Bases And Basis Elements Of Quantized-enveloping Algebras

We study the elements in the crystal bases of quantum groups which correspondto the exceptional modules and construct integral bases of symmetric affine envelopingalgebras. The main results we obtained are as following:First, for any finite dimensional hereditary algebra A, the composition subalgebraof its Hall algebra is isomorphic to the positive part of a quantum group U_v⁺ . Weconsider an exceptional A-module V_λ, i.e. ExtA1(V_λ,V_λ) = 0. We denote by u_λthecorresponding element of V_λin the Hall algebra. And the dimension vector of V_λisdenoted byα. We know that the set of all A-modules with dimension vectorαis analgebraic variety (actually it is an affine space). The orbit OV_λof V_λunder the action ofthe algebraic group G_αis a dense open subset. Thus intuitively, the element uλshouldbe the leading term of a canonical basis element of U_v⁺ . However, this result can not bededuced easily from the geometric construction of the canonical basis given by Lusztig.We give a direct proof of this result by the Hall algebra approach. More precisely, weshow that uλlies in the integral form U_v⁺,Z using Lusztig's symmetries and the braidgroup action on the set of exceptional sequences. And then, by a characterization ofthe crystal basis given by Kashiwara using a bilinear form, we prove that u_λlies inthe crystal basis (up to a sign). Finally, we show that the sign could be removed usinggeometric method. Note that our results are valid for any type of quantum group.Secondly, for any tame quiver Q, it corresponds to a symmetric affine envelopingalgebra U(n⁺). By the Hall algebra approach, we construct a Z-basis of the integralform of the enveloping algebra U_Z(n⁺). The difference between our method and oth-ers is that the representation theory of tame quivers, especially the Auslander-Reiten-quiverΓQ, is essentially used in our construction. We construct basis elements fromthe preprojective, regular and preinjective component ofΓ_Q respectively. In the regularcomponent, we construct basis elements from each non-homogeneous tube, and by anembedding of the representation category of the Kronecker quiver we obtain basis ele-ments arising from homogeneous tubes. Finally the desired basis consists of the orderedproducts of these basis elements where the order is given by theΓ_Q. Also we obtain a Z-triangular decomposition: , whereC_Z(Q)^Prep, C_Z(Q)^Reg and C_Z(Q)^Prei are the subalgebras generated by the basis elementsarising from the preprojective, regular and preinjective component respectively.