我们曾完成了一篇173页的关于量子仿射gln和仿射q-Schur代数的文章, 该文章已被London Mathematical Society Lecture Note Series作为专著接受发表. 我们在该文章中通过引入Double Hall代数的方法证明了有理函数域上的量子仿射gln到仿射q-Schur代数的自然代数同态是满射, 利用该结果, 以及仿射A型Hecke代数和量子仿射gln的表示理论, 我们给出了仿射q-Schur代数在非单位根时的有限维不可约模分类, 并利用仿射对称群的理论解决了仿射gln的普遍包络代数的BLM实现问题. 我们将继续研究量子仿射gln和仿射q-Schur代数的结构和表示理论. 另外我们还将研究A型的量子超代数的BLM实现问题, 小q-Schur代数和无穷小q-Schur代数的表示理论以及李共形代数的理论.

We have ever written a 173 pages article on quantum affine gln and affine q-Schur algebras,which is accepted by London Mathematical Society Lecture Note Series as a monograph. In this monograph, we proved that the natural algebra homomorphism from quantum affine gln to affine q-Schur algebras is surjective over rational function field by a double Hall algebra approach. Combining this result with representation theory of affine Hecke algebras of type A and quantum affine gln, we classified finite dimensional irreducible modules for affine q-Schur algebras at non-roots-of-unity. Furthermore, using the theory of affine symmetric group, we have solved the BLM realization problem of the universal enveloping algebra of affine gln. We will continue studying the structure theory and representation theory of quantum affine gln and affine q-Schur algebras. Furthermore we will study BLM realization problem of quantum hyperalgebra of type A, representation theory of little q-Schur algebras and infinitesimal q-Schur algebras, and the theory of Lie conformal algebra.