Skip to main content
SpringerLink
Log in

Constructing a Lie group analog for the Monster Lie algebra

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

Let \(\mathfrak {m}\) be the Monster Lie algebra. We summarize several interrelated constructions of Lie group analogs for \(\mathfrak {m}\). Our constructions are analogs for \(\mathfrak {m}\) of Chevalley and Kac–Moody groups and their generators and relations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

This manuscript has no associated data.

References

  1. Ali, A., Carbone, L., Murray, S. H., Yap, C.: Integral structure in the automorphism group of the Monster Lie algebra. In preparation (2022)

  2. Ali, A., Carbone, L., Liu, D., Murray, S. H.: Infinite dimensional Chevalley groups and Kac-Moody groups over \(\mathbb{Z}\). arXiv:1803.11204 [math.RT]

  3. Borcherds, R. E.: Introduction to the Monster Lie algebra, Groups, combinatorics and geometry (Durham, 1990), 99–107, London Math. Soc. Lecture Note Ser., 165, Cambridge Univ. Press, Cambridge (1992)

  4. Borcherds, R.E.: Generalized Kac-Moody algebras. J. Algebra 115, 501–512 (1988)

    Article  MathSciNet  Google Scholar 

  5. Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109(2), 405–444 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  6. Carbone, L., Chen, H., Jurisich, E., Murray, S. H., Paquette, N.: Imaginary reflections and automorphisms of the Monster Lie algebra. In preparation (2022)

  7. Carbone, L., Ginory, A., Jurisich, E., Murray, S. H.: Automorphism groups of standard induced modules for the Monster Lie Algebra. In preparation (2022)

  8. Carbone, L., Jurisich, E., Murray, S. H.: A Lie group analog for the Monster Lie algebra. In preparation (2022)

  9. Carbone, L., Paquette, N.: Imaginary reflections and discrete symmetries in the heterotic Monster (2022), arXiv:2202.09720v3 [hep-th]

  10. Carbone, L., Garland, H.: Existence of lattices in Kac-Moody groups over finite fields. Commun. Contemp. Math. 5(5), 813–867 (2003)

    Article  MathSciNet  Google Scholar 

  11. Carnahan, S.: Fricke Lie algebras and the genus zero property in Moonshine. J. Phys. A 50, 40 (2017)

    Article  MathSciNet  Google Scholar 

  12. Conway, J.H., Norton, S.P.: Monstrous Moonshine. Bull. Lond. Math. Soc. 11, 308–339 (1979)

    Article  MathSciNet  Google Scholar 

  13. Frenkel, I.B., Lepowsky, J., Meurman, A.: A natural representation of the Fischer-Griess Monster with the modular function J as character. Proc. Nat. Acad. Sci. USA 81(10), 3256–3260 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  14. Frenkel, I., Lepowsky, J., Meurman, A.: Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134. Academic Press Inc, Boston, MA (1988)

    MATH  Google Scholar 

  15. Garland, H.: The arithmetic theory of loop groups. Inst. Hautes tudes Sci. Publ. Math. No. 52, 5–136 (1980)

    Article  MathSciNet  Google Scholar 

  16. Goodman, R., Wallach, N.R.: Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle. J. Reine Angew. Math. 347, 69–133 (1984)

    MathSciNet  MATH  Google Scholar 

  17. Harvey, J., Moore, A.G.: On the algebras of BPS states. Commun. Math. Phys. 197 197(3), 489–519 (1998)

    Article  MathSciNet  Google Scholar 

  18. Jurisich, E.: An exposition of generalized Kac-Moody algebras. Lie algebras and their representations, (Seoul, 1995), 121–159, Contemp. Math., 194, Am. Math. Soc., Providence, RI, (1996)

  19. Jurisich, E.: Generalized Kac-Moody Lie algebras, free Lie algebras and the structure of the Monster Lie algebra. J. Pure Appl. Algebra 126(1–3), 233–266 (1998)

    Article  MathSciNet  Google Scholar 

  20. Jurisich, E., Lepowsky, J., Wilson, R.L.: Realizations of the Monster Lie algebra. Selecta Math. (N.S.) 1(1), 129–161 (1995)

    Article  MathSciNet  Google Scholar 

  21. Kac, V., Peterson, D.: Defining relations of certain infinite dimensional groups. Astérisque Numéro Hors Série, 165–208 (1985)

  22. Kumar, S.: Kac-Moody groups, their flag varieties and representation theory, progress in mathematics, vol. 204. Springer, New York (2012)

    Google Scholar 

  23. Lepowsky, J., Li, H.: Introduction to vertex operator algebras and their representations, progress in mathematics, 227. Birkhaüser Boston, Inc., Boston, MA, (2004). xiv+318 pp

  24. Marcuson, R.: Tits’ systems in generalized nonadjoint Chevalley groups. J. Algebra 34, 84–96 (1975)

    Article  MathSciNet  Google Scholar 

  25. Marquis, T.: An introduction to Kac-Moody groups over fields, EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, (2018). xi+331 pp

  26. Mathieu, O.: Sur la construction de groupes associés aux algébre s de Kac–Moody. (French) [On the construction of groups associated with Kac–Moody algebras] C. R. Acad. Sci. Paris Sér. I Math. 299(6): 161–164 (1984)

  27. Moody, R.V., Teo, K.L.: Tits’ systems with crystallographic Weyl groups. J. Algebra 21, 178–190 (1972)

    Article  MathSciNet  Google Scholar 

  28. Paquette, N.M., Persson, D., Volpato, R.: BPS algebras, genus zero and the heterotic monster. J. Phys. A 50(41), 17 (2017)

    Article  MathSciNet  Google Scholar 

  29. Rousseau, G.: Groupes de Kac-Moody déployś sur un corps local II. Masures ordonnés. (French) [Split Kac-Moody groups over a local field II. Ordered hovels]. Bull. Soc. Math. France 144(4), 613–692 (2016)

    Article  MathSciNet  Google Scholar 

  30. Slodowy, P.: An adjoint quotient for certain groups attached to Kac–Moody algebras, infinite dimensional groups with applications (Berkeley, Calif., 1984), 307–333, Math. Sci. Res. Inst. Publ., 4, Springer, New York (1985)

  31. Steinberg, R.: Générateurs, relations et revêtements de groupes algébriques, (French) 1962 Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) pp. 113–127 Librairie Universitaire, Louvain; Gauthier-Villars, Paris

  32. Tits, J.: Uniqueness and presentation of Kac-Moody groups over fields. J. Algebra 105, 542–573 (1987)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854-8019, USA

    Lisa Carbone

  2. Department of Mathematics, College of Charleston, 66 George Street, Charleston, SC, 29424, USA

    Elizabeth Jurisich

  3. University of Toronto Mississauga, 3359 Mississauga Road, Mississauga, ON, L5L 1C6, Canada

    Scott H. Murray

Authors
  1. Lisa Carbone
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elizabeth Jurisich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Scott H. Murray
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lisa Carbone.

Ethics declarations

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author’s research is partially supported by the Simons Foundation, Mathematics and Physical Sciences-Collaboration Grants for Mathematicians, Award Number 422182.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carbone, L., Jurisich, E. & Murray, S.H. Constructing a Lie group analog for the Monster Lie algebra. Lett Math Phys 112, 43 (2022). https://doi.org/10.1007/s11005-022-01531-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11005-022-01531-4

Keywords

Search

Navigation