Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 4 2 2023 07 01 A generalization of the Heisenberg group 105 119 172322 10.30504/jims.2023.396137.1113 EN L. Székelyhidi Institute of Mathematics, University of Debrecen 0000-0001-8078-6426 Journal Article 2023 05 06 In our former paper we studied spectral synthesis on the Heisenberg group. This problem is closely connected with the finite dimensional representations of the Heisenberg group on the space of continuous complex valued functions. In this paper we make an attempt to generalise the Heisenberg group over any commutative topological group. Starting with a basic commutative topological group we define a non-commutative topological group whose elements are triplets consisting of an element of the basic group, an exponential on the basic group, and a nonzero complex number which serves as a scaling factor. The group operation is a combination of the addition on the basic group, the multiplication of the exponentials and the multiplication of complex nonzero numbers. Although there is no differentiability, our generalised Heisenberg group shares some basic properties with the classical one. In particular, we describe finite dimensional representations of this group on the space of continuous functions, and we show that finite dimensional translation invariant function spaces over this group consist of exponential polynomials. https://jims.ims.ir/article_172322_9feb729370eb981497006b45c944bb2b.pdf