In our thesis we study the algebras of differential operators in algebraic and geometric terms. We consider two
problems in which the algebras of differential operators naturally arise. The first one deals with the algebraic
structure of differential and pseudodifferential operators. We define the Krichever-Novikov type Lie algebras of
differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and
central extensions. We show that the corresponding algebras of meromorphic differential operators and
pseudodifferential symbols have many invariant traces and central extensions, given by the logarithms of meromorphic
vector fields. We describe which of these extensions survive after passing to the algebras of operators and symbols
holomorphic away from several fixed points. We also describe the associated Manin triples, emphasizing the
similarities and differences with the case of smooth symbols on the circle.
The second problem is related to the geometry of differential operators and its connection with representations of
semi-simple Lie algebras. We show that the semiregular module, naturally associated with a graded semi-simple
complex Lie algebra, can be realized in geometric terms, using the Brion's construction of degeneration of
the diagonal in the square of the flag variety. Namely, we consider the Beilinson-Bernstein localization
of the semiregular module and show that it is isomorphic to the D-module obtained by applying the
Emerton-Nadler-Vilonen geometric Jacquet functor to the D-module of distributions on the square of the flag variety
with support on the diagonal.
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/16779 |
| Date | 20 January 2009 |
| Creators | Donin, Dmitry |
| Contributors | Khesin, Boris, Arkhipov, Sergey |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
| Format | 506075 bytes, application/pdf |
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