We use the Le Bruyn–Procesi theorem to prove several results on quiver schemes for non-generic stability conditions: We show that the stability-zero finite ADE-type Nakajima quiver schemes are reduced points and give an example of a closely related non-reduced quiver scheme. In broader generality, we prove that adding modules supported on stability-zero vertices induces closed embeddings of quiver schemes, and show how in many cases the quiver scheme associated with the cornered algebra defines a limit to this system of embeddings. As an application, we show that there is an isomorphism between the underlying reduced schemes of certain equivariant Quot schemes and Nakajima quiver varieties.

In this note, we construct a family of non-commutative, non-cocommutative semisimple Hopf algebras $ H_{n,m} $ of dimension $ n^m m! $ over a field of characteristic zero containing a primitive $ n $th root of unity, where $ n, m \geq 2 $ are integers. The well-known eight-dimensional Kac–Paljutkin algebra arises as the special case $ H_{2,2} $, while the Hopf algebras previously constructed by Pansera correspond to the instances $ H_{n,2} $. Each algebra $ H_{n,m} $ is defined as an extension of the group algebra $ \mathbb{K} \Sigma_m $ of the symmetric group by the $ m $-fold tensor product $ R = \mathbb{K} \mathbb{Z}_n^{\otimes m} $, where $ \mathbb{Z}_n $ denotes the cyclic group of order $ n $. This extension admits a realization as a crossed product: $H_{n,m} = \mathbb{K} \mathbb{Z}_n^{\otimes m} \#_\gamma \Sigma_m.$ In the final section, we construct a family of irreducible $ m $-dimensional representations of $ H_{n,m} $ that are inner faithful as $ R $-modules and exhibit a non-trivial inner-faithful action of a subalgebra of $ H_{n,m} $ on a quantum polynomial algebra.