Tight Monomials Of Quantum Enveloping Algebras

In [32] and [34], Lusztig constructed canonical bases for quan-tum enveloping algebras. Meanwhile, Kashiwara [19] defined global crystal bases for quantum enveloping algbras. And then Lusztig [33] proved that the canonical bases coincide with the global crys-tal bases. In 1993, Lusztig introduced tight monomials in [36]-monomials in the canonical bases.As to tight monomials, Lusztig, Reineke, Deng and Du gave the criterion respectively. Recently, M. Khovanov and A. Lauda categorified the quantum enveloping algebras and found that the tight monomials had close relationship with discomposable modules.In this paper, based on the results known, we discuss the tight monomials for the quantum enveloping algebra of type A3 and the quantum enveloping algebras associated to rank 2 (generalized) Car-tan matrices. Then we generalize the definition of Lusztig cone in order to reveal the relation between the tight monomials and Lusztig cone. We also found that the length of the tight monomials is closely related to the length of the longest element in the Weyl group. Nat-urally, we have the following conjecture for all quantum enveloping algebras:When C is the Cartan matrix of finite type, the length of the longest tight monomial of U is finite, equal to the length of the longest word in the corresponding Weyl group. Otherwise, there exist tight monomials of any length.In [9], Carter and Marsh raised some questions about the linear regions of quantum enveloping algebras of type A. Now making use of Lusztig's work, we can raise the same questions about the linear regions of quantum enveloping algebras of some related types.