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Nakajima varieties provide a natural home for geometric representation theory of simply-laced complex simple Lie algebras. Ultimately, Nakajima theory is a theory about the interaction of symplectic geometry and representation theory.
Yiqiang Li
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The demands of university teaching, addressed to students ⌠with [only] a modest (and frequently less than modest) mathematical baggage, led me to ⌠start from an intuitive baggage common to everyone, independent of any technical language used to express it, and anterior to any such languageâit turned out that the geometric and topological intuition of shapes, particularly two-dimensional shapes, formed such a common ground. These themes can be grouped under the general names âtopology of surfacesâ or âgeometry of surfacesâ, ⌠the main emphasis being on the ⌠combinatorial aspects which form the most down-to-earth technical expression of themâand not on the differential, conformal, Riemannian, holomorphic, [Kähler, contact, symplectic, Moishezon] aspectsâand from there on to â algebraic curves.
Alexandre Grothendieck, 1988, in a letter to
translated by Michael Barr
Henry Horton, A Functorial Symplectic Instanton Homology via Traceless Character Varieties
The Maslov Index - Part 2: The Index for Î(n) and its Geometric Meaning
Okay, so, now that we have a rough understanding of the Lagrangian Grassmannian Î(n), weâre almost ready to write down the definition of the Maslov index. Since we know that Î(n)~U(n)/O(n), we have a map Ď:Î(n)â>U(1) given by Ď(V)=det(a^2), where we identify V with the orbit aO(n). The Maslov index can then be defined as the map Îź:Ď_1(Î(n))â>Z given by the degree of the map Ďογ:U(1)â>U(1) for Îł a generic representative of [Îł] in Ď_1(Î(n)). At this point, youâre probably wondering: âThis is all fine and dandy but, at the end of the day, what does it mean?â What this means is that if we have a loop Îł of Lagrangian subspaces of R^2n and we fix L=Îł(0), then Îź([Îť]) is the number of times (counted with sign, depending on whether or not the intersections are positive) Îł(t) fails to intersect L transversely, i.e. in more than just a single point since Lagrangian subspaces have dimension equal to half that of the ambient symplectic vector space.
The Maslov Index - Part I: the Lagrangian Grassmannian
Iâve decided my first proper post should be hardcore math so here goes:
As everybody whoâs seen a bit of topology ought to know, there are these wonderful objects called âprojective spacesâ: for a (real or complex) vector space V, the associated projective space P(V) is the space (in fact smooth manifold) of all 1-dimensional linear subspaces of V, i.e. the quotient of V-{0} under the identification x~y if there exists some nonzero scalar c such that y=cx, endowed with the quotient topology. âWell,â you might be wondering, âwhat about 2-dimensional subspaces or 3-dimensional subspaces?â Indeed, such things exist and they, too, are smooth manifolds in a natural way: if m is less than or equal to n, then we denote by Gr(m,n) the space of all m-dimensional linear subspaces of R^n and call this the m-Grassmannian of R^n.
The Lagrangian Grassmannian Î(n) is an (at least superficially) analogous construction in Symplectic topology: if we endow R^2n with the standard symplectic form Ď(u,v)=u^tJv, where J is the 2x2 block matrix whose diagonal entries are the nxn zero matrix and whose top right and bottom left entries are -I and I, respectively, where I is the nxn identity matrix, then a Lagrangian subspace of R^2n is an n-dimensional linear subspace of R^2n on which Ď vanishes identically. The Lagrangian Gradsmannian Î(n) is, then, the space of all Lagrangian subspaces of R^2n. As it happens, Î(n) can be identified with the homogeneous space U(n)/O(n) and, hence, is itself a smooth manifold: given a Lagrangian subspace L, write an orthogonal basis for L as the columns of a 2x1 block matrix with entries A and B and identify L with the 2x2 block matrix with diagonal entries A and off diagonal entries -B and B. Since the basis was chosen to be orthogonal, this construction gives us an element of U(n). Since the natural action of O(n) by change of basis preserves the subspace L, the quotient U(n)/O(n) then gives us the Lagrangian Grassmannian.
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The Fight to Fix Symplectic Geometry | Quanta Magazine
It's a good thing that finite-dimensional vector spaces are not canonically isomorphic to their duals. Such an iso. would amount to choosing an inner product (when the field is appropriate), and that would mean that vector spaces come with some intrinsic geometry, which is a terrifying notion. Related to that: the non-degeneracy of symplectic structures picks an isomorphism between the tangent and cotangent spaces at a point, which is what leads to one being able to define a hamiltonian as a vector field, leading to time evolution being the flow of a vector field. so basically there not being a canonical isomorphism leads right to classical mechanics,lol. That leads to the question: what does it mean in terms of classical mechanics when two symplectic forms are in the same cohomology class?
John Paprocki (@jpoprox)
