Vertex $F$-Algebras and Their Associated Lie Algebra

Markus Upmeier

Department of Mathematics, University of Aberdeen, Fraser Noble Building, Elphinstone Rd, Aberdeen, AB24 3UE, UK

Abstract: Vertex $F$-algebras are a deformation of the concept of an ordinary vertex algebra in which the additive formal group law is replaced by an arbitrary formal group law $F$. The main theorem of this paper constructs a Lie algebra from a vertex $F$-algebra – for the additive formal group law, this extends Borcherds' well-known construction for ordinary vertex algebras. Our construction involves the new concept of an $F$-residue and some other new algebraic concepts, which are deformations of familiar concepts for the special case of an additive formal group law.

Keywords: vertex algebras, formal group laws, Lie algebras.

MSC: 17B69, 17B65

Received: April 18, 2025; in final form January 3, 2026; Published online January 15, 2026

Language: English

DOI: 10.3842/SIGMA.2026.004

ArXiv: 2503.21390