Chevalley Groups Over Commutative Rings

Introduction

Chevalley groups are important families of simple groups of Lie type that arise from the construction of simple Lie algebras. First systematically developed by Claude Chevalley in the 1950s, these groups provide a unified approach to understanding many finite simple groups and their algebraic generalizations. Named after their discoverer, Chevalley groups represent one of the most significant contributions to the classification of finite simple groups.

The fundamental insight of Chevalley was that one could construct linear algebraic groups from root data associated with simple Lie algebras, and these constructions could be performed over arbitrary fields, not just the complex numbers. This allowed for the creation of finite simple groups of Lie type, which played a crucial role in the overall classification of finite simple groups.

Construction of Chevalley Groups

Chevalley's construction begins with a complex semisimple Lie algebra L with a Cartan subalgebra H and root system . The key steps in the construction are:

1. Choose a Chevalley basis for L, consisting of root vectors e_ ( ) and an h-basis of H.
2. Exhibit the structure constants of L with respect to this basis as integers.
3. Consider the Lie algebra L_Z generated by this basis over the integers.
4. For any field k of characteristic 0, form the Lie algebra L_k = L_Z _Z k.
5. Construct the group G(k) as the group of automorphisms of L_k generated by exponentials of root vectors.

This construction yields a group that is defined by polynomial equations over k, hence making G(k) an algebraic group.

Basic Properties

Chevalley groups possess several fundamental properties that make them particularly important in group theory and algebraic geometry:

• Generators: Chevalley groups are generated by root subgroups X_(k) = {x_(t) : t k}, each isomorphic to the additive group of k.
• Chevalley-Demazure relations: The relations between the generators can be made explicit, providing a presentation of the group.
• Natural action: G(k) acts naturally on the Lie algebra L_k, preserving its structure.
• Tori: The maximal torus T consists of the "diagonal" elements of the group.
• Borel subgroup: There exists a Borel subgroup B = TU, where U is generated by positive root subgroups.

Classification of Chevalley Groups

Chevalley groups are classified according to the type of the underlying simple Lie algebra, which in turn corresponds to the associated Dynkin diagram. There are four infinite families and five exceptional types:

• Type A_n (n 1): G(k) SL_{n+1}(k)
• Type B_n (n 2): G(k) SO_{2n+1}(k)
• Type C_n (n 3): G(k) Sp_{2n}(k)
• Type D_n (n 4): G(k) SO_{2n}(k)
• Exceptional types: G_2, F_4, E_6, E_7, E_8

Another important classification is based on the field k:

• Universal Chevalley groups: When k is a field, the group G(k) is called universal.
• Adjoint Chevalley groups: These are quotients of universal groups by their centers.
• Finite Chevalley groups: When k = GF(q) is a finite field, G(k) is a finite group.

From the original Chevalley groups, Steinberg, Suzuki, and Ree later derived the twisted Chevalley groups, which include the unitary groups, Suz_u(2), and the Ree groups. Together with the ordinary Chevalley groups, they form almost all groups of Lie type.

Chevalley Groups Over Commutative Rings

The construction of Chevalley groups can be extended beyond fields to arbitrary commutative rings. This generalization has important applications in algebraic K-theory and number theory.

For a commutative ring R with identity, we can construct a Chevalley group G(R) as follows:

1. Start with the integer form L_Z of the simple Lie algebra as before.
2. Form the Lie algebra L_R = L_Z _Z R.
3. Define elementary generators x_(r) for , r R.
4. Define G(R) as the subgroup of Aut(L_R) generated by these elementary elements.

The resulting group G(R) is functorial in R, meaning that any ring homomorphism R S induces a group homomorphism G(R) G(S).

Applications in Mathematics

Chevalley groups over commutative rings have numerous applications across various areas of mathematics:

• Algebraic K-theory: Chevalley groups provide a way to define higher K-groups. For instance, Milnor used Steinberg groups to define K_2(R).
• Number theory: Chevalley groups over rings of integers play a role in studying Galois representations and automorphic forms.
• Topology: The topology of automorphism groups of various mathematical objects can be studied through Chevalley groups.
• Geometric group theory: Some infinite groups arising in topology can be approximated by Chevalley groups over commutative rings.
• Representation theory: Representations of Chevalley groups over commutative rings are important in the study of modular representations.

Historical Context and Development

The study of Chevalley groups has evolved significantly since Chevalley's original work in the 1950s:

• 1950s: Claude Chevalley developed his construction of groups from simple Lie algebras over arbitrary fields.
• 1960s: Steinberg, Suzuki, and Ree discovered twisted groups, expanding the family of groups of Lie type.
• 1970s: The classification of finite simple groups was largely completed, with groups of Lie type (including Chevalley groups) playing a central role.
• 1980s-1990s: Mathematicians began studying Chevalley groups over more general rings, with applications to K-theory.
• 2000s-present: Current research explores further generalizations, including Kac-Moody groups and supergroups, as well as computational aspects of these groups.

The profound impact of Chevalley's work cannot be overstated. By providing a uniform construction that works over arbitrary fields and later over commutative rings, Chevalley created a powerful framework that continues to be central to modern research in algebra, geometry, and number theory.