Down-up algebras were introduced by \textit{G. Benkart} and \textit{T. Roby} [J. Algebra 209, No. 1, 305-344 (1998; Zbl 0922.17006)] as a generalization of the algebra determined by the down and up operators on a partially ordered set. Specifically, if \(K\) is a field, then \(A=A(\alpha,\beta,\gamma)\) is the \(K\)-algebra generated by the elements \(d\) and \(u\) subject to two defining relations of degree 3 that depend upon the parameters \(\alpha,\beta,\gamma\in K\). Since \(A(\alpha,\beta,\gamma)\cong A(\alpha,\beta,1)\) for all \(\gamma\neq 0\), there are essentially just two cases to consider, namely \(\gamma=0\) or 1. This paper is concerned with the \(\gamma=0\) situation, under the additional assumption that \(\beta\neq 0\). When \(\gamma=0\), the defining relations of \(A\) become homogeneous and hence \(A\) admits scalar-type automorphisms. In particular, if \(r,s\in K\) are the roots of the quadratic equation \(t^2-\alpha t-\beta=0\), then \(r\) and \(s\) are nonzero and they determine scalar-type automorphisms \(\omega_1\) and \(\omega_2\) of \(A\). Furthermore, if \(G\) is the Abelian subgroup of \(\Aut A\) generated by \(\omega_1\) and \(\omega_2\), then it is shown here that the skew group ring \(B=B(\alpha,\beta,0)=A*G\) is a Hopf algebra, with the elements of \(G\) being group-like and with \(d\) and \(u\) being \(\omega_1\)-primitive and \(\omega_2\)-primitive, respectively. At this point, the authors assume that \(K\) is an algebraically closed field of characteristic 0. They carefully describe the irreducible modules of \(A\) and then, via Clifford theory, the irreducible \(B\)-modules. This lifting process, from \(A\) to \(B\), is tedious and requires separate arguments for the various possible structures of \(G\). Finally, the authors describe the tensor products of those finite-dimensional simple \(B\)-modules on which both \(d\) and \(u\) act in a nilpotent fashion.