Talk:Symplectic group

Real or complex entries[edit]

Symmetric matrices are defined for complex entries only in the Wikipedia. --Andres.

Are we talking about Symmetric matrix or Symplectic matrix here? Phys

Sorry, I mean symplectic matrices. --Andres

Fundamental groups[edit]

Does anyone know which of the symplectic groups are simply connected? I know that Sp(1) and Sp(2, C) are, and that Sp(2, R) is not. I suspect this holds for all n, but I have no proof. -- Fropuff 16:04, 2004 May 20 (UTC)

Nevermind. What I said is correct, I know how to prove it now. -- Fropuff 14:28, 2004 May 28 (UTC)

Simplicity and center[edit]

I'm trying to pull together information on matrix groups and was wondering if someone could answer the following questions.

When is Sp(2n,F) simple?
What is the center of Sp(2n,F)?

In particular I'm interested in the case when F is finite, however if the results include R,C or all fields it would be useful to include them. TooMuchMath 17:47, 15 April 2006 (UTC)[reply]

Expert attention needed[edit]

Can someone please flag this page as "in need of attention by an expert"? The last section is rather rough, and in particular, it implies that "O(3) is isomorphic SU(2)", which is wrong. Thanks. 98.223.87.110 (talk) 06:17, 10 May 2008 (UTC)[reply]

Unitary symplectic group[edit]

This topic redirects to this page; presumably this corresponds to Sp(n) (the compact group). This should be make clear.128.32.107.24 (talk) 21:18, 17 February 2009 (UTC)[reply]

Projective symplectic groups?[edit]

Hello. There is a well known isomorphism between the groups Sp(2) and Spin(3) and also one between Sp(4) and Spin(5). Now as everybody knows, Spin(3) is the double cover of SO(3), while Spin(5) is the double cover of SO(5), so you could see SO(3) as a kind of projective version of Sp(2) (and likewise SO(5) as a projective version of Spin(4)).

What I'm wondering is:

Do all the (unitary) symplectic groups have projective versions, and
if so, do these projective versions inherit some (non-projective) irreducible representations from their covers, in a similar way that SO(3) inherits the 3-dimensional irrep of Spin(2), and SO(5) inherits the 5-dimensional irrep of Sp(4).

85.224.19.225 (talk) 12:49, 4 April 2009 (UTC)[reply]

Reference or plug[edit]

I have removed the following strange plug from the article. It might just be a clumsily-added relevant reference. I'm not sure, but I would prefer it if someone (besides the original IP who added it) verify the contents of the article against it. Obviously, this will require a willingness to read the cited paper before restoring the reference to the article:

Use the Wikipedia search for Roger Howe to read his comments on "elegance" in a PDF-version of his "Remarks on Classical Invariant Theory" in Trans. Amer. Math. Soc. 313 (1989) p. 539-570 and the reference given therein. The graded generalization of Weyl and Clifford algebras allow the basis-free formulation of the canonical commutation and anticommutation relations in terms of a symplectic and a symmetric non-degenerate bilinear form. In addition the binary elements in this graded Weyl-algebra give a basis-free version of the commutation relations of the symplectic and pseudo-orthognal Lie algebras.

--Sławomir Biały (talk) 23:44, 10 November 2010 (UTC)[reply]

Unclear section of article[edit]

The section titled Important subgroups reads as follows, in full:

" The symplectic group Sp(n) is sometimes written as USp(2n) which is convenient for the following equations. The symplectic group comes up in quantum physics as a symmetry on poisson brackets so it is important to understand its subgroups. Some main subgroups are:

{\mathit {USp}}(2n)\supset {\mathit {USp}}(2n-2)

{\mathit {USp}}(2n)\supset {\mathit {U}}(n)

{\mathit {USp}}(4)\supset O(4)

The symplectic groups are also subgroups of various Lie groups:

{\mathit {SU}}(n)\supset {\mathit {USp}}(n)

F_{4}\supset {\mathit {USp}}(8)

G_{2}\supset {\mathit {USp}}(2)

There are also the isomorphisms of the Lie algebras usp(4) = so(5) and usp(2) = so(3) = su(2). "

Three comments:

1) It is important for us to learn alternative notations, such as what physicists use. That is different from saying that the article should suddenly switch notations, as it has done in this section: That is a very bad idea.

2) There is no reason given that the physicists' notation is "convenient for" the inclusion equations displayed. If there is a good reason for this, it would be helpful to say what it is.

3) How the symplectic groups (presumably the quaternionic ones, Sp(n) — but this is not clear) "come[s] up in quantum physics as a symmetry on Poisson brackets" should be either explained, or provided with a link to an explanation.Daqu (talk) 22:02, 14 May 2014 (UTC)[reply]

4) Since numerous series of symplectic groups are discussed in this article, it should never use the phrase "the symplectic group" without specifying which one is meant. Also: Since for each n there is an nth symplectic group in each series, it should be clearly stated whether an entire series or a single group is being referred to. Regardlesss of the jargon used by experts.Daqu (talk) 21:47, 15 May 2014 (UTC)[reply]

Using alternative notations in a math article is a very bad idea[edit]

Because it is a very bad idea to switch notations in the middle of a math article, I have removed the "USP" and "usp" notations for the symplectic Lie groups and Lie algebras (in the Subgroups section), other than just mentioning that such notations exist and can be useful in some circumstances.

If such notations are used in physics, then maybe they are appropriate for a physics article. But this is a math article, and the "USP" and "usp" notations are not modern mathematical notation. Nor is it "convenient" for any reader to suddenly experience different notation in the middle of an article.Daqu (talk) 21:08, 25 July 2014 (UTC)[reply]

Notation[edit]

Why not use a notation that conforms to the size of the matrices in all cases (including the compact case)? I have seen no reference mixing like this. Also,

"Although the

Sp(n)

notation is more common (and hence used here), it can be confusing in that

Sp(2n, C)

is often denoted

Sp(2 n)

.""

is something I have never seen. YohanN7 (talk) 05:24, 18 December 2014 (UTC)[reply]

Terrible definition[edit]

If you don't have enough time, energy, or knowledge to do a good job, then: please let someone else do it. The definition of symplectic group in the section Sp(2n, F) begins as follows:

"The symplectic group of degree $2 n$ over a field $F$ , denoted $Sp(2 n, F)$ , is the group of $2 n \times 2 n$ symplectic matrices with entries in $F$ , and with the group operation that of matrix multiplication. Since all symplectic matrices have determinant $1$ , the symplectic group is a subgroup of the special linear group $SL(2 n, F)$ .

"Notational warning: What is here called $Sp(2 n, F)$ is often referred to as $Sp(n, F)$ .

"More abstractly, the symplectic group can be defined as the set of linear transformations of a $2 n$ -dimensional vector space over $F$ that preserve a non-degenerate, skew-symmetric, bilinear form, see classical group for this definition."

Wait: You are definining the symplectic group here, but you're too busy to tell readers what the definition of a symplectic matrix is???

Do not make people click to other articles to learn the most basic aspects of the subject of the current article. That makes articles unreadable and therefore useless.

I hope someone knowledgeable on this subject will take the trouble to rewrite this so it is not a useless article.2600:1700:E1C0:F340:1913:1136:8EEE:A852 (talk) 00:42, 13 November 2018 (UTC)[reply]

What you're pointing out seems to be a general plague that is sweeping over wikipedia articles. The people writing the articles don't know how to explain the topics in notation that is comprehensible to someone without a graduate degree in mathematics. As someone with a PhD in physics, it is very frustrating that I would ever have to look up terms like "symplectic manifold" and "h-cobordism" to understand the introduction of a simple physics topic.

...and if you're reading this and asking "which topics can be explained in simple terms?", the answer is "all of them" (if you are a good physicist), and this should be the default viewpoint if you are editing wikipedia articles.2001:480:91:3304:0:0:0:658 (talk) 16:39, 25 June 2021 (UTC)[reply]