In this paper we define an action of the Weyl group on the quiver varieties $M_{m,\lambda }\left(v\right)$ with generic $(m,\lambda )$.

In the cases $𝔸$ and $\widetilde{𝔸}$, we describe the seeds obtained by sequences of mutations from an initial seed. In the $\widetilde{𝔸}$ case, we deduce a linear representation of the group of mutations which contains as matrix entries all cluster variables obtained after an arbitrary sequence of mutations (this sequence is an element of the group). Nontransjective variables correspond to certain subgroups of finite index. A noncommutative rational series is constructed, which contains all this information.

The paper considers a fuzzification of the notion of quantaloid of K. I. Rosenthal, which replaces enrichment in the category of $\bigvee$-semilattices with that in the category of modules over a given unital commutative quantale. The resulting structures are called quantale algebroids. We show that their constitute a monadic category and prove a representation theorem for them using the notion of nucleus adjusted for our needs. We also characterize the lattice of nuclei on a free quantale algebroid. At...

We prove that generalized Verma modules induced from generic Gelfand-Zetlin modules, and generalized Verma modules associated with Enright-complete modules, are rigid. Their Loewy lengths and quotients of the unique Loewy filtrations are calculated for the regular block of the corresponding category 𝒪(𝔭,Λ).

Let $R$ be a fusion ring and $R_{ℂ}:=R{\otimes }_{\mathbb{Z}}ℂ$ be the corresponding fusion algebra. We first show that the algebra $R_{ℂ}$ has only one left (right, two-sided) cell and the corresponding left (right, two-sided) cell module. Then we prove that, up to isomorphism, $R_{ℂ}$ admits a unique special module, which is 1-dimensional and given by the Frobenius-Perron homomorphism FPdim. Moreover, as an example, we explicitly determine the special module of the interpolated fusion algebra $R\left(\mathrm{PSL}(2,q)\right):=r\left(\mathrm{PSL}(2,q)\right){\otimes }_{\mathbb{Z}}ℂ$ up to isomorphism, where $r\left(\mathrm{PSL}\right(2,q\left)\right)$ is the interpolated...

We investigate the structures of Hopf $*$-algebra on the Radford algebras over $ℂ$. All the $*$-structures on $H$ are explicitly given. Moreover, these Hopf $*$-algebra structures are classified up to equivalence.