Geometry of Affine Actions

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Kilmurray, Donough

Contributors dc:contributor

• Maarten Bergvelt

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI9990040

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/87004