Some subalgebras of the Monster Lie algebra and the modular polynomial
type: Item
instance of: scholarly article
title: Some subalgebras of the Monster Lie algebra and the modular polynomial
zbMATH Open document ID: 0929.17028
author: Urmie Ray
published in: Journal of Algebra
publication date: 2000-01-30T00:00:00Z
full work available at URL: http://cds.cern.ch/record/304038
review text: Let \(I\) be a finite or countable index set. A generalized Cartan matrix \(A = (a_{ij})_{i, j \in I}\) is a real symmetric matrix with nonpositive off-diagonal entries such that \(2 a_{ij}/a_{ii} \in {\mathbb Z}\) if \(a_{ii} > 0\). The universal generalized Kac-Moody (GKM) algebra \(\widehat G\) is a Lie algebra with generators \(e_i, f_i, h_{ij}\), where \(i, j \in I\). The generators satisfy the standard structural identities \[ [e_i, f_j] = h_{ij},\quad [h_{ij}, e_k] = \delta_{ij} a_{ik} e_k, \quad [h_{ij}, f_k] = - \delta_{ij} a_{ik} f_k, \quad [e_i, e_j] = 0 = [f_i, f_j] \text{ if } a_{ij} = 0, \] and for \(a_{ii} > 0\) the \((1 - 2 a_{ij}/a_{ii})\)th power of ad\( e_i\) (respectively, ad\( f_i\)) annihilates \(e_j\) (resp., \(f_j\)). A GKM algebra \(G\) is a semidirect product \((\widehat G K) \cdot D\), where \(D\) is an abelian subalgebra of outer derivations for which all \(e_i\) and \(f_j\) are simultaneous eigenvectors and \(C\) is a central subalgebra of \(\widehat G\). \textit{R.~Borcherds} [Invent.~Math. 109, 405-444 (1992; Zbl 0799.17014)] constructed the monster Lie algebra \(M\) and showed it to be a GKM with root lattice the rank two Lorentzian lattice. Roots of \(M\) are representable as pairs \((m, n)\) of integers. For a fixed positive integer \(r\), the root spaces corresponding to roots of \(M\) of the form \((m, r n)\) for integers \(m\) and \(n\) generate a GKM subalgebra \(M_r\). Let \(c(mrn)\) represent the multiplicity of such a root. The author first studies the dimension of the subspace of a positive root space that cannot be generated by positive root spaces for roots of smaller height (the restricted simple root multiplicities). This leads to an analytic continuation of the product \(\prod(1 - p^m q^n)^{c(r m n)}\) for \(m > 0, n \in {\mathbb Z}\) and \(p, q \in {\mathbb C}\) such that \(| p |, | q^{1/r} | < e^{-2 \pi}\) (corresponding to the denominator formula for \(M_r\)) to the product of two copies f the open unit disk. The ana…
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 11F22
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 20H20
zbMATH DE Number: 1019686
zbMATH Keywords: monster Lie algebra
zbMATH Keywords: modular polynomial
zbMATH Keywords: generalized Kac-Moody algebra
zbMATH Keywords: root space
zbMATH Keywords: Lorentzian lattice
zbMATH Keywords: analytic continuation
Wikidata QID: Q115396034
reviewed by: James F. Hurley
MaRDI profile type: Publication
OpenAlex ID: W2034732108
cites work: Generalized Kac-Moody algebras
cites work: Monstrous moonshine and monstrous Lie superalgebras
cites work: Q3693664
cites work: Q4723876
cites work: Q5631207
DOI: 10.1006/JABR.1996.6908
description: scientific article; zbMATH DE number 1019686