A family of generalized Kac-Moody algebras and deformation of modular forms
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.
scientific article; zbMATH DE number 6073794