2020/01/12 by Aleksandr Fadeev, Fadeev, Aleksandr
Mathematics · #Combinatorics #Commutative Algebra and Its Applications #FOS: Mathematics #Mathematics #Physics #Pure mathematics #Representation Theory (math.RT) #Rings, Modules, and Algebras #math.RT
paper · pdf · doi:10.48550/arxiv.2001.03858
published in arXiv (Cornell University) (Cornell University) · PhD thesis
arxiv created 2020/01/12 · openalex publication_date 2020/01/12 · arxiv updated 2020/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of this Ph.D. thesis is to study and classify primitive ideals of the enveloping algebras U(\mathfrako(∞)) and U(\mathfraksp(∞)). Let \mathfrakg(∞) denote any of the Lie algebras \mathfrako(∞) or \mathfraksp(∞). Then\break \mathfrakg(∞)=\bigcupn≥ 2 \mathfrakg(2n) for \mathfrakg(2n)=\mathfrako(2n) or \mathfrakg(2n)=\mathfraksp(2n), respectively. We show that each primitive ideal I of U(\mathfrakg(∞)) is weakly bounded, i.e., I∩ U(\mathfrakg(2n)) equals the intersection of annihilators of bounded weight \mathfrakg(2n)-modules. To every primitive ideal I of \mathfrakg(∞) 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(\mathfrakg(∞)) 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 \mathfrako(∞), 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 \mathfraksp(∞), only 'half' of the primitive ideals are integrable, and nonintegrable primitive ideals correspond to triples (x,y,Z) where y is a half-integer.