Guanglian Zhang

Representations of Tame Quivers and Affine Canonical Bases

An integral PBW-basis of type

Setwise Homotopy Category

A_1^{(1)}

Linear quivers, generic extensions and Kashiwara operators

has been constructed by Zhang [Z] and Chen [C] using the Auslander-Reiten quiver of the Kronecker quiver. We associate a geometric order to elements in this basis following an idea of Lusztig [L1] in the case of finite type. This leads to an algebraic realization of a bar-invariant basis of

Derived Categories and Lie Algebras

\uq2

BGP reflection functors in root categories

. For any affine symmetric type, we obtain an integral PBW-basis of the generic composition algebra, by using an algebraic construction of the integral basis for a tube in [DDX], an embedding of the module category of the Kronecker quiver into the module category of the tame quiver, and a list of the root vectors of indecomposable modules according to the preprojective, regular, and preinjective components of the Auslander-Reiten quiver of the tame quiver. When the basis elements are ordered to be compatible with the geometric order given by the dimensions of the orbit varieties and the extension varieties, we can show that the transition matrix between the PBW-basis and a monomial basis is triangular with diagonal entries equal to 1. Therefore we obtain a bar-invariant basis. By a orthogonalization for the PBW-basis with the inner product, we finally give an algebraic way to realize the canonical bases of the quantized enveloping algebras of all symmetric affine Kac-Moody Lie algebras.

Equivariant vector bundles on quantum homogeneous spaces

We introduce the setwise homotopy relation and prove that two chain maps induce the same cohomology map if and only if they are setwise homotopic.

The Green formula and heredity of algebras

Abstract In the present paper, we introduce the generic extension graph G of a Dynkin or cyclic quiver Q and then compare this graph with the crystal graph C for the quantized enveloping algebra associated to Q via two maps ℘ Q , Q : Ω → Λ Q induced by generic extensions and Kashiwara operators, respectively, where Λ Q is the set of isoclasses of nilpotent representations of Q , and Ω is the set of all words on the alphabet I , the vertex set of Q . We prove that, if Q is a (finite or infinite) linear quiver, then the intersection of the fibres ℘ Q −1 (λ) and K Q −1 (λ) is non-empty for every λ ∈ Λ Q . We will also show that this non-emptyness property fails for cyclic quivers.