User francois ziegler - MathOverflow

Answer by Francois Ziegler for The orbit $(G\cdot X) \cap \mathfrak{t}$ for $X\in \mathfrak{t}$ singular

2013-06-20T04:25:31Z

Yes, it's still true that $G\cdot X\cap\mathfrak t=W\cdot X$.

The inclusion $\supset$ is clear. Conversely, suppose $g\cdot X\in\mathfrak t$. Since $\mathfrak t$ consists exactly of all $T$-fixed points in $\mathfrak g$, it follows that $t\cdot g\cdot X=g\cdot X$ for all $t\in T$. Hence $g^{-1}Tg$ is contained in the stabilizer $G_X$. So $T$ and $g^{-1}Tg$ are two maximal tori in $G_X$. Hence they are conjugate by some $h\in G_X$: $T = h^{-1}g^{-1}Tgh$. So now $gh$ normalizes $T$, and the Weyl group element it represents still sends $X$ to $g\cdot X$. QED.

Answer by Francois Ziegler for Where did Sophus Lie write the group commutator for two one parameter groups.

2013-06-19T16:04:39Z

I believe you're not going to find exactly what you want in Lie, because he never formalized flows (or finite transformations) and their commutation as you do. Maybe the closest would be this, from Über Differentialinvarianten, Math. Ann. 24 (1884) 537-578:

... erhalten wir folgenden Fundamentalsatz, den ich 1872 entdeckt habe:

Satz 3. Enthält eine kontinuierliche Gruppe die beiden infinitesimalen Transformationen: $$ Bf=\sum\xi_\varkappa\frac{\partial f}{\partial x_\varkappa} \quad\textit{und:}\quad Cf=\sum\eta_\varkappa\frac{\partial f}{\partial x_\varkappa}, $$ so enthält sie ebenfalls die infinitesimale Transformation: $$ \sum_i(B\eta_i-C\xi_i)\frac{\partial f}{\partial x_i}, $$ deren Symbol bekanntlich auf die beiden äquivalenten Formen: $$ B(C(f)) - C(B(f)) = (B, C) $$ gebracht werden kann.

As you can see, his definition of the bracket of vector fields is always as the commutator of the derivations they define on functions (something that goes back to Jacobi). What this Satz states, then, is that the finite transformations (or flow) generated by the infinitesimal commutator $(B,C)$ belong to the group generated by (the flows of) $B$ and $C$. Not surprisingly, Lie's proof is by expanding the flows to second order.

Lie may or may not have stated this Satz elsewhere before 1884, but I doubt he ever wrote a formula for, much less definition of, the bracket as limit of commutators of finite transformations.

Update As to your question of who (esp. first) expressed the bracket as a derivative of commutators of flows: I don't know (my impression is that these things developed slowly in a sort of consensus), but one might argue that the formula $$ [V,T]=\frac{d}{ds}\frac{d}{dt}e^{-sV}e^{tT}e^{sV}\Bigr|_{s=t=0} $$
is on p. 240 of Poincaré, Sur les groupes continus, Trans. Cambridge Philos. Soc. 18 (1900) 220-255.

Further update Trotter's formula that you also mention now is indeed called "Lie-Trotter" by e.g. Chernoff [1968,1974] or Chorin et al. [1978]. The latter write (sic):

... the equation $dx/dt=Ax+Bx$ leads to the 1875 formula of S. Lie [38]: $$ \exp\{A+B\} = \lim_{n\to\infty}(\exp\{A/n\}\exp\{B/n\})^n.\tag{$*$} $$ [38] Lie, S., and Engel, F., Theorie der Transformationsgruppen, 3 Vols., Teubner, Leipzig, 1888.

The problem is that [38] is not from 1875, nor does it contain anything remotely like formula ( $*$ ) (I am ready to bet a lot of money). I may be wrong but until someone finds that elusive 1875 paper, I would tend to date ( $*$ ) from around von Neumann [1929, p. 19].

Answer by Francois Ziegler for Calculus book in the spirit of the 18th century

2013-06-13T18:20:22Z

I like Analysis by its History, by Hairer and Wanner. Chapter I "Introduction to Analysis of the Infinite" covers precalculus as Euler would. Chapter II "Differential and Integral Calculus" proceeds essentially as you desire (through envelopes, caustics, curvature, and differential equations). Then Chapters III and IV 'redo' everything with Weierstrassian rigor, and move on to several variables.

Answer by Francois Ziegler for Reference request - localisation de g-modules

2013-05-01T17:43:57Z

http://gallica.bnf.fr/ark:/12148/bpt6k6226873r/f29

Answer by Francois Ziegler for surjective submersion and fibrations

2013-04-08T15:58:50Z

No. To see this, take a bona fide fibration such as $\operatorname{pr}_2:\mathbf{R}^2\to\mathbf{R}$ and remove a point from the domain.

Meigniez on p. 3778 lists a number of sufficient conditions that a submersion $f$ be a fibration. The best known, already noted by Damian and Donu, is that $f$ be proper as in Ehresmann's Theorem (proved e.g. in Bröcker and Jänich, (8.12)).

Answer by Francois Ziegler for Reference request: Lascoux's formulas for Chern classes of tensor products and symmetric powers

2013-03-08T22:03:11Z

You say you don't have access to Lascoux's paper, but it is actually available online at the BNF:

http://gallica.bnf.fr/ark:/12148/bpt6k62341359/f397.image

Answer by Francois Ziegler for spectacular applications of functional analysis in resolutions of apparently unrelated problems

2013-02-24T16:22:09Z

One should probably mention Gelfand's proof of Wiener's theorem that, if a nowhere zero periodic $f$ has absolutely convergent Fourier series, then so does $1/f$.

There is also Kantorovich's famous note "On a problem of Monge":

Monge, in his memoirs of 1781, considering the problem of the most rational ways of transporting earth from an embankment to an excavation, proposed the following problem: divide two equal volumes into infinitesimal particles and associate them one to another so that the sum of the path lengths multiplied by the volumes of the particles be minimum possible.

In connection with this problem, Monge created the geometrical theory of congruences. As to the problem itself, he conjectured, but did not proved rigorously, that the paths of the mass translocation form a family of normals to a certain family of surfaces.

The same problem was studied later by Dupin, but a rigorous proof of the Monge theorem was given only a century later, in 1884, in a 200-page memoirs by Appell. (...)

Meanwhile, this assertion follows immediately from the abstract theorem mentioned above. (...)

Answer by Francois Ziegler for Stabilizers for Nilpotent Adjoint Orbits of Semisimple Groups

2013-02-21T10:33:16Z

The component groups $\pi_0(C_G(X))$ are known by work of Alekseevskiĭ, MR557505 = MR2140712, which contains additional references and can be read on Google books. Another good place to look is the book by Collingwood & McGovern, Nilpotent orbits in semisimple Lie algebras.

(Out of curiosity, what other topological invariants are you after?)

Answer by Francois Ziegler for Why all irreducible representations of compact groups are finite-dimensional ? [EDIT: Subtleties: AC,etc]

2013-01-20T20:16:22Z

(Addressing only the title question.) There is a short proof avoiding Peter-Weyl and the theory of compact operators. It is due to Nachbin and is reproduced in Hewitt-Ross, Abstract Harmonic Analysis 1, p. 344. A slight simplification of it runs as follows: Pick a unit vector $v$ in your representation space $V$. Schur's lemma gives $$ \int_G gv(gv,\cdot)dg = \lambda1 \tag 1 $$ where $\lambda = \int_G|(v,gv)|^2dg>0$ (sandwiching (1) with $(v,\cdot v)$). Now let $W\subset V$ be finite-dimensional, and write $E=E^2$ for the orthogonal projection $V\to W$. We get $$ \int_GEgv(gv,E\cdot)dg = \lambda E, \tag 2 $$ whence, taking traces in (2), $\lambda\dim(W)=\int_G||Egv||^2dg\leqslant\operatorname{vol}(G)$. Thus, the dimension of any finite-dimensional subspace is bounded, as was to be shown.

Answer by Francois Ziegler for Is there a "geometric" intuition underlying the notion of normal varieties?

2012-12-27T15:03:40Z

Regarding the question "Who was the person who invented this notion?", a paper of H. T. Muhly provides interesting background (as well as a geometric interpretation) for projectively normal:

In the terminology of the Italian School an algebraic variety is called "normal" if its system of hyperplane sections is complete. O. Zariski applies the term "normal" to an algebraic variety whose associated ring of homogeneous coordinates is integrally closed. The two concepts are not equivalent. Zariski refers to a variety which satisfies the former condition as "normal in the geometric sense" and to one which satisfies the latter condition as "normal in the arithmetic sense".

(...)

The object of this note is to characterize geometrically those algebraic varieties which are normal in the arithmetic sense. To this end we propose the following theorem: A necessary and sufficient condition that the $r$-dimensional algebraic variety $V_r$ be normal in its ambient projective space $P_n$ is that for every integer $m$ the linear system cut out on $V_r$ by the hypersurfaces of order $m$ in $P_n$ be complete.

Answer by Francois Ziegler for The origin of sets?

2012-12-23T02:26:24Z

Euler in Lettres à une princesse d'Allemagne sur divers sujets de physique et de philosophie, 17-24 feb 1761, writes about objects he calls spaces (my emphasis):

As a general notion encompasses an infinity of individual objects, one regards it as a space within which all these individuals are enclosed: thus, for the notion of man, one makes a space (fig. 39) in which one conceives that all men are comprised. For the notion of mortal, one also makes a space (fig. 40), where one conceives that everything mortal is comprised. Then, when I say that all men are mortal, that comes down to the former figure being contained in the latter.

(...)

These round figures or rather these spaces (for it doesn't matter what shape we give them) are very well-suited to facilitating our reflections (...)

etc., and illustrates this with what we would call ensemblist diagrams (fig. 39 to 89), famously reproduced on Swiss banknotes. The applications he gives here are to everyday logic, so perhaps less mathematical than intended by the question. (I don't know if he ever wrote again on the subject.)

Answer by Francois Ziegler for What is the structure of the space of solutions of a non linear ODE?

2012-12-08T22:50:47Z

As Robert Bryant observed, something like the solubility of $F$ for $u^{(n)}$ needs to be assumed. Then by the trick (due I believe to D'Alembert) of setting $x=(t,u,\dots,u^{(n-1)})$ and dt/ds=1, the equation can always be rewritten $$ \frac{dx}{ds}=f(x). $$ This is what most geometers would call the "standard ODE", wherein $f$ is a smooth vector field on the manifold where $x$ evolves.

In this setting, the space of (maximal connected) solution curves is indeed always a (not necessarily Hausdorff) manifold. I don't know who first wrote this, but according to this recent obituary a contender would be J.-M. Souriau in his 1970 book "Structure des systèmes dynamiques". Translation of the relevant passage:

The manifold of motions of a system

Jean-Marie Souriau has observed that the set of maximal solutions of a differential system on a differentiable manifold possesses itself a natural structure of differentiable manifold, not always separated: thus one can speak of the system's manifold of motions. This very simple property is rarely mentioned in the usual courses on differential calculus.

Answer by Francois Ziegler for maximal subgroups of finite simple groups

2012-11-20T23:20:28Z

The recent book of Malle and Testerman, Linear algebraic groups and finite groups of Lie type, has several chapters on the subject. From the MR review:

some important recent developments are treated here for the first time. For instance, the authors describe the classification of the maximal subgroups of simple algebraic groups, and this is used in their subsequent analysis of the subgroup structure of finite groups of Lie type.

(etc. The review goes on to describe the Aschbacher/Liebeck-Seitz results mentioned in other replies.)

Also Robert A. Wilson's The finite simple groups addresses the question, for the classical groups in a devoted section (3.10), for the others by systematically giving references to original papers classifying the maximal subgroups.

Answer by Francois Ziegler for Fundamental motivation for several complex variables

2012-11-27T06:54:59Z

To say that $F$ is holomorphic is to say in a sense that it depends on only "half" of the variables $(\partial F/\partial\bar z_j=0)$. This is analogous to the wave functions of quantum mechanics depending only on $q$, i.e. only on "half" the phase space variables $(p,q)$. The analogy is made precise by the theory of polarizations in geometric quantization and the orbit method.

For instance, a reductive Lie group has some of its irreducible unitary representations attached to hyperbolic coadjoint orbits (consisting of matrices with real eigenvalues), others to elliptic orbits (consisting of matrices with imaginary eigenvalues). While the former admit real polarizations which lead to representations in $L^2$ sections over a space half the dimension (i.e., real analysis), the latter admit complex polarizations which lead to representations in holomorphic sections over the orbit (or Dolbeault cohomology thereof; i.e., complex analysis).

A classic example of the elliptic story is the Borel-Weil(-Bott) realization of all irreducible representations of compact Lie groups. This involves holomorphic functions on the complexified group in an essential way.

Answer by Francois Ziegler for Weakest assumption for pointwise convergence of Fourier series

2012-11-12T02:07:21Z

The function must be integrable in a certain sense defined by Denjoy and others. Here is an interesting survey paper on the subject:

One of the problems in the theory of trigonometric series $$\frac12a_0+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx)\tag{1.1}$$ is that of suitably defining a trigonometric integral with the property that, if the series (1.1) converges everywhere to a function $f(x)$, then $f(x)$ is necessarily integrable and the coefficients, $a_n$ and $b_n$, given in the usual Fourier form. It is well known that a series may converge everywhere to a function which is not Lebesgue summable nor even Denjoy integrable (...) The problem has been solved by Denjoy [4; 5], Verblunsky [14], Marcinkiewicz and Zygmund [10], Burkill [1; 2], and James [8]. (...) The solutions are described, mainly in the order in which they were published, in §§2-7 below.

Answer by Francois Ziegler for Polarisation in a nighbourhood of a Lagrangian submanifold

2012-11-13T16:14:15Z

Arnol'd (p. 3314) puts it that way:

Weinstein's Theorem. Some neighborhood of any Lagrangian submanifold in any symplectic manifold is symplectomorphic to some neighborhood of this Lagrangian submanifold in any other symplectic manifold, for instance in its own cotangent bundle space.

(The resulting neighborhood then has the obvious transverse polarization by fibers of the cotangent bundle.) Unless I am mistaken, Weinstein proves this in Symplectic manifolds and their lagrangian submanifolds, Theorem 6.1 and Corollary 6.2 (which he points out goes back to Souriau).

Answer by Francois Ziegler for Reference request for the list of maximal subgroups of SU(3,1)

2012-11-05T07:44:02Z

Mohamed Selim Taufik, On maximal subalgebras in classical real Lie algebras: "This paper is concerned with the classification of irreducible maximal subalgebras of the classical real Lie algebras su(p,q), sv(p,q) and si(p,q). We use the results of E. B. Dynkin, who classified the maximal subalgebras of the classical Lie algebras in the complex case. (...)"

Boris P. Komrakov Maximal subalgebras of real Lie algebras and a problem of Sophus Lie: "A classification of the maximal proper subalgebras of the simple real finite-dimensional Lie algebras is presented without proof. Contributions by A. A. Morozov, E. B. Dynkin, M. Berger and M. S. Taufik are mentioned."

Answer by Francois Ziegler for Jacobi method on first order partial differential equations

2012-10-10T01:52:41Z

Chapter VII of É. Goursat's book, Leçons sur l'intégration des équations aux dérivées partielles du premier ordre, exposes the method and ends with 14 examples of applying it (pp. 168-169).

Answer by Francois Ziegler for Synthetic approach to hyperbolic geometry?

2012-10-08T19:36:04Z

I'd say your best bet is with works from the early 20th century, when this sort of thing was in fashion:

Julian Lowell Coolidge, The elements of non-Euclidean geometry (1909)
Horatio Scott Carslaw, The elements of non-Euclidean plane geometry and trigonometry (1916)
Duncan M'Laren Young Sommerville, The elements of non-Euclidean geometry (1919)

Answer by Francois Ziegler for Examples of non-Kahler compact symplectic manifolds.

2012-09-22T02:45:28Z

I would recommend the Tralle-Oprea book, Symplectic manifolds with no Kähler structure.

Answer by Francois Ziegler for Structure of the unitary representation $L^2(N/M)$ when $N$ is a nilpotent Lie group

2012-08-31T21:43:37Z

The representation you're looking at is $\mathrm{Ind}_M^N1$ and as such, its decomposition into irreducibles is very well understood using Kirillov's orbit method. (Essentially, the irreducibles that enter correspond to the coadjoint orbits in the image of the moment map $T^*(N/M)\to\mathfrak n^*$ .)

I'd say the basic paper on the subject is this one by Corwin, Greenleaf, and Grélaud. It has references to the earlier work by Kirillov himself, and you'll find more in mathscinet's forward references to reviews citing it.

Answer by Francois Ziegler for History Question: AUTObiography of Mathematicians

2012-08-26T01:08:58Z

Analyse des travaux scientifiques de Henri Poincaré faite par lui-même, Acta Math. 38 (1921), 3-135.

Apparently written in 1901 at the request of Mittag-Leffler, this is not quite an autobiography but more in the style of the "Notices sur les travaux scientifiques" that many French scientists wrote, often as candidates to the Academy of Sciences. A 19th century example is Darboux (1884).

Answer by Francois Ziegler for Symplectic formulation of statistical physics

2012-07-19T14:40:29Z

You want to read Chapter IV "Statistical Mechanics" in Structure of Dynamical Systems (1970 French original available here) by J.-M. Souriau, one of the pioneers of symplectic mechanics.

Given a symplectic manifold $X$ on which a Lie group $G$ acts with moment map $\Psi$, Souriau defines the Gibbs states as the probability measures on $X$ that maximize entropy for a given mean value of $\Psi$. He shows (thm 16.219) that they have the form
$$ \text{const}\times e^{-\langle\Psi(\cdot),\beta\rangle}\lambda, \qquad (\lambda=\text{Liouville measure}) $$ for some $\beta\in\mathfrak g$ which generalizes the "inverse temperature" when $G=\mathbf R=$ {time translations}. The rest of the Chapter is devoted to the study of these states; in particular when $G=$ SO(3), $\beta$ can be interpreted as a rotation vector, and the fact that planets revolve in a common plane as the equality of their equilibrium $\beta$s. See also Définition covariante des équilibres thermodynamiques, Suppl. Nuovo Cimento 1 (1966), 203–216.

Later Souriau developed a general-relativistic viewpoint on dissipative processes which explains why they preserve the mean value of $\Psi(x)$. Thus, quoting from this summary to whet your appetite: "the first principle of thermodynamics [loses] its primitive status and [becomes] a necessary consequence of the invariance of the symplectic structure in gravitational gauge transformations." For more details, see Thermodynamique et géométrie, Lecture Notes in Math. 676 (1978), 369–397 or scanned preprint.

Answer by Francois Ziegler for the use of parentheses to mean "I won't tell you this again"

2012-07-20T19:57:13Z

Re: "Does this use of parentheses have a name?",

preterition |ˌpretəˈri sh ən|

noun (...) the rhetorical technique of making summary mention of something by professing to omit it.

ORIGIN late 16th cent.: from late Latin praeteritio(n-), from praeterire ‘pass, go by.’

Answer by Francois Ziegler for determinants and polynomials in matrices

2012-06-07T01:35:45Z

This is going to sound like massive overkill, but it is "very well known" that the only 1-dimensional polynomial representations of $GL(V)$ (which is what you're looking at) are the nonnegative powers of $\mathrm{det}$.

Reference (I assume from the mention of statistics that you are OK working with base field $\mathbf{R}$ or $\mathbf{C}$): e.g. Procesi on p.278 of Lie Groups lists all irreducible rational representations as all $$ S_\lambda(V)\otimes\mathrm{det}^k,\qquad k\in\mathbf{Z}, $$ where $\lambda$ runs over a certain set of partitions or Young tableaux; and on p.270 he gives a dimension formula for $S_\lambda(V)$ which is $>1$ unless $S_\lambda(V)$ is trivial.

Answer by Francois Ziegler for Papers whose title defines a new terminology

2012-06-06T18:40:12Z

Diener, Francine; Diener, Marc Chasse au canard. I. Les canards. (French) [Duck hunt. I. The ducks] Collect. Math. 32 (1981), no. 1, 37–74.

Benoît, Éric Chasse au canard. II. Tunnels—entonnoirs—peignes. (French) [Duck hunt. II. Tunnels—funnels—combs] Collect. Math. 32 (1981), no. 2, 77–97.

Callot, Jean-Louis Chasse au canard. III. Les canards ont la vie brève. (French) [Duck hunt. III. Ducks have a short life] Collect. Math. 32 (1981), no. 2, 99–114.

Benoît, Éric; Callot, Jean-Louis Chasse au canard. IV. Annexe numérique. (French) [Duck hunt. IV. Numerical appendix] Collect. Math. 32 (1981), no. 2, 115–119.

Answer by Francois Ziegler for Papers whose title defines a new terminology

2012-06-06T18:21:42Z

Jean-Pierre Serre: Gèbres, Enseign. Math. (2) 39 (1993), 33–85.

Answer by Francois Ziegler for Deformation of Lagrangian manifolds

2012-06-01T16:18:20Z

You already know that the pair $(M,L)$ of a symplectic manifold and a Lagrangian submanifold is locally isomorphic to $(T^*L, L)$. This is the beginning of Corollary 6.2 of Weinstein, who continues: "and the lagrangian submanifolds of $M$ "near" $L$ are in 1-1 correspondence with "small" closed forms on $L$."

The correspondence in question (explained on the previous page of Weintein's paper) is that "a submanifold of $T^*L$ transversal to the fibres is locally the graph of a 1-form $\sigma:L\to T^*L$. The graph of $\sigma$ is isotropic if and only if... $\sigma$ is a closed 1-form."

In short, the map you want attaches to a closed 1-form (on $L$!) its graph in $M\simeq T^*L$.

Update: This construction identifies a neighborhood of $f_0:L\hookrightarrow M$ in the space of embeddings (Whitney C$^1$ topologized), with a neighborhood of zero in the space of closed 1-forms on $L$. See Thm II.3.8 in Michèle Audin's notes (available here). She concludes that $Z^1(L)$ "can be considered as a neighbourhood of $f_0$ in the “manifold” of deformations of $f_0$, or as its tangent space at $f_0$."

Answer by Francois Ziegler for Steinmetz, Laplace and Fourier Transforms

2012-06-04T21:10:58Z

(Too long for a comment.) I think the italian Wiki page is wrong. It says the transform "was conceived by the author in 1893 [probably this text] and exposed in his treatise Theory and Calculation of Alternating Current Phenomena four years later."

However, as you can check there is not a trace in these texts (nor in the others authored by Steinmetz and available on archive.org) of what Wikipedia calls "la transformata di Steinmetz" -- i.e. the Fourier isomorphism $L^2(S^1)\to\ell^2(\mathbf{Z})$ -- which by the way, was written explicitly much before 1893.

These texts are famous for introducing complex numbers (in particular the notation $j=\sqrt{-1}$) into electrical engineering, but my impression is that naming the transform in Steinmetz's honor happened much later, perhaps in Italy, with little regard to what he himself actually did (or as the case may be, never did) with it.

Answer by Francois Ziegler for When do commuting Hamiltonian flows have commuting generators?

2012-06-01T05:05:30Z

This is not so much an answer as a suggestion to change the question. When $(P,\Omega)$ is prequantizable, i.e. there exists over $P$ a hermitian line bundle with connection $(L,\nabla)$ having curvature $\Omega$ (see e.g. Kostant 1970), then your hamiltonian vector fields $X^g, X^h$ and their flows $\varphi^g, \varphi^h$ lift canonically to $\nabla$-preserving vector fields $\xi^g, \xi^h$ and flows $\psi^g, \psi^h$ on $L$. These commute if and only if $[g,h]=0$.

That, I believe, is the correct "classical analogue" of the quantum facts you allude to.

Conversely, the true quantum analogue of looking at $\varphi^g, \varphi^h$ is looking at the action of $e^{ibG}, e^{iaH}$ not on Hilbert space $\mathcal{H}$ but on projectivized Hilbert space $\mathbb{P}\mathcal{H}$. There they commute iff (disregarding the usual domain questions) $[G,H]$ is a constant multiple of the identity.

In fact the analogy is good enough that $\mathbb{P}\mathcal{H}$ is a (usually infinite-dimensional) symplectic manifold, to which the first paragraph above applies, with $g, h$ the expectation values of $G, H$ and $L^\times \to P$ the tautological projection $\mathcal{H}\setminus\lbrace0\rbrace\to \mathbb{P}\mathcal{H}$. Moreover $\xi^h$ and $\psi^h(a)$ are just $H$ and $e^{iaH}$ (acting on $\mathcal{H}\setminus\lbrace0\rbrace$) -- so we've come full circle.

[P.S.: Regarding functions whose Poisson brackets are constant, you might be interested in this paper of Roels and Weinstein.]

Update regarding your extra question ("Isn't $e^{ibG}e^{iaH}=e^{iaH}e^{ibG}\Leftrightarrow[G,H]=0$ also true on $\mathbb{P}\mathcal{H}$?"): This is a statement about transformations of $\mathcal{H}$, not $\mathbb{P}\mathcal{H}$. Write $\underline{e^{iaH}}$ for the diffeo of $\mathbb{P}\mathcal{H}$ induced by $e^{iaH}\in\mathrm{U}(\mathcal{H})$, and likewise $\underline{iH}$ for the vector field on $\mathbb{P}\mathcal{H}$ induced by $iH\in\mathrm{End}(\mathcal{H})$. Then (exercise!) $e^{iaH}\mapsto\underline{e^{iaH}}$ is a group morphism with kernel the multiples of the identity, and likewise $iH\mapsto\underline{iH}$ is a Lie algebra morphism with kernel the multiples of the identity. Therefore we have

\begin{array}{cccl} \underline{e^{ibG}}.\underline{e^{iaH}}=\underline{e^{iaH}}.\underline{e^{ibG}} & \Leftrightarrow & [\underline{iG},\underline{iH}]=0 &\text{(actions on }\mathbb{P}\mathcal{H})\\ \Updownarrow & & \Updownarrow\\ e^{ibG}e^{iaH}e^{-ibG}e^{-iaH}\in\mathbb{C}\cdot\mathbf{1} & \Leftrightarrow & [iG,iH]\in\mathbb{C}\cdot\mathbf{1} & \text{(actions on }\mathcal{H}) \end{array}

and my claim is that these, not $[G,H]=0$, are the quantum analogs of $\varphi^g(b)\varphi^h(a)=\varphi^h(a)\varphi^g(b)$.

Comment by Francois Ziegler

2013-06-18T17:56:12Z

@unknown: I don't know. But what I sketched in my comment to your question above gives you the solution anyway. The point is that whenever the minimal polynomial factors without multiplicities, as in your case: $(R-1)(R-\lambda_+)(R-\lambda_-)=0$, then one has explicit formulas (which I have now added) for the eigenprojectors and hence a square root.

Comment by Francois Ziegler

2013-06-18T17:51:01Z

Here the eigenprojectors are, explicitly: $$E_1=\frac{R-\lambda_+}{1-\lambda_+}\frac{R-\lambda_-}{1-\lambda_-}$$ $$E_+=\frac{R-1}{\lambda_+-1}\frac{R-\lambda_-}{\lambda_+-\lambda_-}$$ $$E_-=\frac{R-1}{\lambda_--1}\frac{R-\lambda_+}{\lambda_--\lambda_+}$$

Comment by Francois Ziegler

2013-06-18T14:11:11Z

I think the OP wants a square root of $I+P(r,r,r-1)$, not $I+P(r,r,0)$.

Comment by Francois Ziegler

2013-06-18T14:04:17Z

Unless I'm mistaken, your matrix has minimal polynomial $(R-1)(R^2-(nr+1)R+r)=0$. From there it is easy to get its spectral form $E_1+\lambda_+E_++\lambda_-E_-$ where $\lambda_\pm=\frac12\left(nr+1\pm\sqrt{(nr+1)^2-4r}\right)$, and from there the square root $E_1+\sqrt{\lambda_+}E_++\sqrt{\lambda_-}E_-$.

Comment by Francois Ziegler

2013-06-15T04:22:17Z

The downvotes and closure of this question seem based on misunderstanding. Indeed as Joel David points out, it is literally true that the naive "set" of standard integers is not a set in the formal sense. See e.g. page 5 of "Analyse Non Standard et Représentation du Réel" here: hal.inria.fr/inria-00163365 . (Whence the title of a famous book by the same author, "Et pourtant... ils ne remplissent pas N!")

Comment by Francois Ziegler

2013-05-28T15:52:27Z

Apart from the title, this seems an exact duplicate of mathoverflow.net/questions/131911/…

Comment by Francois Ziegler

2013-04-29T01:53:59Z

Also, Weyman's "Cohomology of Vector Bundles and Syzygies", §3.1, seems to do what you want. books.google.com/…

Comment by Francois Ziegler

2013-01-21T03:54:39Z

@Aakumadula: You are quite right about Schur's Lemma. (A detailed reference for it would be Hewitt-Ross, p. 324.) So I'd call this proof short and simple, but not quite "elementary".

Comment by Francois Ziegler

2012-12-27T22:22:57Z

Matthew -- Thanks; you are quite right and I have edited accordingly. (To me the main interest here was the statement that normality has origins earlier than Zariski.)

Comment by Francois Ziegler

2012-12-09T02:34:31Z

Not likely, by "dimension counting". Given $f$, the second integral defines a linear form on part of $f^\perp$, which is still going to have a huge kernel.

Comment by Francois Ziegler

2012-12-09T02:23:00Z

For $g\in L_2$, $\int g(x)\,dx$ need not exist...

Comment by Francois Ziegler

2012-12-09T01:58:15Z

<google.com/search?q=Π4(S3)>

Comment by Francois Ziegler

2012-11-14T20:28:33Z

No, there is no reason that the symplectomorphism send your $\alpha$ to the standard 1-form.

Comment by Francois Ziegler

2012-11-13T19:08:18Z

I don't know. By "preserve" do you mean "map $\alpha$ to the standard cotangent bundle 1-form"? Why should it? I will admit that I couldn't tell what you mean by a polarization "whose one form is $\alpha$". That may also be why I don't see why you'd expect uniqueness of the transverse polarization.

Comment by Francois Ziegler

2012-11-12T00:50:10Z

@Yemon, what's wrong about the ambient class being all functions $S^1\to\mathbf{C}$? Therein the poster wants to characterize the set of all pointwise limits of pointwise convergent Fourier series.