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Title:Geometry of Affine Actions
Author(s):Kilmurray, Donough
Doctoral Committee Chair(s):Maarten Bergvelt
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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.
Issue Date:2000
Description:85 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2000.
Other Identifier(s):(MiAaPQ)AAI9990040
Date Available in IDEALS:2015-09-28
Date Deposited:2000

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