[Paper Review] Category O for the Schrödinger algebra

Brendan Dubsky, Rencai Lü+2 more|arXiv (Cornell University)|Feb 3, 2014

Algebraic structures and combinatorial models15 references3 citations

TL;DR

This paper establishes the structure of Category 𝒪 for the centrally extended Schrödinger algebra, determining quivers and relations for blocks with nonzero central charge, describing the finite-dimensional part of 𝒪, and classifying primitive ideals and annihilators of Verma modules. It corrects errors in prior work and proves that primitive ideals with nonzero central charge are precisely the annihilators of simple highest weight modules.

ABSTRACT

We study category O for the (centrally extended) Schrödinger algebra. We determine the quivers for all blocks and relations for blocks of nonzero central charge. We also describe the quiver and relations for the finite dimensional part of O. We use this to determine the center of the universal enveloping algebra and annihilators of Verma modules. Finally, we classify primitive ideals with nonzero central charge.

Motivation & Objective

• To resolve inconsistencies between [WZ1] and [Pe] regarding annihilators of Verma modules over the Schrödinger algebra.
• To develop a comprehensive understanding of Category 𝒪 for the centrally extended Schrödinger algebra.
• To classify primitive ideals in the universal enveloping algebra of the Schrödinger algebra that intersect the center trivially.
• To describe the center of the universal enveloping algebra and the structure of annihilators of Verma modules.
• To analyze the finite-dimensional part of Category 𝒪, showing it has wild representation type.

Proposed method

• Constructing a graded version of Category 𝒪 for the zero central charge case to facilitate analysis.
• Using the BGG reciprocity and quiver techniques to determine quivers and relations for blocks with nonzero central charge.
• Applying Harish-Chandra bimodule theory to classify primitive ideals in the universal enveloping algebra.
• Analyzing the structure of Verma modules and their annihilators via tensor products with simple highest weight modules over 𝔰𝔩₂.
• Employing the universal enveloping algebra's decomposition into weight spaces and analyzing the action of the central element z.
• Using the simplicity of the quotient algebra B_ż = U(𝔰)/ (z−ż) to deduce properties of annihilators and bimodules.

Experimental results

Research questions

• RQ1 What is the structure of Category 𝒪 for the Schrödinger algebra, particularly for blocks with nonzero central charge?
• RQ2 How do the quivers and relations of blocks in Category 𝒪 for the Schrödinger algebra depend on the central charge?
• RQ3 What is the center of the universal enveloping algebra of the Schrödinger algebra, and how is it related to annihilators of Verma modules?
• RQ4 Which primitive ideals in the universal enveloping algebra of the Schrödinger algebra have trivial intersection with the center?
• RQ5 What is the structure of the finite-dimensional part of Category 𝒪, and why does it have wild representation type?

Key findings

• The quiver and relations for blocks of Category 𝒪 with nonzero central charge are fully determined, with the quiver being a Dynkin diagram of type A∞.
• The finite-dimensional part of Category 𝒪 for the Schrödinger algebra has wild representation type, implying no finite classification of its indecomposable modules.
• The center of the universal enveloping algebra of the Schrödinger algebra is isomorphic to the polynomial algebra ℂ[z], generated by the central element z.
• For nonzero central charge, primitive ideals in the universal enveloping algebra are exactly the annihilators of simple highest weight modules.
• The annihilator of a Verma module over the Schrödinger algebra is determined by its highest weight and central charge, and the classification corrects an error in [WZ1].
• The universal enveloping algebra admits a graded structure for zero central charge, enabling analysis of the finite-dimensional part of Category 𝒪.