The Grothendieck ring of vector spaces with two idempotent endomorphisms (Q1173881)

Let \(k\) be an algebraically closed field of characteristic 0. The author computes the Grothendieck ring of \(\mathbb{Z}/e\mathbb{Z}\)-graded \(k[x]\)-modules for any integer \(e\geq 2\) by explicitly determining the decomposition of the tensor product of two indecomposable modules. The results are applied to compute the Grothendieck ring of a particular bialgebra over \(k\), which is related to the universal measuring bialgebra of \(k\times k\).