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Natural Construction of Ten Borcherds-Kac-Moody Algebras Associated with Elements in \(M_{23}\)

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Abstract

Borcherds-Kac-Moody algebras generalise finite-dimensional, simple Lie algebras. Scheithauer showed that there are exactly ten Borcherds-Kac-Moody algebras whose denominator identities are completely reflective automorphic products of singular weight on lattices of square-free level. These belong to a larger class of Borcherds-Kac-Moody (super)algebras Borcherds obtained by twisting the denominator identity of the Fake Monster Lie algebra. Borcherds asked whether these Lie (super)algebras admit natural constructions. For the ten Lie algebras from the classification we give a positive answer to this question, i.e. we prove that they can be realised uniformly as the BRST cohomology of suitable vertex algebras.

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Notes

  1. The term Monster Lie algebra was later recoined to denote the Borcherds-Kac-Moody algebra obtained as quantisation of \(V^\natural \otimes V_{I\!I_{1,1}}\) (with the Moonshine module \(V^\natural \) [30]), which was used by Borcherds in his proof of the Monstrous Moonshine conjecture [6].

  2. Here, \({\text {O}}(I\!I_{26,2})^+\) denotes the subgroup of \({\text {O}}(I\!I_{26,2})\) of elements preserving the (choice of continuously varying) orientation on the 2-dimensional positive-definite subspaces of \(I\!I_{26,2}\otimes _\mathbb {Z}\mathbb {R}\). See, for example, Section 13 in [8].

  3. Except for \(m=23\) this is also the unique conjugacy class. When \(m=23\), \(\nu \) and \(\nu ^{-1}\) represent two distinct conjugacy classes.

  4. Indeed, every quadratic form Q (on some finite, abelian group D) has a unique associated bilinear form \(B_Q\). On the other hand, given a finite bilinear form B, there are |D/2D| many quadratic forms Q with \(B_Q=B\).

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Acknowledgements

The author would like to thank Jethro van Ekeren, Gerald Höhn, Ching Hung Lam, Jim Lepowsky, Maximilian Rössler and Nils Scheithauer for helpful discussions. The author thanks the two anonymous referees for their detailed comments. The author was partially supported by a scholarship from the Studienstiftung des deutschen Volkes and by the Deutsche Forschungsgemeinschaft through the project “Infinite-dimensional Lie Algebras in String Theory”.

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    Sven Möller

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Möller, S. Natural Construction of Ten Borcherds-Kac-Moody Algebras Associated with Elements in \(M_{23}\). Commun. Math. Phys. 383, 35–70 (2021). https://doi.org/10.1007/s00220-021-04018-w

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