A family of generalized Kac-Moody algebras and deformation of modular forms
type: Item
instance of: scholarly article
zbMATH Open document ID: 1321.11049
DOI: 10.1142/S1793042112500674
author: Henry H. Kim
author: Kyu-Hwan Lee
published in: International Journal of Number Theory
publication date: 2012-08-29T00:00:00Z
Mathematics Subject Classification ID: 11F22
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 6073794
zbMATH Keywords: generalized Kac-Moody algebra
zbMATH Keywords: modular form
zbMATH Keywords: denominator identity
MaRDI profile type: Publication
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title: A family of generalized Kac-Moody algebras and deformation of modular forms
review text: Let \(b(n)\), \(n \geq 1\) be a sequence of positive integers and \(b(-1) = 1\). Let \(M\) be a symmetric matrix of blocks indexed by \(\{-1,1,2,\ldots\}\) whose \((i,j)\) block is a \(b(i) \times b(j)\) matrix with entries \(-(i+j)\). Then \(M\) has only one positive diagonal entry. Let \(\mathfrak{g}(M)\) be a generalized Kac-Moody algebra associated to the matrix \(M\) in the sense of \textit{E. Jurisich} [J. Pure Appl. Algebra 126, No. 1--3, 233--266 (1998; Zbl 0898.17011)]. The authors first consider an analogue of the Gindikin-Karpelevich formula for \(\mathfrak{g}(M)\), which expresses a product over a set of positive roots as a sum over a certain PBW basis. It contains a parameter \(t\). Next, the authors take a specialization of the formula to obtain a deformation of a partial denominator identity. Applications of those deformations related to a weakly holomorphic modular form of weight \(1/2\) with respect to \(\Gamma_0(4)\) in the Kohnen plus space, the modular \(j\)-function and Ramanujan's \(\tau\)-function are also discussed.
reviewed by: Hiromichi Yamada
description: scientific article; zbMATH DE number 6073794