The centre of the modular affine vertex algebra

Tomoyuki Arakawa; Lewis Topley; Juan Villarreal

doi:10.1016/j.aim.2024.110052

The centre of the modular affine vertex algebra

Tomoyuki Arakawa, Lewis Topley, Juan Villarreal

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

The Feigin–Frenkel theorem states that, over the complex numbers, the centre of the universal affine vertex algebra at the critical level is an infinite rank polynomial algebra. The first author and W. Wang observed that in positive characteristics, the universal affine vertex algebra contains a large central subalgebra known as the p-centre. They conjectured that at the critical level the centre should be generated by the Feigin–Frenkel centre and the p-centre. In this paper we prove the conjecture for classical simple Lie algebras for p larger than the Coxeter number, and for exceptional Lie algebras in large characteristics. Finally, we give an example which shows that at non-critical level the center is larger than the p-centre.

Original language	English
Article number	110052
Journal	Advances in Mathematics
Volume	461
Early online date	29 Nov 2024
DOIs	https://doi.org/10.1016/j.aim.2024.110052
Publication status	Published - 1 Feb 2025

Funding

We would like to thank Dmitriy Rumynin and Alexander Molev for useful correspondence on the subject of this paper, Andrew Linshaw for sharing his knowledge of the OPEdefs 3.1 Mathematica package. We are grateful to University of Bath, RIMS, Ningbo University and SUS Tech international center for mathematics for the excellent working conditions were this work was done. We offer special thanks to Gurbir Dhillon for numerous enlightening conversations. The first author is partially supported by JSPS KAKENHI Grant Number J21H04993 and 19KK0065. The second and third author are supported by a UKRI Future Leaders Fellowship, grant numbers MR/S032657/1, MR/S032657/2, MR/S032657/3.

Funders	Funder number
Ningbo University
University of Bath
Japan Society for the Promotion of Science London	19KK0065, J21H04993
Japan Society for the Promotion of Science London
UK Research and Innovation	MR/S032657/3, MR/S032657/1, MR/S032657/2
UK Research and Innovation

Keywords

Affine Lie algebra
Centre
Characteristic p
Positive characteristic
Vertex algebra

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2024.110052Licence: CC BY

Cite this

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N2 - The Feigin–Frenkel theorem states that, over the complex numbers, the centre of the universal affine vertex algebra at the critical level is an infinite rank polynomial algebra. The first author and W. Wang observed that in positive characteristics, the universal affine vertex algebra contains a large central subalgebra known as the p-centre. They conjectured that at the critical level the centre should be generated by the Feigin–Frenkel centre and the p-centre. In this paper we prove the conjecture for classical simple Lie algebras for p larger than the Coxeter number, and for exceptional Lie algebras in large characteristics. Finally, we give an example which shows that at non-critical level the center is larger than the p-centre.

AB - The Feigin–Frenkel theorem states that, over the complex numbers, the centre of the universal affine vertex algebra at the critical level is an infinite rank polynomial algebra. The first author and W. Wang observed that in positive characteristics, the universal affine vertex algebra contains a large central subalgebra known as the p-centre. They conjectured that at the critical level the centre should be generated by the Feigin–Frenkel centre and the p-centre. In this paper we prove the conjecture for classical simple Lie algebras for p larger than the Coxeter number, and for exceptional Lie algebras in large characteristics. Finally, we give an example which shows that at non-critical level the center is larger than the p-centre.

KW - Affine Lie algebra

KW - Centre

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KW - Positive characteristic

KW - Vertex algebra

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The centre of the modular affine vertex algebra

Abstract

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Keywords

ASJC Scopus subject areas

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