Schur-Weyl Categories and Non-quasiclassical Weyl Type Formula

Details Schur-Weyl Categories and Non-quasiclassical Weyl Type Formula Schur-Weyl Categories and Non-quasiclassical Weyl Type Formula

1999-11-18

arxiv.org/abs/math/9911139v1

Mathematics

Quantum Algebra

Latex file, 35 pages

Scientific paper

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 $\Vl$ and compute its dimension. The functor $V\mapsto \Vl$ is an analogue of the well-known Schur functor. The category generated by the objects $\Vl$ 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 $N(\la)=\{\sharp \la_i\leq\la\}$, where $\la_i$ are the eigenvalues of this operator.

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