Chevalley Basis

In the theory of Lie algebras, a Chevalley basis is a specific basis of a simple Lie algebra over the complex numbers with the property that all structure constants are integers. Introduced by Claude Chevalley in the 1950s, this basis provides a canonical integer form of any simple Lie algebra, allowing the algebra to be "reduced" to arbitrary fields — including finite fields — and serving as the foundation for the construction of Chevalley groups.

The Chevalley basis exists for every simple Lie algebra. It consists of a Cartan subalgebra together with root vectors, chosen so that the commutators of root vectors yield integer multiples of other root vectors. This integrality is the key: it means the Lie algebra can be defined over the integers, and by extension over any commutative ring.

Construction and Properties

The construction of a Chevalley basis begins with the root system of a simple Lie algebra. For each root α, one selects a root vector e_α in the corresponding root space. The Chevalley basis is then the set {h_1, ..., h_r} ∪ {e_α : α ∈ Φ} where the h_i form a basis of the Cartan subalgebra and Φ is the set of roots. The structure constants are integers because the Serre relations — which completely characterize simple Lie algebras — can be realized with integer coefficients.

The Chevalley basis is not merely a computational convenience. It reveals that the continuous simple Lie groups classified by Élie Cartan and Wilhelm Killing possess a discrete, arithmetic skeleton. This skeleton is what Chevalley exploited to construct finite simple groups: by evaluating the matrix entries of the Lie algebra in a Chevalley basis over a finite field, one obtains the Chevalley groups.

Significance and Generalizations

The Chevalley basis has been generalized in several directions. In Kac-Moody algebras — infinite-dimensional Lie algebras that generalize simple Lie algebras — an analogous Chevalley basis exists and plays a similar role in the construction of Kac-Moody groups over arbitrary fields. The theory of quantum groups also begins with a deformation of the universal enveloping algebra of a Lie algebra in a Chevalley basis.

The Chevalley basis exemplifies a recurring pattern in modern mathematics: the discovery that analytic or continuous structures possess canonical arithmetic forms that reveal their behavior under reduction to discrete settings. The same impulse that led Chevalley to the adele ring in number theory — the search for a structure that is simultaneously local and global, continuous and discrete — led him to the Chevalley basis in Lie theory.

The Chevalley basis is not a computational convenience but a metaphysical claim: that the infinite-dimensional symmetry of a simple Lie algebra is not merely a property of the complex numbers, but a structural fact that persists when the field itself is stripped away. The integers are not a stepping-stone to the real numbers; they are the bones that hold the flesh in place.