Unlike many other terms in mathematics which have a universally understood meaning (for instance, "group"), the term *classical group* seems to have a fuzzier definition. Apparently it originates with Weyl's book *The Classical Groups* but doesn't make it into the index there. It was propagated by Dieudonne and others. But I'm never sure exactly what groups are included/excluded by this label. Weyl himself seems to have been interested in general (and perhaps special) linear groups, together with orthogonal (or special orthogonal) and symplectic groups attached to bilinear/quadratic forms. Initially questions were raised mainly in characteristic 0, usually over $\mathbb{C}$ but sometimes other fields as well.

Obviously it helps mathematical communication to have words and symbols which need no further explanation. But ambiguity tends to creep in. For example, what does one mean by "natural numbers" or the symbol $\mathbb{N}$? (Is 0 a natural number or not?) What is a "ring"? (Does it have an identity element or
not?) By now a number of book titles and thousands of research papers refer to
*classical groups*. But which groups are included? Spin or half-spin groups? Projective versions of the linear groups mentioned above?

Is there any precise definition of

classical groups?

ADDED: The answers and comments have been enlightening, though like some other people I lean more toward a "no" answer to my basic question. The underlying concern on my part is whether the notion of "classical group" has become too vague to be useful, which I sometimes suspect is the case with newer umbrella terms like "quantum group". It seems that the only safe usage nowadays is "classical groups, by which I mean one of those in the following list ....", at which point the original label has lost most of its purpose.

However ... the careful treatment by Porteous (which I wasn't familiar with) strikes me as well focused even if it omits some groups of interest. Weyl himself wanted a direct and concrete approach to representations and invariants of certain specific matrix groups, mainly over $\mathbb{C}$. That's clearly much too narrow for later purposes, where the geometry of various kinds of forms over various kinds of rings gets more attention, along with internal group structure. But some of the geometric viewpoints might suggest paying more attention to PGL than to GL, contrary to the matrix group emphasis in most other work.

In any case, while the Killing-Cartan classification for Lie algebras still makes it natural to view A-D types as "classical" and the rest as "exceptional", I'm reluctant to go too far in fitting classical groups into the framework of semisimple Lie or algebraic groups based heavily on differential or algebraic geometry. That framework already has to be stretched to admit general linear groups or rings coming from number theory. And spin or half-spin or adjoint groups, however natural in Lie theory, probably don't fit so well into the familiar world of matrix groups.

One viewpoint I resist is the attempted definition given by Popov in the Springer encyclopedia. This doesn't really cover the ground consistently or comprehensively, besides which the short reference list is totally unbalanced.

P.S. The views expressed in the various answers and comments are mostly quite reasonable, but leave me with the sense that everyday usage won't tend to converge. Maybe I should sum up my lingering uncertainty about the value of the term "classical group" by quoting one of Emil Artin's 1955 papers on the orders of finite simple groups: *The notion of classical groups is taken in such a wide sense as to embrace all finite simple groups which are known up to now.*

despitetheir slight vagueness butbecauseof their slight vagueness. $\endgroup$ – Timothy Chow Dec 29 '10 at 15:16