SIGMA 22 (2026), 004, 19 pages      arXiv:2503.21390      https://doi.org/10.3842/SIGMA.2026.004
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

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

Received April 18, 2025, in final form January 03, 2026; Published online January 15, 2026

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.

Key words: vertex algebras; formal group laws; Lie algebras.

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References

1. Borcherds R.E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. USA 83 (1986), 3068-3071.
2. Bourbaki N., Éléments de mathématique. Algèbre commutative. Chapitres 1 à 4, Masson, Paris, 1985.
3. Creutzig T., McRae R., Yang J., Ribbon tensor structure on the full representation categories of the singlet vertex algebras, Adv. Math. 413 (2023), 108828, 79 pages, arXiv:2202.05496.
4. Gross J., Upmeier M., Vertex $F$-algebra structures on the complex oriented homology of H-spaces, J. Pure Appl. Algebra 226 (2022), 107019, 13 pages, arXiv:2022.10701.
5. Hazewinkel M., Formal groups and applications, AMS Chelsea Publishing, Providence, RI, 2012.
6. Huang Y.-Z., Two-dimensional conformal geometry and vertex operator algebras, Progr. Math., Vol. 148, Birkhäuser, Boston, MA, 1997.
7. Huang Y.-Z., Differential equations, duality and modular invariance, Commun. Contemp. Math. 7 (2005), 649-706, arXiv:math.QA/0303049.
8. Joyce D., Enumerative invariants and wall-crossing formulae in abelian categories, arXiv:2111.04694.
9. Joyce D., Ringel-Hall style vertex algebra and Lie algebra structures on the homology of moduli spaces, available at https://people.maths.ox.ac.uk/joyce/publ.html.
10. Lazard M., Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math. France 83 (1955), 251-274.
11. Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progr. Math., Vol. 227, Birkhäuser, Boston, MA, 2004.
12. Li H., Vertex $F$-algebras and their $\phi$-coordinated modules, J. Pure Appl. Algebra 215 (2011), 1645-1662, arXiv:1006.4126.
13. Liu H., Equivariant K-theoretic enumerative invariants and wall-crossing formulae in abelian categories, arXiv:2207.13546.
14. Remmert R., Theory of complex functions, Grad. Texts in Math., Vol. 122, Springer, New York, 1991.