Abstract
We define a new category of quantum polynomial functors extending the quantum polynomials introduced by Hong and Yacobi. We show that our category has many properties of the category of Hong and Yacobi and is the natural setting in which one can define composition of quantum polynomial functors. Throughout the paper we highlight several key differences between the theory of classical and quantum polynomial functors.
Highlights
Hong and Yacobi [10] introduced a category of quantum polynomial functors which quantizes the strict polynomial functors of Friedlander and Suslin [7]
The purpose of this paper is to introduce higher level categories of quantum polynomial functors that extend the construction of Hong and Yacobi and explain why they give a natural quantization of classical polynomial functors
A polynomial functor is defined as a functor between vector spaces which is polynomial on the space of morphisms
Summary
Hong and Yacobi [10] introduced a category of quantum polynomial functors which quantizes the strict polynomial functors of Friedlander and Suslin [7]. The root of unity case is significantly harder than the generic q case, since the domain category of the functors in Pqd,e consists of degree e polynomial representations of Uq (gln), which is more complicated the corresponding category for generic q. We consider another category of polynomial functors, whose definition involves restricting the domain; we denote the new category by Pq◦,,ed This category has a projective generator as shown in Theorem 6.15 and we explain that when e = 1 we get back the main result of [10] in full generality in Remark 6.17. The existence of the projective generators allows us to conclude the equivalence between the categories Pq◦,,ed and Pqd,e and the category of finite dimensional modules of certain Schur algebra. 3, we define the categories Pqd,e, the main objects to be studied in this paper and present several interesting examples of quantum polynomial functors.
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