量子对称对和 Schur-Weyl 对偶

本项目主要研究量子代数实现、典范基构造、及Schur-Weyl对偶。主要内容包括：通过仿射B、D型的Hecke代数实现仿射A型量子群的一类特殊余理想子代数，并构造相应的典范基。这些余理想子代数与仿射A型量子群构成量子对称对。我们还将证明这些余理想子代数与各对应的仿射Hecke代数构成量子化的Schur-Weyl对偶。此外，我们还将考虑与有限B/C型对称对有关的高阶Schur-Weyl对偶。这些预期成果既是对量子对称对的开拓性工作，也是对经典Schur-Weyl对偶理论的重要完善。

This is a proposal on realization of some quantum algebras and their canonical bases, and on Schur-Weyl duality. We shall realize some coideal subalgebras of the quantum affine gl_n via the affine Hecke algebras of types B and D, and shall construct their canonical bases. These coideal subalgebras and the quantum affine gl_n form quantum symmetric pairs. We shall prove such a coideal subalgebra and the associated affine Hecke algebra admit a quantum Schur-Weyl duality. Furthermore, we shall consider higher level Schur-Weyl duality associated with symmetric pairs of finite type B/C. These prospective results are not only fundamental work for quantum symmetric pairs, but also important improvements for the classic Schur-Weyl duality.