Talk:Classical group

Matrix group content

I strongly oppose merging "Matrix group" and "Classical Lie group". They are very different topics. A matrix group is any group of matrices. It may be finite, it may have coefficients in a finite field, it may be a topologically discrete subgroup of a classical Lie group, etc. A classical Lie group is a manifold with group structure; this gives it very particular properties. There are different questions and results that belong to the two topics. Zaslav 20:49, 26 March 2007 (UTC)

I think what is meant here is to merge the section on classical groups from the article "matrix groups" into this article, where it would fit better. Perhaps, the title would have to be changed to "Classical group". This would allow us to concentrate on the general properties of matrix (i.e. linear) groups in the Matrix group article, as opposed to descriptions of classical groups, as is presently the case. Arcfrk 22:08, 4 April 2007 (UTC)

Eh?

Copuls someone perhaps explain the relationship between these four groups and eucledian geometry? I note that one of them mentions "having a deteminant of 1", so it seems to be something to do withh affine transformations that preserve area. Maybe a few paragraphs pitched at a slightly easier level?

Not B-class

There is no way this article is class B, so I gave it a "Start" rating. The subject has inspired much classical research, many books, and is the backbone in any course beyond "abstract group theory". It deserves a much more inspired article.

In today's terminology there are the complex classical groups, the real classical groups and the compact classical groups. Variants exist; the connoisseur might add the exceptional groups, others relax the determinant = 1 condition to determinant = +/-1, yet others include also GL(n, R), GL(n, C) and some quaternion groups (GL(n, H), Sp(p, q), SO*(2n)). The groups U(p, q) and SU(p, q) (coming from indefinite Hermitean forms on Cⁿ, see below) are missing all-together in the article.

The unifying framework is bilinear forms on Rⁿ and Cⁿ, sesquilinear forms on Cⁿ and Hⁿ. These forms can be symmetric or antisymmetric (in the bilinear case), Hermitean or anti-Hermitean (in the sesquilinear case).

On the other hand, the section Classical groups over general fields or rings is a completely unnecessary extension of the subject. It is far from the classical considerations.

An introductory exposition of these forms and their signatures, could explain all terminology and some connections between the groups. A beefier explanation of 'compact real form is also called for.

Most (all actually, and much more) of what I propose could be extracted from §3.1 in Wulf Rossmann's "Lie Groups, an Introduction through Linear Groups". YohanN7 (talk) 13:38, 2 January 2014 (UTC)

Matrix groups over non-commutative rings

There is an ongoing discussion that is relevant to this article, as well as some particular classical groups such as special linear group and special orthogonal group. Articles on particular classical groups do not define them over anything more general than fields, such as non-commutative rings, but this article currently makes wild claims about it. Let’s start from the determinant. How can one define it for, say, a 2×2 matrix $\scriptstyle {\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}$? Would it be a₁₁a₂₂ − a₂₁a₁₂? a₁₁a₂₂ − a₁₂a₂₁? a₂₂a₁₁ − a₂₁a₁₂? ({a₁₁, a₂₂} − {a₂₁, a₁₂})/2? Whichever will you choose, it wouldn’t be respected by the matrix multiplication.

User:YohanN7 argues SL_n(H) is possible. Is it really? I didn’t learn specified sources, but the only possibility I see is to play on the Cayley–Dickson construction that can define quaternions from complex numbers. Namely, if one uses

$q={\begin{bmatrix}A&C\\-{\overline {C}}&{\overline {A}}\end{bmatrix}}$

representation, as it is suggested here, then one can define SL₁(H) as quaternions that have its determinant equal to 1, i.e. unit quaternions. As for SL₂(H), an element of the desired group becomes

${\begin{pmatrix}A_{11}&C_{11}&A_{12}&C_{12}\\-{\overline {C_{11}}}&{\overline {A_{11}}}&-{\overline {C_{12}}}&{\overline {A_{12}}}\\A_{21}&C_{21}&A_{22}&C_{22}\\-{\overline {C_{21}}}&{\overline {A_{21}}}&-{\overline {C_{22}}}&{\overline {A_{22}}}\end{pmatrix}}$

and this 4×4 matrix, of course, has its determinant well-defined and even real, because 8 terms are real by construction and the rest 16 come in 8 conjugated pairs. By applying one real equation to eight complex variables, one can define what is the special linear group of the rank 2 “over” quaternions, a subgroup of SL₄(C).

If it is the construction suggested by YohanN7, then a rhetorical question: has any non-commutative ring a 2×2 matrix representation over a commutative ring? Of course, not any. Incnis Mrsi (talk) 12:07, 18 February 2014 (UTC)

I meant: do all non-commutative rings have such representations? There was something wrong with grammar. Incnis Mrsi (talk) 17:18, 18 February 2014 (UTC)

You are on the right track. I don't have time a t m, but I'll check later in Rossmann's book. B t w, the matrix representation to use for the quaternions in this context is

$q={\begin{bmatrix}A&-{\overline {C}}\\C&{\overline {A}}\end{bmatrix}}.$

Also, the action of quaternion groups are on right vector spaces over H (scalars go to the right). YohanN7 (talk) 15:07, 18 February 2014 (UTC)

The difference is only in the matrix transpose and, consequently, the order of matrix multiplication. Incnis Mrsi (talk) 17:18, 18 February 2014 (UTC)

Exactly. With the latter expression you can multiply using matrix multiplication, qv, where v is a column vector representation of a quaternion. With the former expression, you would need to use v^†q. Disclaimer: This, or something similar, is true, can't check this thoroughly a t m, playing poker;). YohanN7 (talk) 18:47, 18 February 2014 (UTC)

It's hard for me to follow exactly what is being contested, but my take on groups like SL(n,H) is that these are subgroups of SL(4n,R) preserving a pair of anti commuting complex structures. Thus they are quaternionic in the sense that they have a fundamental representation that is quaternionic. I would count these among the classical groups. Sławomir Biały (talk) 16:51, 18 February 2014 (UTC)

Not very different from what I said: linear maps that preserve the quaternionic structure, but their determinant is calculated over a field (real or complex numbers, both are feasible). This case is lucky to admit this possibility. The main question is not quaternions, but generality. Read the section again from the beginning (including the headline) until the “respected by the matrix multiplication” words, please. Incnis Mrsi (talk) 17:18, 18 February 2014 (UTC)

Certainly groups like SL(n,H) are regarded as classical groups. (SL(n,H) has type AII in Cartan's classification if I recall correctly.) Sławomir Biały (talk) 17:29, 18 February 2014 (UTC)

The cases R, C and H can be treated fairly uniformly by using C only. That is to say, vector spaces over R and H can be regarded as vector spaces over C endowed with a little extra structure. YohanN7 (talk) 17:17, 25 March 2014 (UTC)

Not C-class

A while ago, I changed the article rating from B-class to start-class (motivated above). Someone now changed this to C-class with the motivation that "article ratings aren't arbitrary". Right, they aren't—at least they shouldn't be. This article is very incomplete in coverage, covers stuff not relevant to the subject, and contains several errors. There are plenty of articles that are complete, correct, and good reading, but will never get a C-class rating because the subject is narrow and will never motivate more than one page of text. (I disagree with this rating policy b t w.)

This article is one page (at least upon removal of irrelevant stuff) on a fully developed huge classical topic that is extremely useful in modern applications. (You can take it to the extreme point that modern theoretical physics is the theory of particular symmetries without exaggerating very much. These symmetries are often described by classical groups.) With this in mind, the article is, in my opinion, smack dab start-class, even a pretty good start-class article, but it is not C-class. I will not change the rating back unless others agree with me, and I can live with this as an unusually poor C-class article.

Of course, I'm not blaming the article's authors. They, at least, wrote it, giving it a solid start. I'm merely complaining. I wish I had the time to develop the article, but I don't. YohanN7 (talk) 14:43, 29 April 2014 (UTC)

New version

I rewrote the article from scratch but retained the section Classical groups over general fields or algebras. I think it should go elsewhere (or be scrapped) because it isn't in the spirit of classical groups. YohanN7 (talk) 18:33, 11 July 2014 (UTC)

Does YohanN7 like serif \mathrm{$\mathrm {T}$} for transpose due to aversion to any use of sans-serif in math notation? Incnis Mrsi (talk) 07:58, 18 August 2019 (UTC)

Does Incnis Mrsi make unnecessarily aggressive, personalized comments because they are opposed to any forms of polite engagement? --JBL (talk) 23:47, 23 August 2019 (UTC)

Unclear statement

Near the end of the section titled Quaternionic case this passage appears:

"The determinant of a quaternionic matrix is defined in this representation as being the ordinary complex determinant of its representative matrix. The non-commutative nature of quaternionic multiplication would, in the quaternionic representation of matrices, be ambiguous."

But it is not the "non-commutative nature of quaternionic multiplication" that would be ambiguous; it is the definition of the determinant of a quaternionic matrix that would be ambiguous. 2601:200:C000:1A0:1976:AC3F:FBA4:37D4 (talk) 17:17, 4 October 2021 (UTC)

Is it four or is it three "infinite families of Lie groups" ???

The Introduction contains this claim:

"the complex classical Lie groups are four infinite families of Lie groups".

But immediately after the table in the article, a sentence reads as follows:

"The complex classical groups are SL(n, $\mathbb {C}$), SO(n, $\mathbb {C}$) and Sp(n, $\mathbb {C}$)."

So: Which is it, three infinite families, or four infinite families?