Schur-Weyl Categories and Non-quasiclassical Weyl Type Formula

To a vector space V equipped with a non-quasiclassical involutary solution of the quantum Yang-Baxter equation and a partition $\lambda$, we associate a vector space $\text{Vl}$ and compute its dimension. The functor $V\mapsto \text{Vl}$ is an analogue of the well-known Schur functor. The category generated by the objects $\text{Vl}$ is called the Schur-Weyl category. We suggest a way to construct some related twisted varieties looking like orbits of semisimple elements in sl(n)^*. We consider in detail a particular case of such “twisted orbits”, namely the twisted non-quasiclassical hyperboloid and we define the twisted Casimir operator on it. In this case, we obtain a formula looking like the Weyl formula, and describing the asymptotic behavior of the function $N(\text{la})=\{\mathrm{♯}{\text{la}}_i\le \text{la}\}$, where ${\text{la}}_i$ are the eigenvalues of this operator.