Is there a quaternionic algebraic geometry ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T07:43:37Z http://mathoverflow.net/feeds/question/51965 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51965/is-there-a-quaternionic-algebraic-geometry Is there a quaternionic algebraic geometry ? Qfwfq 2011-01-13T14:46:50Z 2011-01-14T11:55:02Z <p>Let $\mathbb{H}$ be the skew-field of quaternions. I'm aware of the </p> <p><em>Theorem 1.</em> A function $f:\mathbb{H}\to\mathbb{H}$ which is $\mathbb{H}$-differentiable on the left (i.e. the usual limit $h^{-1}\cdot (f(x+h)-f(x))$, for $h\to 0$, exists for every $x\in \mathbb{H}$) is a quaternionic affine function on the right (i.e. of the form $x\mapsto x\cdot \alpha + v$).</p> <p>This means that there are no interesting smooth quaternionic funcions, hence no interesting "quaternionic-smooth manifolds" (which is not the same as the quaternionic-Kahler or hyperkahler structures you encounter in differential geometry and complex analytic geometry).</p> <p>I think I can also recall the</p> <p><em>Theorem 2.</em> If a function $f:\mathbb{H}\to\mathbb{H}$ is locally $\mathbb{H}$-analytic (i.e. it can be locally developped in power series, for the suitable noncommutative notion of "power series"), than it corresponds to a real-analytic function $f:\mathbb{R}^4\to \mathbb{R}^4$, and any real-analytic funcion $f:\mathbb{R}^4\to \mathbb{R}^4$ can be obtained in this way.</p> <p>That says that $\mathbb{H}$-analytic functions are essentially the same as quadruples of real-analytic functions of 4 variables. Hence there is no "quaternionic-analytic geometry" distinguishable from $4n$-dimentional real-analytic geometry. I think the same happens with quaternionic (noncommutative) polynomials: they're just 4-tuples of real polynomials in 4 variables. </p> <p>But, is it reasonable that the zero locus on $\mathbb{H}^n$ of a "noncommutative polynomial" with $\mathbb{H}$-coefficients doesn't have any further mathematical structure than it's real-algebraic variety structure? It would be nice to be able to see things such as $\mathbb{HP}^1$ as a "quaternionic <em>curve</em> ", and to speak of a <em>point</em> " $\mathrm{Spec}(\mathbb{H})$ " (whatever it means) if it possible...</p> <blockquote> <p>Is there a theory of "quaternionic algebraic geometry", maybe as a branch or particular case of some noncommutative (algebraic) geometry theory? </p> </blockquote> <p>Of course, if such a theory has some sense, it cannot be the "obvious analog" of complex algebraic or analytic geometry, as theorems 1. and 2. above show. </p> http://mathoverflow.net/questions/51965/is-there-a-quaternionic-algebraic-geometry/51973#51973 Answer by Spinorbundle for Is there a quaternionic algebraic geometry ? Spinorbundle 2011-01-13T16:04:25Z 2011-01-13T22:46:42Z <p>The answer is yes!(at least if quaternionic holomorphic geometry counts) "Qauternionic holomorphic geometry" provides a very elegant description of surfaces in 3- and 4-dimensional space.</p> <p>The first paper in this field is more or less </p> <blockquote> <p>Franz Pedit and Ulrich Pinkall, Quaternionic Analysis on Riemann Surfaces and Differential Geometry, in: Proocedings of the international con- gress of mathematicians, Berlin 1998, II Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, Extra Volume ICM 1998, 389-400. <a href="http://www.math.tu-berlin.de/~pinkall/forDownload/Pinkall.MAN.pdf" rel="nofollow">Click me</a></p> </blockquote> <p>A good introduction to "Quaternionic holomorphic geometry" is given by</p> <blockquote> <p>Francis E. Burstall, Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Conformal geometry of surfaces in S4 and quaternions, Lecture Notes in Mathematics 1772, Springer-Verlag, Berlin, 2002. <a href="http://arxiv.org/abs/math/0002075" rel="nofollow">http://arxiv.org/abs/math/0002075</a></p> </blockquote> <p>and </p> <blockquote> <p>Dirk Ferus, Katrin Leschke, Franz Pedit and Ulrich Pinkall, Quaternionic Holomorphic Geometry: Plucker Formula, Dirac Eigenvalue Esitmates and Energy Estimates of Harmonic 2-Tori, Inventiones Mathematicae 146 (2001),no. 3, 507-593 <a href="http://arxiv.org/abs/math/0012238v1" rel="nofollow">arxiv.org/abs/math/0012238v1</a></p> </blockquote> <p>But just to clarify things: "Of course, if such a theory has some sense, it cannot be the "obvious analog" of complex algebraic or analytic geometry, as theorems 1. and 2. above show."</p> <p>This isn't really true, because the quaternionic holomorphic geometry developed in the papers above, is a kind of generalization of complex geometry, i.e. if you look how a "quaternionic holomorphic structure" is defined, you see that it is a kind of $\overline{\partial}$-Operator with some extra data (a so called Hopf field).</p> <p>If you need more references, there are plenty available (just type "quaternionic holomorphic geometry" into google)</p> http://mathoverflow.net/questions/51965/is-there-a-quaternionic-algebraic-geometry/51996#51996 Answer by Ben Webster for Is there a quaternionic algebraic geometry ? Ben Webster 2011-01-13T19:20:19Z 2011-01-13T19:20:19Z <p>I actually have no idea how this relates to the "quaternionic geometry" mentioned above, but there is also <a href="http://en.wikipedia.org/wiki/Hyperk%C3%A4hler_manifold" rel="nofollow">hyperkähler geometry</a>. A hyperkähler manifold is a real manifold such that every tangent space has an action of the quaternions, and there is a single metric which makes the manifold Kähler in the complex structure induced by $I,J$, or $K$. This sounds a little differential geometric at first, but one can get a purely algebraic object by turning $J$ and $K$ into their respective symplectic forms $\omega_J$ and $\omega_K$. The sum $\omega_J+i\omega_K$ is a holomorphic symplectic form for the complex structure $I$, and many examples of these come from purely algebro-geometric sources. See, for example, the <a href="http://arxiv.org/abs/math/0608143" rel="nofollow">survey of Kaledin</a>.</p> http://mathoverflow.net/questions/51965/is-there-a-quaternionic-algebraic-geometry/52027#52027 Answer by Tony Pantev for Is there a quaternionic algebraic geometry ? Tony Pantev 2011-01-14T02:44:14Z 2011-01-14T02:44:14Z <p>Take a look at Dominic Joyce's paper <a href="http://arxiv.org/abs/math/0010079" rel="nofollow">"A theory of quaternionic algebra, with applications to hypercomplex geometry"</a>.</p> http://mathoverflow.net/questions/51965/is-there-a-quaternionic-algebraic-geometry/52029#52029 Answer by David Ben-Zvi for Is there a quaternionic algebraic geometry ? David Ben-Zvi 2011-01-14T03:27:03Z 2011-01-14T03:27:03Z <p>In a more analytic direction there is also a recent theory of "split quaternionic analysis" developed by Igor Frenkel and Matvei Libine, starting <a href="http://arxiv.org/abs/0711.2699" rel="nofollow">here</a> with a survey <a href="http://arxiv.org/abs/1009.2540" rel="nofollow">here</a>, with applications to representation theory and physics.</p> http://mathoverflow.net/questions/51965/is-there-a-quaternionic-algebraic-geometry/52064#52064 Answer by Roberto Frigerio for Is there a quaternionic algebraic geometry ? Roberto Frigerio 2011-01-14T11:55:02Z 2011-01-14T11:55:02Z <p>You may perhaps be also interested in the quite recent paper</p> <p>Gentili, Graziano; Stoppato, Caterina Zeros of regular functions and polynomials of a quaternionic variable. Michigan Math. J. 56 (2008), no. 3, 655–667</p> <p>(and other papers by the same authors).</p>