Representations of the Lie ring \(sl_2(\mathbb{Z})\) over the ring of integers (Q5945485)

Denote by \(S\) the Lie ring \(sl_2(\mathbb{Z})\), by \(\{ e,h,f\}\) the standard base of \(S\), and by \(\mathcal O\) the category of finite-dimensional \(S\)-modules without torsion. Given \(V\in \mathcal O\), denote by \(V_d\) the submodule generated by \(\{ v\in V\mid \exists i\in \mathbb{Z}\), \(vh=iv \}\). An \(S\)-module \(V\) is called diagonal if \(V=V_d\) and semisimple if it is a direct sum of irreducible modules. Let \(V_s\) be a maximal semisimple submodule of \(V\). A diagonal \(S\)-module \(V\) is called extremal if, for every diagonal \(S\)-module \(W\) such that \(V \subseteq W\), we have \(V_s\neq W_s\). The main problem of the theory of diagonal \(S\)-modules is the description of the structure of extremal modules. In the article under review, the author describes the structure of the tensor product \(V\otimes W\) and computes the index \(|V\otimes W:(V\otimes W)_s|\) of diagonal irreducible \(S\)-modules \(V\), \(W\).