Root vectors for Hecke algebras and quantized enveloping algebras (Q1895638)

Let \(C\) be the Cartan matrix of a simply semisimple Lie algebra. Let \(U\) be the corresponding quantized enveloping algebra associated to \(C\), and \(B\) its braid group, \(H\) its Hecke algebra. In this paper, the author defines the root vectors for \(U\) using the action of \(B\) on \(U\) defined by Lusztig and the root vectors of \(H\) using its reflection representation. He shows that these two notions are in bijection. In particular, he considers the case of type \(A_n\), and gives the decomposition of the positive root vectors of \(U\) in terms of a PBW basis of \(U^+\) and also in terms of Lusztig's canonical basis of \(U^+\). Finally he gives the number of root vectors associated to a root \(\alpha\) for type \(A_n\) \((n \geq 1)\), \(D_n\) \((n \geq 4)\), \(E_6\), and \(E_7\) and a partial description of the root vectors for the Hecke algebra of the affine Weyl group of type \(\widetilde A_n\).