Infinite Dimensional Grassmannians
نویسندگان
چکیده
We study the differentiable structure and the homotopy type of some spaces related to the Grassmannian of closed linear subspaces of an infinite dimensional Hilbert space, such as the space of Fredholm pairs, the Grassmannian of compact perturbations of a given space, and the essential Grassmannians. We define a determinant bundle over the space of Fredholm pairs. We lift the composition of Fredholm operators to the Quillen determinant bundle, and we show how this map can be used in several constructions involving the determinant bundle over the space of Fredholm pairs. We deduce some properties of suitable orientation bundles.
منابع مشابه
On Infinite Dimensional Grassmannians and Their Quantum Deformations
An algebraic approach is developed to define and study infinite dimensional grassmannians. Using this approach a quantum deformation is obtained for both the ind-variety union of all finite dimensional grassmannians G∞, and the Sato grassmannian ̃ UGM introduced by Sato in [Sa1], [Sa2]. They are both quantized as homogeneous spaces, that is together with a coaction of a quantum infinite dimensi...
متن کاملGeometric structures on finite- and infinite-dimensional Grassmannians
In this paper, we study the Grassmannian of n-dimensional subspaces of a 2n-dimensional vector space and its infinite-dimensional analogues. Such a Grassmannian can be endowed with two binary relations (adjacent and distant), with pencils (lines of the Grassmann space) and with so-called Zreguli. We analyse the interdependencies among these different structures. Mathematics Subject Classificati...
متن کاملFramed Moduli and Grassmannians of Submodules
In this work we study a realization of moduli spaces of framed quiver representations as Grassmannians of submodules devised by Markus Reineke. Obtained is a generalization of this construction to finite dimensional associative algebras and for quivers with oriented cycles over an arbitrary infinite field. As an application we get an explicit realization of fibers for the moduli space bundle ov...
متن کاملThe Algebraic Formalism of Soliton Equations over Arbitrary Base Fields
The aim of this paper is to offer an algebraic construction of infinitedimensional Grassmannians and determinant bundles. As an application we construct the τ -function and formal Baker-Akhiezer functions over arbitrary fields, by proving the existence of a “formal geometry” of local curves analogous to the geometry of global algebraic curves. Recently G. Anderson ([A]) has constructed the infi...
متن کاملQuantum Hadrondynamics in Two Dimensions
A nonlocal and nonlinear theory of hadrons, equivalent to the color singlet sector two dimensional QCD, is constructed. The phase space space of this theory is an infinite dimensional Grassmannian. The baryon number of QCD corresponds to a topological invariant ('virtual rank') of the Grassmannian. It is shown that the hadron theory has topological solitons corresponding to the baryons of QCD. ...
متن کامل