In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups.
The name "groups of Lie type" is due to the close relationship with the (infinite) Lie groups, since a compact Lie group may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers. and are standard references for groups of Lie type.
Classical group
An initial approach to this question was the definition and detailed study of the so-called classical groups over finite and other fields by . These groups were studied by L. E. Dickson and Jean Dieudonné. Emil Artin investigated the orders of such groups, with a view to classifying cases of coincidence.
A classical group is, roughly speaking, a special linear, orthogonal, symplectic, or unitary group. There are several minor variations of these, given by taking derived subgroups or central quotients, the latter yielding projective linear groups. They can be constructed over finite fields (or any other field) in much the same way that they are constructed over the real numbers. They correspond to the series An, Bn, Cn, Dn,2An, 2Dn of Chevalley and Steinberg groups.
Chevalley groups can be thought of as Lie groups over finite fields. The theory was clarified by the theory of algebraic groups, and the work of on Lie algebras, by means of which the Chevalley group concept was isolated. Chevalley constructed a Chevalley basis (a sort of integral form but over finite fields) for all the complex simple Lie algebras (or rather of their universal enveloping algebras), which can be used to define the corresponding algebraic groups over the integers.
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In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups.
Jacques Tits (tits) (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Tits was born in Uccle to Léon Tits, a professor, and Lousia André. Jacques attended the Athénée of Uccle and the Free University of Brussels. His thesis advisor was Paul Libois, and Tits graduated with his doctorate in 1950 with the dissertation Généralisation des groupes projectifs basés sur la notion de transitivité.
In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type. The more specialized theory of Bruhat–Tits buildings (named also after François Bruhat) plays a role in the study of p-adic Lie groups analogous to that of the theory of symmetric spaces in the theory of Lie groups.
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