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every matrix A in GL(n,K), you define A^* to be the matrix formed from A by first taking the transpose and then applying f to each entry. (A^*)^* = A, and the matrices satisfying AA^* = 1 form a group called U(n,K/F). When K is finite, then F and K uniquely define each other and every F has a K (but not every K has an F); notation varies about whether the group is called U(n,q) or U(n,q^2) where q is the size of F and q^2 is the size of K.
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one k x k (unique), and only one inseparable algebra, k/(x^2) (also unique; has a unique auto of degree 2 in char not 2, but lots of other autos). It seems silly to use such language when the only interesting example is a degree two field extension. Also to make sure, there is no need to consider K=k x k for your calculation of U(n,C/R,1)(C), right? I only ask because the calculation for U(n,(C x C)/C,1)(C)=GL(n,C) is fairly similar.
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There are also some exception isomorphisms, but I think they are incredibly generic (and weren't obviously visible when I was transcribing Grove into here). I think it is really easy like PSU(2,q^2)=SL(2,q) or so, and the isomorphism is natural for all commutative rings with involution, not just the finite field case. I have an article lying around the office with conjugacy class information, but I can't remember if it was legible.
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construction as well. This is similar to the classification of central simple algebras of dimension 4: there are all the various division algebras (quaternion algebras) and then there's the 2x2 matrices (and this is related to the current situation as one can construct a unitary group as the units in a central simple algebra).
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these unitary groups and the theory of their automorphic representations, Shimura varieties, and associated Galois representations. They are important in the study of the
Langlands program as one of the examples of reductive algebraic groups that give rise to Shimura varieties (unlike the the groups GL(n) themselves).
1209:. Tracing the definition of everything, and element (A,B) in GL(n,K)xGL(n,K) is in U(n,K/k)(K) if AB=I and BA=I, which are both the same equation and are satisfied if, and only if, B=(A). So U(n,K/k)(K)={(A,(A)):A is in GL(n,K)} and this is clearly just GL(n,K). I've said quite a bit, so I'll leave it there for now.
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I guess 1 is already done, 2 I think you know about, 3 either of us could write (I have a reference handy), 4 I can do, 5 and 6 could be combined if you like. For 6, if K,F are algebraically independent extensions of k, then U(n,K/k,Φ)(F) is U(n,KF/F,Φ)(F) which is somewhat pretty. In particular,
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For applications to finite simple groups, Φ is required to be non-degenerate and K is required to be a field (that is, K=k × k and K=k/(x) are not allowed), and then all such groups U(n, K/k, Φ) for a fixed finite field k are isomorphic and depend only on the order q of k, or on the order q^2 of K,
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One method to motivate U(n,C)=GL(n,C) would be to say that U is the twisted group of Lie type with Dynkin diagram of type A, and over an algebraically closed field there are no possible twists, so instead ofa twisted version of GL(n,C), you just get GL(n,C). Both GL(n,C) and O(n,C) as answers seem
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I read over Milne's notes (I had been reading some books on linear algebraic groups and those sufficed yesterday). His notes are quite clear and answered my silly question: Surely there are only two types of separable k-algebras of dimension 2, one a field extension (possibly many isoclasses) and
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One reason I hadn't tried writing this up before was a lack of references which I still have. Milne does a nice job, but does not address the more general Φ situation, and the only references I've found so far only address this situation when k is a local or global field, and though everything is
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Looks good to me. I was painfully precise about which field where in the finite field section (hopefully correct too, it's late). I added SU, PU, and PSU, and included the details on perfection and simplicity. I need to add orders, and check if I know the Sylow and conjugacy class structure.
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I think it is definitely important to be able to view U(n) as an algebraic group and be able to evaluate it on k-algebras other than k. For example, the recent proof of the Sato-Tate conjecture, as well as the attacks on the main conjecture of
Iwasawa theory for GL(2) all use in an essential way
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I think someone has done this to an extent, but it is not clear if the contributions were correct. From the standpoint of finite groups, you do not definite the unitary group for just one field, but for a degree two field extension F < K with some field automorphism f of order two. Then for
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if k is the rationals, and K is any imaginary quadratic extension and F is the reals, then U(n) is just the R points of U(n,K/k,Φ). When they are not algebraically independent it just looks silly to me, but probably the point is that the theory is uniform, not that the examples are interesting.
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One could avoid saying degree two separable algebra and just consider degree two extensions, but I find it aesthetically better to do the more general thing and mention that really there's only the two cases (degree two extensions and kxk). I think it's nice that one can obtain GL(n) from this
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I think your definition should work well in all of the contexts I care about. The only minor problem is the full notation would be quite long: U(n,K/k,Φ)(L) is defined for all fields k, all separable 2 dimensional k-algebras K (which defines the automorphism f and the matrix operator *), all
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At any rate, back to the current revision. I don't think U(n,C) has a well-defined meaning. For A in U(n,K/F), 1 = det(AA^*) = det(A)*det(A)^f, so Norm_{K/F}(det(A))=1. When K=F, that means f=1, and det(A)^2=1, so U(n,C/C) is at best O(n,C), but really should be undefined since =1, not
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Would it better then to require K to be a field extension of degree 2? I think including Φ is a good idea. Me personally, I don't see much interesting about evaluating the group scheme thingy on other k-algebras than k, but in some sense the U(n,K/k,1)(K)=GL(n,K) is
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There are of course many many unitary groups. I've personally mainly only dealt with unitary groups related to a totally complex extension of a totally real field, though just looking some stuff up now here's what seems to be going on (see page 103 of Milne's notes
627:(K) (or to avoid picking a basis, pick an n-dimensional K-vector space and a non-degenerate hermitian form Ψ on V) and define the unitary group in n-variables over k with respect to K/k and Φ (or Ψ) as the algebraic group over k whose points over a k-algebra R are
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Thanks for being bold! I have a dream for writing up the finite simple groups on wikipedia in a sane, comprehensive, and consistent way, and it really helps to have skeleton (or even partially fleshy zombie) articles around to improve.
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I have a soft spot for the trivial element of the Brauer group, and apparently group schemes are quite popular, so we might as well include all of it. To make it more digestible though, how about presenting it in stages:
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fine with what I said above, I kinda wanted to get an all-encompassing reference before including it in the article. Sorry I'm pretty busy right now, so I won't be able to do much, but I definitely think this page needs
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Does it sound reasonable to include the "full" definition and notation, and then a nice list of examples? I am fine with writing up the finite field case, but I'd want you to handle the case where Φ actually mattered.
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and so they can be denoted either U(n, q) or U(n, q). The independence on Φ is interesting and often discussed in books on classical groups, so it seems quite reasonable to mention Φ even in the finite field case.
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I've taken a stab at writing it up, largely following Jack's outline (which outlines the development of the above discussion). My write-up could surely use some work and references, but hopefully it's a good
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As described in the article "Symplectic group", there are various meanings to the symbol Sp(2n): it could mean Sp(2n,R), Sp(n,C), or USp(n). It is not clear in this article which of these it refers to.
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In other words, I'm leaning more towards the "standard" definition of a unitary space of a k-vector space and a K/k sesqui-linear form. Grove's
Classical Groups text has such a point of view.
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I think the huge advantage of this way is that it is easy to write a small part, find sources for it, etc. without compromising either the special case or the general case.
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I have deleted, that compactness of U(n) is consequence of Heine-Borel theorem, because Heine-Borel theorem is for real spaces. Přemysl Šťastný 09:46, 27 June 2019 (UTC)
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Could someone add a definition of a unitary group over a finite field? These are pretty important in group theory however I don't have the background to do this.
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The hard part will be to do this in a way that will be readable and understandable. Perhaps one could start with just the usual U(n) as a matrix group (over
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2283:{\displaystyle \operatorname {U} (n)=\operatorname {O} (2n)\cap \operatorname {GL} (n,\mathbf {C} )\cap \operatorname {Sp} (2n,\mathbf {R} ).}
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I just noticed this as well, this really needs to be fixed. It's a math article, so the first thing that should pop up is the definition. --
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Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be
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more than it currently has, and this is definitely the first step. I'll keep trying to find references (unless you already have one)
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There is not definition of U(n) in the article! Not in explicit form at least. Shouldn't it be given in the introduction? --
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for example). Milne says take a field k and let K be a degree 2 separable k-algebra, then there's a unique k-automorphism
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Finite field interlude (all Φ are equivalent, only one quadratic extension of any finite field, so U(n,K/k,Φ)=U(n,q)).
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My modification of the section "Other Fields" was in the following vein: there is an algebraic group over
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AΦA*=Φ is just a set of polynomial equations, so this is an algebraic group (not group scheme)
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and Φ the diagonal (m+n)x(m+n) matrix with m +1's and n -1's, this would give the group over
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Polynomial equations make sense over any k-algebra, so this is a group scheme
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2) We know the definition of a "unitary matrix" and a "symplectic matrix".
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AΦA*=Φ over any dim2 sep K/k, especially K a quadratic extension field of k
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Presumably for your applications it would be interesting to mention U(m+n,
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Because this agrees with how matrices representing forms transform: if
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of classical unitary matrices, as both equations are equivalent to
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in the complex structure and the symplectic form, and that this
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But what is a "unitary structure" or a "symplectic structure"?
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Now, this doesn't make a difference for the definition of the
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Would someone please mention the role U(1) plays in the
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2152:
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1987:
1985:
1900:
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1431:interesting.
1405:
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778:
774:
770:
246:
232:
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161:
158:Other fields
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
2063:92.50.107.1
2057:—Preceding
1647:JackSchmidt
1591:JackSchmidt
1436:JackSchmidt
1414:JackSchmidt
1408:) = U(m,n).
967:is sent to
524:(where for
183:JackSchmidt
164:TooMuchMath
112:Mathematics
103:mathematics
59:Mathematics
2347:Categories
2297:compatible
2172:symplectic
2164:orthogonal
2108:Definition
569:we define
207:points G(
2059:unsigned
1524:probably
2174:groups:
2168:complex
2114:Cokaban
1080:. Then
920:, then
810:)=GL(n,
139:on the
30:C-class
2170:, and
2050:Sp(2n)
2037:Nbarth
1943:, not
1901:second
1624:Nbarth
1620:start.
1533:RobHar
1211:RobHar
798:)=U(n,
781:whose
36:scale.
1897:first
1529:a lot
773:over
2335:talk
2132:talk
2118:talk
2099:talk
2077:U(1)
2067:talk
2041:talk
1651:talk
1628:talk
1595:talk
1537:talk
1440:talk
1418:talk
1215:talk
213:real
205:real
187:talk
2320:1)
2085:of
1988:set
1826:is
878:to
229:the
131:Mid
2349::
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2249:∩
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1372:⊗
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440::=
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177:2.
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223:/
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185:(
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
