Borcherds' 1990 problem on Lie algebras for the fixed-point lattices of $\mathit{II}_{25,1}$: automorphic-product generalized Kac-Moody algebras and their natural constructions

The source is the closing problem section of Borcherds' 1990 paper "The monster Lie algebra", which constructed what is now called the fake monster Lie algebra. The source asks whether, for a finite subgroup $G$ of $\mathrm{Aut}(\mathit{II}_{25,1})$ fixing the Weyl vector $\rho$, there is a "nice" Lie algebra whose root lattice is the fixed-point lattice $L=\mathit{II}_{25,1}^{G}$, and instantiates this with two conjectures: a family of ten algebras indexed by cycle shapes of elements of $M_{24}\subset\mathrm{Aut}(\Lambda)$ with multiplicities governed by eta-products, and a rank-10 Lie superalgebra; it also asks whether such algebras admit direct connections to sporadic simple groups. Both concrete conjectures have since been proved. The twisted denominator identities of the fake monster algebra, established by Borcherds in 1992 for elements of the Conway group $\mathrm{Co}_{0}$, construct the $g$-twisted generalized Kac-Moody superalgebras; Scheithauer (2000-2008) proved the associated "moonshine conjecture for Conway's group" at squarefree level, and Wang and Williams (2025) proved it for all of $\mathrm{Co}_{0}$: each twisted identity is a singular-weight Borcherds product on $U(N_{g})\oplus U\oplus\Lambda^{g}$. Natural constructions as BRST cohomology of explicit vertex algebras are known for the ten squarefree-level $M_{23}$ algebras (Möller 2021, after partial constructions by Höhn-Scheithauer 2003 and Creutzig-Klauer-Scheithauer 2007) and for the adjacent rank-14 algebra of the $2^{12}$ twist (Höhn-Scheithauer 2014), and the rank-10 superalgebra was realized through the Enriques moduli space (Borcherds 1996, 1998) and as superstring physical states (Scheithauer 2000). The surviving open core is the general problem: classify the generalized Kac-Moody (super)algebras with Weyl vector whose denominator functions are automorphic forms - equivalently, classify reflective and singular-weight Borcherds products - and construct them naturally, together with the source's structural subquestion on direct connections to sporadic simple groups; only partial classifications exist, and the natural-construction conjecture is open beyond the known cases.

Background

An even lattice is a free abelian group of finite rank equipped with an integer-valued nondegenerate symmetric bilinear form in which every vector has even norm; it is unimodular if its discriminant group is trivial. Up to isomorphism there is a unique even unimodular lattice of Lorentzian signature $(25,1)$, denoted $\mathit{II}_{25,1}$; it splits as the orthogonal sum of the Leech lattice $\Lambda$ (the unique even unimodular positive-definite lattice of rank 24 with no vectors of norm 2, also called rootless) and the rank-2 even unimodular Lorentzian lattice $\mathit{II}_{1,1}$, commonly denoted $U$. Conway (1983) showed that the reflection group of $\mathit{II}_{25,1}$, generated by the reflections in the norm-2 vectors, has a Weyl vector $\rho$: a norm-0 vector with $(\rho,r)=-r^{2}/2$ for every simple real root $r$. Borcherds' 1990 paper constructed on this lattice the fake monster Lie algebra, a generalized Kac-Moody algebra with root lattice $\mathit{II}_{25,1}$ whose real simple roots are the norm-2 vectors $(\lambda,1,\lambda^{2}/2-1)$ for $\lambda\in\Lambda$ (writing vectors of $\Lambda\oplus U$ as $(\lambda,m,n)$ with $\lambda\in\Lambda$, $m,n\in\mathbb{Z}$ and squared norm $\lambda^{2}-2mn$), whose imaginary simple roots are the positive multiples $n\rho$ each of multiplicity 24, and whose root multiplicities are $p_{24}(1-r^{2}/2)$, where $p_{24}(n)$ counts partitions of $n$ into 24 colors. (The 1990 paper called this the "monster Lie algebra"; the name was later transferred to the rank-2 algebra built from the monster vertex algebra, the moonshine vertex operator algebra on which the monster finite simple group acts by automorphisms, and the present algebra was renamed the fake monster Lie algebra.)

A generalized Kac-Moody algebra (also called a Borcherds-Kac-Moody algebra) is defined like a Kac-Moody algebra but from a symmetrizable generalized Cartan matrix whose entries may be zero or negative, which forces the appearance of imaginary simple roots of nonpositive norm alongside the usual real ones of positive norm (normalized to norm 2). Every such algebra satisfies a denominator identity generalizing the Weyl denominator formula, and the structure is most explicit precisely when the reflection group on the real roots admits a Weyl vector and the root multiplicities have closed form, in which case the identity becomes an infinite product over roots with the multiplicities as exponents.

Borcherds' regularized theta lift (1995, 1998) attaches to a weakly holomorphic modular form for the Weil representation of $\mathrm{SL}_{2}(\mathbb{Z})$ on the discriminant form of an even lattice $M$ of signature $(l,2)$ (the discriminant form being the discriminant group $M'/M$, with $M'$ the dual lattice, carrying the induced quadratic form) a meromorphic automorphic form on the orthogonal group of $M$ carrying an infinite product expansion, called a Borcherds product. Its zeros lie along rational quadratic divisors; the product is called reflective when those divisors correspond to roots of $M$, and it has singular weight when its weight is minimal for its representation, in which case the form is essentially determined by its divisor. The fake monster denominator is the Borcherds form $\Phi_{12}$ of singular weight 12 on $\mathit{II}_{26,2}$, and the denominator identities of the nicest generalized Kac-Moody algebras are exactly reflective Borcherds products of singular weight. The 1990 problem asks which fixed-point sublattices $L=\mathit{II}_{25,1}^{G}$ support such structure.

Elements $g$ of the Conway group $\mathrm{Co}_{0}=\mathrm{Aut}(\Lambda)$ have eigenvalues on the underlying 24-dimensional real vector space encoded by a cycle shape $\prod_{d}d^{n_{d}}$ with $\sum_{d}n_{d}d=24$, to which one attaches the eta-product $\eta_{g}(\tau)=\prod_{d}\eta(d\tau)^{n_{d}}$, where $\eta(\tau)=q^{1/24}\prod_{i>0}(1-q^{i})$ is Dedekind's eta function. The ten cycle shapes listed in the source are those of elements of the Mathieu sporadic simple group $M_{24}\subset\mathrm{Aut}(\Lambda)$, in its natural degree-24 permutation action, having squarefree order and level; they are the shapes $\prod_{t\mid m}t^{24/\sigma_{1}(m)}$ for $m$ in $\{1,2,3,5,6,7,11,14,15,23\}$, with $\sigma_{1}$ the sum-of-divisors function. For these the fixed-point sublattice $\Lambda^{g}$, of rank $24\sigma_{0}(m)/\sigma_{1}(m)$ with $\sigma_{0}$ the divisor-count function, is rootless - indeed it is the unique even lattice in its genus without roots - and, the order and level being squarefree and equal to $m$, the twisted denominator identities take their cleanest form. Two further $M_{24}$ cycle shapes, $1^{4}2^{2}4^{4}$ and $1^{2}2^{1}4^{1}8^{2}$, of non-squarefree level 4 and 8, were noted in the source not to work for the conjectured multiplicity formula; the later literature covers them by generalized Borcherds lifts on other lattices (Scheithauer 2009).

Beyond the 1990 source's own notion of "nice", Borcherds also required that the algebras be constructible naturally, not merely by generators and relations. The known mechanism is string-theoretic: the physical states of a chiral string compactification are the BRST cohomology (the cohomology of the Becchi-Rouet-Stora-Tyutin ghost differential, the standard complex of string quantization) of an explicit vertex (super)algebra, a vertex algebra being the algebraic structure underlying a chiral conformal field theory, and the no-ghost theorem, a result from string theory, identifies the resulting graded Lie (super)algebra with a free Lie (super)algebra on the physical state spaces, from which simple roots and multiplicities can be read off. The fake monster Lie algebra itself is the algebra of physical states of the bosonic string on a 26-dimensional torus, and the monster Lie algebra arises likewise from the monster vertex algebra. Borcherds conjectured (1992, 2001) that the other generalized Kac-Moody algebras with automorphic denominator identities, including all twisted algebras attached to elements of $\mathrm{Co}_{0}$, should admit comparable natural constructions, which would in particular make finite symmetry groups, such as sporadic groups, visible as automorphism groups.

Problem Statement

The source question (Borcherds 1990, final section, quoted in substance):

$\text{Let }G\text{ be a finite subgroup of }\mathrm{Aut}(\mathit{II}_{25,1})\text{ fixing }\rho\in\mathit{II}_{25,1}\text{, and let }L\text{ be the Lorentzian lattice of vectors fixed by }G.\text{ Is there a nice Lie algebra with root lattice }L\text{?}$

Here "nice" means, in the source's own elaboration for its conjectural cases: a generalized Kac-Moody (super)algebra with an explicit simple-root system, root multiplicities given in closed form by coefficients of modular forms (eta-products and theta series), a Weyl vector, and a closed-form denominator identity - in the strongest cases an automorphic product. The two concrete conjectures with which the source instantiated the question, the ten $M_{24}$ cycle-shape algebras and the rank-10 Lie superalgebra conjectured in the source, have been confirmed, so the open core of the problem, in the form recognized by the later literature, is the following. Each of (1)-(3) is an independently standing part of this open core, and a solution is any independently standing result that settles (1) or (2) for a new class of algebras or lattices, or resolves (3); no part is contingent on the others.

(1) Classification: determine all generalized Kac-Moody (super)algebras that have a Weyl vector and whose denominator functions are automorphic forms (Borcherds 1995 Problem 3, 1998 Problem 16.2, 2001 Problem 6); equivalently, classify the reflective automorphic forms and singular-weight Borcherds products that can serve as denominator identities, on the lattices arising here - for $G=\langle g\rangle$ with $g\in\mathrm{Co}_{0}$, the lattices $U(N_{g})\oplus U\oplus\Lambda^{g}$ (with $N_{g}$ the level of $g$, namely the level of the subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ under which the eta-product $\eta_{g}$ transforms, and $U(N)$ the lattice $U$ with one basis vector rescaled by $N$), together with the overlattices on which the corresponding products become reflective, and beyond the cyclic $\mathrm{Co}_{0}$-twist cases the fixed-point lattices $L=\mathit{II}_{25,1}^{G}$ for general finite $G$ fixing $\rho$. Partial classifications are known (squarefree and prime level; maximal lattices, meaning even lattices not properly contained in an even lattice of the same rational span; simple lattices, meaning those on which Borcherds products can be constructed without integrality obstruction; and small signatures); no full classification is available.

(2) Natural constructions: for every generalized Kac-Moody (super)algebra whose denominator identity is a singular-weight automorphic product, the distinguished case of the classification target in (1), construct it naturally so that its finite symmetry groups act visibly. This is established for the ten squarefree-level algebras attached to elements of $M_{23}$, the point stabilizer in $M_{24}$, in which the relevant automorphisms of squarefree order lie, and conjectured in general.

(3) The structural subquestion recorded in the source, whether these algebras connect directly to sporadic simple groups (the source noted there was no known direct connection even between the monster Lie algebra and the monster group), remains open.

References

1. Richard E. Borcherds, The monster Lie algebra, Advances in Mathematics 83 (1990) 30-47, DOI 10.1016/0001-8708(90)90067-W, https://doi.org/10.1016/0001-8708(90)90067-W
2. Richard E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Inventiones Mathematicae 109 (1992) 405-444, DOI 10.1007/BF01232032, https://doi.org/10.1007/BF01232032
3. Richard E. Borcherds, Automorphic forms on O_{s+2,2}(R) and infinite products, Inventiones Mathematicae 120 (1995) 161-213, DOI 10.1007/BF01241126, https://doi.org/10.1007/BF01241126
4. Richard E. Borcherds, The moduli space of Enriques surfaces and the fake monster Lie superalgebra, Topology 35 (1996) 699-710, DOI 10.1016/0040-9383(95)00036-4, https://doi.org/10.1016/0040-9383(95)00036-4
5. Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Inventiones Mathematicae 132 (1998) 491-562, DOI 10.1007/s002220050232, https://doi.org/10.1007/s002220050232
6. Richard E. Borcherds, Problems in moonshine, in First International Congress of Chinese Mathematicians (Beijing, 1998), AMS/IP Studies in Advanced Mathematics 20, American Mathematical Society, 2001, ISBN 978-0-8218-2652-2, volume DOI 10.1090/amsip/020, https://math.berkeley.edu/~reb/papers/iccm/iccm.pdf
7. J. H. Conway, The automorphism group of the 26-dimensional even unimodular Lorentzian lattice, Journal of Algebra 80 (1983) 159-163, DOI 10.1016/0021-8693(83)90025-x, https://doi.org/10.1016/0021-8693(83)90025-x
8. Valeri A. Gritsenko and Viacheslav V. Nikulin, Automorphic forms and Lorentzian Kac-Moody algebras, Part I, International Journal of Mathematics 9 (1998) 153-199, DOI 10.1142/S0129167X98000105, https://doi.org/10.1142/S0129167X98000105
9. Nils R. Scheithauer, The fake monster superalgebra, Advances in Mathematics 151 (2000) 226-269, DOI 10.1006/aima.1999.1886, arXiv math/9905113, https://arxiv.org/abs/math/9905113
10. Nils R. Scheithauer, Automorphic forms, fake monster algebras and hyperbolic reflection groups, Advances in Mathematics 164 (2001) 301-324, DOI 10.1006/aima.2001.2021, arXiv math/0007163, https://arxiv.org/abs/math/0007163
11. Nils R. Scheithauer, Twisting the fake monster superalgebra, Advances in Mathematics 164 (2001) 325-348, DOI 10.1006/aima.2001.2020, arXiv math/0001160, https://arxiv.org/abs/math/0001160
12. Nils R. Scheithauer, Generalized Kac-Moody algebras, automorphic forms and Conway's group I, Advances in Mathematics 183 (2004) 240-270, DOI 10.1016/S0001-8708(03)00088-4, https://doi.org/10.1016/S0001-8708(03)00088-4
13. Nils R. Scheithauer, On the classification of automorphic products and generalized Kac-Moody algebras, Inventiones Mathematicae 164 (2006) 641-678, DOI 10.1007/s00222-006-0500-5, https://doi.org/10.1007/s00222-006-0500-5
14. Nils R. Scheithauer, Generalized Kac-Moody algebras, automorphic forms and Conway's group II, Journal für die reine und angewandte Mathematik 625 (2008) 125-154, DOI 10.1515/crelle.2008.092, https://doi.org/10.1515/crelle.2008.092
15. Nils R. Scheithauer, The Weil representation of SL_2(Z) and some applications, International Mathematics Research Notices 2009 (8) 1488-1545, DOI 10.1093/imrn/rnn166, https://doi.org/10.1093/imrn/rnn166
16. Nils R. Scheithauer, Some constructions of modular forms for the Weil representation of SL_2(Z), Nagoya Mathematical Journal 220 (2015) 1-43, DOI 10.1017/S0027763000027161, https://doi.org/10.1017/S0027763000027161
17. Nils R. Scheithauer, Automorphic products of singular weight, Compositio Mathematica 153 (2017) 1855-1892, DOI 10.1112/S0010437X17007266, https://doi.org/10.1112/S0010437X17007266
18. Gerald Höhn and Nils R. Scheithauer, A natural construction of Borcherds' fake baby monster Lie algebra, American Journal of Mathematics 125 (2003) 655-667, DOI 10.1353/ajm.2003.0016, arXiv math/0312106, https://arxiv.org/abs/math/0312106
19. Thomas Creutzig, Alexander Klauer and Nils R. Scheithauer, Natural constructions of some generalized Kac-Moody algebras as bosonic strings, Communications in Number Theory and Physics 1 (2007) 453-477, DOI 10.4310/CNTP.2007.v1.n3.a1, arXiv 0801.1829, https://arxiv.org/abs/0801.1829
20. Gerald Höhn and Nils R. Scheithauer, A generalized Kac-Moody algebra of rank 14, Journal of Algebra 404 (2014) 222-239, DOI 10.1016/j.jalgebra.2014.01.028, arXiv 1009.5153, https://arxiv.org/abs/1009.5153
21. Sarah M. Harrison, Natalie M. Paquette and Roberto Volpato, A Borcherds-Kac-Moody superalgebra with Conway symmetry, Communications in Mathematical Physics 370 (2019) 539-590, DOI 10.1007/s00220-019-03518-0, arXiv 1803.10798, https://arxiv.org/abs/1803.10798
22. Sven Möller, Natural construction of ten Borcherds-Kac-Moody algebras associated with elements in M_23, Communications in Mathematical Physics 383 (2021) 35-70, DOI 10.1007/s00220-021-04018-w, arXiv 1905.09629, https://arxiv.org/abs/1905.09629
23. Haowu Wang and Brandon Williams, The fake monster algebra and singular Borcherds products, Advances in Mathematics 461 (2025) 110083, DOI 10.1016/j.aim.2024.110083, arXiv 2207.14518, https://arxiv.org/abs/2207.14518