contains interact with the theory of weights, roots, and the action of the Weyl group.

Applications and interpretation

Representation theory: The Cartan matrix provides a compact description of the simple roots and their interactions, which in turn determine the structure of all finite-dimensional representations of the corresponding semisimple Lie algebra. See representation theory and highest weight module.
Physics and symmetry: Lie algebras and their root data appear in gauge theories, particle physics, and crystallography. Cartan matrices help organize the possible symmetry types and their representations, contributing to model building and the analysis of conserved quantities.
Generalizations: The idea of a Cartan matrix extends beyond finite-type Lie algebras to more general Kac–Moody algebras, where generalized Cartan matrices play a central role. See Kac–Moody algebra for the broader framework.

Controversies and contemporary perspectives

From a tradition-minded viewpoint, the strength of the Cartan matrix lies in its economy and universality: a single matrix encapsulates a wide array of structural information about symmetries, their representations, and their geometric underpinnings. Critics within broader academic discussions sometimes argue that modern mathematics can become too abstract or overly reliant on categorical language, potentially obscuring concrete computations and applications. Proponents counter that the compact encoding provided by the Cartan matrix and its Dynkin-diagram companion fosters cross-disciplinary insights—bridging algebra, geometry, and physics—and yields a robust classification that makes subsequent work more tractable.

A related debate centers on the role of mathematical structures in physics. Some physicists rely on Lie theory and the associated Cartan data to propose models of fundamental interactions, while others caution that mathematical elegance does not guarantee empirical validity. In this landscape, the Cartan matrix is often cited as a case where deep structural insight leads to powerful, testable frameworks (via representation theory and symmetry principles), even as speculative physics reminds researchers to stay grounded in phenomenology. Critics of overreliance on abstract frameworks sometimes argue for a greater emphasis on computation, numerical methods, and concrete models, while supporters highlight the long-term payoff of a principled, unified approach to symmetry.

See also: the discussion of how these matrices tie into broader classification schemes, and how they interface with the algebraic and geometric apparatus that underpins much of modern mathematical physics. See also root system, Dynkin diagram, Lie algebra, Weyl group, and Kac–Moody algebra for related developments.