Quantum polynomial functors

We define the quantum analogue of polynomial functors, and show that the category of quantum polynomial functors of fixed degree is equivalent to category of modules over q-Schur algebra. We construct quantum analogue of Schur functors in the sense of Akin-Buchsbaum-Weyman and show it gives Schur modules of quantum general linear group after evaluations. As an application, we obtain a proof of quantum Howe duality, and hence proofs of quantum Schur-Weyl duality and the FFT of quantum invariant theory. This talk is based on the ongoing joint work with Oded Yacobi.