Classification of primitive ideals of U(o(\infty)) and U(sp(\infty))

  • The purpose of this Ph.D. thesis is to study and classify primitive ideals of the enveloping algebras U(o(\infty)) and U(sp(\infty)). Let g(\infty) denote any of the Lie algebras o(\infty) or sp(\infty). Then g(\infty)=\Bigcap_{n\geq 2}g(2n) for g(2n) = o(2n) or g(2n) = sp(2n), respectively. We show that each primitive ideal I of U(g(1)) is weakly bounded, i.e., I \ U(g(2n)) equals the intersection of annihilators of bounded weight g(2n)-modules. To every primitive ideal I of g(\infty) we attach a unique irreducible coherent local system of bounded ideals, which is an analog of a coherent local system of finite-dimensional modules, as introduced earlier by A. Zhilinskii. As a result, primitive ideals of U(g(\infty)) are parametrized by triples (x;y;Z) where x is a nonnegative integer, y is a nonnegative integer or half-integer, and Z is a Young diagram. In the case of o(1), each primitive ideal is integrable, and our classification reduces to a classification of integrable ideals going back to A. Zhilinskii, A. Penkov and I. Petukhov. In the case of sp(\infty), only 'half' of the primitive ideals are integrable, and nonintegrable primitive ideals correspond to triples (x;y;Z) where y is a half-integer.

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Publishing Institution:IRC-Library, Information Resource Center der Jacobs University Bremen
Granting Institution:Jacobs Univ.
Author:Aleksandr Fadeev
Referee:Ivan Penkov, Alan Huckleberry, Alexey Petukhov
Advisor:Ivan Penkov
Persistent Identifier (URN):urn:nbn:de:gbv:579-opus-1009005
Document Type:PhD Thesis
Date of Successful Oral Defense:2019/12/04
Date of First Publication:2019/12/17
Academic Department:Mathematics & Logistics
PhD Degree:Mathematics
Focus Area:Mobility
Other Countries Involved:Russian Federation
Call No:2019/21

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