Geometry of Affine Actions

Kilmurray, Donough

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https://hdl.handle.net/2142/87004 Copy

Description

Title

Geometry of Affine Actions

Author(s)

Kilmurray, Donough

Issue Date

2000

Doctoral Committee Chair(s)

Maarten Bergvelt

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

We introduce a new factorization for elements of the loop group LSL2 = SL2( C ((lambda-1))). We then define a new decomposition for the flag manifold FL2 = LSL2/ B+ (= SL2&d14;/B&d14; + ), into isomorphic overlapping cells. This allows us to examine globally the left vector field action of the homogeneous Heisenberg subalgebra Hsl2 on FL2. On each cell of the quotient H -\FL2, we find coordinates for the modified non-linear Schrodinger (mNLS) hierarchy, with Miura, maps to NLS coordinates on the Grassmannian. We also find transformation rules relating coordinates across cells. The right vector field action of n&d14;-⊂ sl2&d14; attaches to the coordinate ring of each cell an object e +/-2&phis;. We show that this gives a special line bundle structure on H-\FL2. Finally, we define on each cell a Hamiltonian structure. This leads to a vertex algebra structure, and from the n&d14;- -action, a vertex algebra module structure, on each cell. We establish the compatibility of these structures across cells to show that H -\FL2 is a vertex variety, i.e. a variety whose structure sheaf is a sheaf of vertex algebras.

Graduation Semester

2000

Type of Resource

text

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http://hdl.handle.net/2142/87004

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