What are "classical groups"?

Question

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.

Answer 1

A classical group means one whose Dynkin diagram is one of the 4 infinite series A, B, C, D whose elements can be extended indefinitely, as opposed to the exceptional groups G2, F4, E6, E7, E8 whose Dynkin diagrams cannot be extended indefinitely (assuming everything is finite dimensional...). Alternatively the classical groups are the ones that (up to abelian pieces) can be defined by messing around with "degree 2" forms (sesquilinear, symmetric, alternating, trivial, etc); the exceptional groups can be defined using forms of degree at least 3.

Answer 2

Such a definition (but not the definition, I suppose) can be found in Clifford Algebras and the Classical Groups by Ian Porteous (see Chapt. 13).

It is based on the classification of real algebra anti-involutions of $A(n)$ where $A$ is equal to $K$ or $^2K$, and $K = \mathbb R$, $\mathbb C$ or $\mathbb H$. By the theorem below there are ten cases. In each case there is a corresponding family of groups of correlated automorphisms analogous to the orthogonal groups.

Theorem. Let $\xi$ be an irreducible correlation on a right $A$-linear space of finite dimension $> 1$, and therefore equivalent to a symmetric or skew correlation. Then $\xi$ is equivalent to one of the following ten types, these being mutually exclusive.

1. A symmetric $\mathbb R$-correlation;
2. A symmetric, or equivalently a skew, $^2\mathbb R^\sigma$-correlation;
3. A skew $\mathbb R$-correlation;
4. A skew $\mathbb C$-correlation;
5. A skew $\tilde{\mathbb H}$- or equivalently a symmetric $\overline{\mathbb H}$-correlation;
6. A skew, or equivalently a symmetric,$^2\overline{\mathbb H}^\sigma$ -correlation;
7. A symmetric $\tilde{\mathbb H}$-, or equivalently a skew, $\overline{\mathbb H}$-correlation;
8. A symmetric $\mathbb C$-correlation;
9. A symmetric, or equivalently a skew, $\overline{\mathbb C}$-correlation;
10. A symmetric, or equivalently a skew, $^2\overline{\mathbb C}^\sigma$-correlation.

The ten families of classical groups are as follows, where $p+q=n$:

1. $O(p, q; \mathbb R)$ or $O(p, q)$, with $O(n) = 0(0, n)$;
2. $GL(n;\mathbb R)$;
3. $Sp(2n;\mathbb R)$;
4. $Sp(2n;\mathbb C)$;
5. $Sp(p,q;\mathbb H)$ or $Sp(p,q)$, with $Sp(n)= Sp(0,n)$;
6. $GL(n;\mathbb H)$;
7. $O(n;\mathbb H)$;
8. $O(n;\mathbb C)$;
9. $U(p,q)$, with $U(n)=U(0,n)$;
10. $GL(n;\mathbb C)$.

Answer 3

I think the question probably won't ever have a precise answer. In the context of linear algebraic groups over arbitrary fields, I happen to like the point of view provided by "algebras with involution" -- see chapter VI of [The Book of Involutions], which the authors describe as giving "the classification of semisimple algebraic groups of classical type without any field characteristic assumptions..."

This point of view sees "classical groups" using the following sort of data. Let $A$ be a (finite dimensional) $k$ algebra which is semisimple and separable (separable means that the center of $A$ is an etale commutative $k$-algebra), and let $\sigma$ be a $k$-involution of $A$. Using this data, one constructs families of algebraic groups.

For an example, consider the algebraic group $G=\operatorname{Iso}(A,\sigma)$ whose functor of points is given by the rule $G(R) = ${$a \in A \otimes_k R \mid a\cdot\sigma(a) = 1$ }. When $A$ is simple then -- depending on the nature of $\sigma$ -- $G=\operatorname{Iso}(A,\sigma)$ can be a (twisted form of a) unitary group, an orthogonal group, a symplectic group. (Further care is needed to get special unitary or orthogonal groups...)

There are related constructions -- $\operatorname{Sim}(A,\sigma)$, $\operatorname{Aut}(A,\sigma)$,... -- to account for isogenies etc.

These constructions give groups which "geometrically" (over an algebraic closure of $k$) have the Dynkin diagrams mentioned in other answers.

Well, I doubt this point of view will given a universally accepted definition of the notion of "classical groups", but it does give a fairly uniform account.

Answer 4

IMHO this definition need not limit itself to groups over $\mathbb{C}$ and its relatives; e.g. in the theory of finite simple groups people usually talk about a classical group as being a member of one of the 4 series: linear, symplectic, orthogonal, and unitary, defined over a finite field. See e.g. http://brauer.maths.qmul.ac.uk/Atlas/v3/lin/ and http://brauer.maths.qmul.ac.uk/Atlas/v3/clas/ (here for some reason linear groups are split from the rest).

More generally, one can even work with classical groups over rings: see e.g. the book by Hahn and O'Meara http://www.springer.com/mathematics/algebra/book/978-3-540-17758-6

Answer 5

A classical group is the isotropy subgroup of an open orbit in a representation of GL(n).

Answer 6

Hmm... I must admit that I never asked myself this question. I have always taken for granted that classical groups are the special linear groups over $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{H}$, with the usual caveat about the definition of quaternionic special linear group, as there is no quaternionic determinant, and their intersection with the automorphism groups of inner products on $\mathbb{R}^n$, $\mathbb{C}^n$ and $\mathbb{H}^n$: namely, symmetric and skewsymmetric inner products on $\mathbb{R}^n$ and $\mathbb{C}^n$; hermitian inner products on $\mathbb{C}^n$ and $\mathbb{H}^n$; and skewhermitian inner products on $\mathbb{H}^n$.

So in summary, the following are (for me) the classical groups: $\mathrm{SL}(n,\mathbb{R})$, $\mathrm{SL}(n,\mathbb{C})$, $\mathrm{SL}(n,\mathbb{H})$, $\mathrm{SO}(p,q;\mathbb{R})$, $\mathrm{SO}(p,q;\mathbb{C})$, $\mathrm{Sp}(2n;\mathbb{R})$, $\mathrm{Sp}(2n;\mathbb{C})$, $\mathrm{SU}(p,q)$, $Sp(p,q)$, and $SO^*(n)$.

I'm afraid that since I didn't come up with this, I have no wise words as to why the general linear groups are excluded. One could argue that including the spin groups would then lead to including other covering groups and we probably we would not want to count the universal cover of $\mathrm{SL}(2,\mathbb{R})$ or the metaplectic group as classical groups.

I think that this definition of classical group agrees with, say, Wolf Rossmann's book Lie groups: an introduction through linear groups, which is the current recommended book for our UG students here in Edinburgh.

Answer 7

Just a piece of information from the french side. Dieudonn\'e, in ``La g\'om\'etrie des groupes classiques'' (Springer, 1970), takes the definition of a classical group for granted. But browsing through the table of contents, it's clear he means $GL_n(K),SL_n(K),O_n(K,f),U_n(K,f),Sp_{2n}(K)$ plus variants (e.g. the projectivized versions).

In the book ``Groupes de Lie classiques''(Hermann, 1986), R. Mneimn\'e and F. Testard define classical Lie groups in their introduction: same list as in Dieudonn\'e, but assuming of course $K=\mathbb{R}$ or $\mathbb{C}$.

Answer 8

It is a hint, when making statements hold is left as an exercise to the reader.

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A longer answer:

Perhaps most terms in mathematics admit various definitions.

• There is the set-theoretic construction aspect (like for the definition of a pair).
• Then the more important question of what properties are assumed in a definition (irreducible for an algebraic variety?).
• We could even consider the underlying logical language and axiom system part of a definition (for instance for most discussions "most mathematicians" assume first-order logic, and that their objects are sets, which e.g. have a well-defined cardinality -I guess this could be argued, for various reasons).

"Classical group" is perhaps an outlier in the variety of interpretations, as proposed in the answers here.

The case of "variety", mentioned by B. Conrad, is similar but also different, a variety can be "reduced", "irreducible", etc., but it is easy to add those adjectives to clarify what we mean. The term "variety" is also used in universal algebra, with a very different meaning, but the 2 uses are easy to differentiate. This issue also arises with "field" either in algebra, or in physics (e.g. scalar field). Also with the term "algebra", in various fields of mathematics.

Returning to "classical groups", part of the unease may be due to tight connections between areas where it may be used with different meanings, e.g. researchers in group theory may pass in their own research from considering finite-dimensional to infinite-dimensional groups, while algebraic geometers are less likely to talk to universal algebraists -though important connections to model theory may prove me wrong here. So if we nonobviously switch definitions the exercise of finding which one, if any, makes a statement hold, may frustrate.

A further idea is that "classical groups" emerged to name various things arrived at from different routes, which were found similar in several respects, after nontrivial thinking, perhaps with more desire to unify than when defining "algebraic variety", which captures a more obvious property (say, being "algebraically locally" like affine space). The definition of "classical group" is "stretched" from the beginning. This happens in history as a scholarly discipline, where trends are identified a posteriori, and fitting can be difficult, but still useful. If we start from a blanker sheet, with less prejudice we will have cleaner definitions. In a sense "classical groups" is ambitious, we have an intuition which happens to be more simplistic than for "variety". The term is attractive, it evokes art, beauty, we want to use it, and those groups, we wish we could describe them simply. In the case of varieties there happens to be more rigidity, we arrive at the concept from fewer places in a sense (perhaps always with "algebraically locally affine" in mind).

It happens that group theory is quite complicated (some subjects have to be, I suppose this could be justified using ideas from computational complexity theory/information theory), in the sense of having "many (irreducible) lists", "many facts", things that must be learned, memorized, its classifications have many cases, so if we do not want to use more terms than in other areas we are bound to oversimplify, to unify too much. Several definitions proposed here of "classical group" rely on classifications with technical hypotheses which can be argued, which have been found after efforts to unify.

The origin of the question seems to be this feeling of bias: why do we want to use "classical groups"? Is it fair?

To return to the idea I started with: definitions are dynamical, they rely on history, on context, on the readers ability, etc., and they are bound to be because they are crucial in attempts to optimize our thinking, by all means. Sometimes this optimization will be stressful.

PS: Borcherds' definition is my favorite.

Answer 9

It is worth mentioning that when talking about Lie algebras classical in positive characteristic is somewhat different than classical in zero characteristic: "By an algebra of classical type is meant an analogue over a field of characteristic $p$ of one of the simple Lie algberas (including the five exceptional Lie algberas) of characteristic 0", see http://www.jstor.org/stable/2034779.