# Ge Li (University of Sydney)

## Friday 16th November, 12:05-12:55pm, Carslaw 173

### An integral basis theorem for cyclotomic KLR algebras of type $$A$$

Khovanov and Lauda and Rouquier have introduced a remarkable new family of algebras $$R_n$$, the quiver Hecke algebras, for each oriented quiver. The algebras $$R_n$$ are naturally $$\mathbb{Z}$$-graded. Brundan and Kleshchev proved that over a field $$F$$, the cyclotomic Khovanov-Lauda-Rouquier algebras $$R_n^\Lambda$$ are isomorphic to the cyclotomic Hecke algebras of type $$A$$, $$H_n^\Lambda$$ by constructing an explicit isomorphic mapping, which gives a $$\mathbb{Z}$$-grading to the cyclotomic Hecke algebras. Based on Brundan and Kleshchev's work, Hu and Mathas constructed a graded cellular basis with some restriction. In this talk I will show that such restriction can be removed and the graded cellular basis introduced by Hu and Mathas can be extended to $$R_n^\Lambda$$ over $$\mathbb{Z}$$. Furthermore we will show that the graded cellular basis can be extended to affine Khovanov-Lauda-Rouquier algebras and it gives a classification of all simple $$R_n$$-modules.