Some subalgebras of the Monster Lie algebra and the modular polynomial
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 analytic continuation is a multivalued function whose coefficients are the multiplicities of the simple roots of \(M_r\). It is a rapidly converging infinite product of modular functions of weight 0 and level \(r^n, n \in {\mathbb N}\). With computer assistance it is possible to calculate those multiplicities, and the author displays a table of multiplicities of the root \((m, 2 n)\) for small \(m\) and \(n\). For \(r > 1\) the orders of the zeros are fractional, so no power of the function is expressible as a polynomial in modular functions. Letting \(j\) denote the normalized Hauptmodul of weight 0 and level 1 from the upper half plane \({\mathcal H}\) to \({\mathbb C}\) with \(q\)-expansion \(\sum_{n \in {\mathbb Z}}c(n) q^n\) with \(c(-1) = 1, c(n) = 0\) for \(n < -1\) or \(n = 0\), and \(c(1) = 196884\), the author studies functions \(j \circ \alpha\) for rational matrices \(\alpha\). This leads to the modular polynomial \(\Phi_n(x, j)\) of order \(n\) belonging to \({\mathbb Z}[x, j]\). A product expansion is derived for \(\Phi_n(j(\tau), j)(\sigma)\), where \(\tau, \sigma \in {\mathcal H}\).
scientific article; zbMATH DE number 1019686