Hopf structures on the Hopf quiverQ(⟨g⟩,g)

Abstract

We investigate pointed Hopf algebras via quiver methods. We classify all the possible Hopf structures arising from the simplest Hopf quiver Q( g , g), which serves as a basic ingredient for the general ones. This provides the very local structure information for the general pointed Hopf algebras.

Actions of some pointed Hopf algebras on path algebras of quivers

“…1,3 for component (3). Putting this all together, the action of Z 3 on Q given by Step 1 extends to an action of T (3) on kQ, as follows: 12 x · f 11 = γ ω 2 f 11 − ωf 10 + λ ′′′ ωf 9 x · f 9 = γ (ωf 9 − ωf 7 ) + λ ′′ ω 2 f 11…”

The Green Rings of Minimal Hopf Quivers

“…The quiver Z is a cyclic quiver of form If n = 1, then Z is the one-loop quiver, that is, it consists of one vertex and one loop. It is easy to see that such a quiver provides only the familiar divided power Hopf algebra in one variable, which is isomorphic to the polynomial algebra in one variable [7].…”

“…By reconstruction theorem [4], a pointed tensor category with a fibre functor can be presented as the category of finite-dimensional right comodules over a pointed Hopf algebra. On the other hand, the tame hereditary condition is equivalent to saying that such a category is equivalent to the category of locally nilpotent finite-dimensional representations of a cyclic quiver or the infinite linear quiver (see [7,12]). Cyclic and infinite linear quivers are called minimal Hopf quivers as they are basic building blocks of general Hopf quivers [3,7].…”

“…On the other hand, the tame hereditary condition is equivalent to saying that such a category is equivalent to the category of locally nilpotent finite-dimensional representations of a cyclic quiver or the infinite linear quiver (see [7,12]). Cyclic and infinite linear quivers are called minimal Hopf quivers as they are basic building blocks of general Hopf quivers [3,7]. Note that finite-dimensional indecomposable representations of cyclic and infinite linear quivers are explicitly classified in quiver representation theory [1] and the Hopf structures over such quivers are given in [7].…”

Single-crystalline Bi19Br3S27 nanorods with an efficiently improved photocatalytic activity

“…[16][17][18] Recently, thanks to their narrow band gap and layered structure, many 45 bismuth-containing oxide materials, such as Bi 2 WO 6 , Bi 2 MoO 6 and BiVO 4 have shown admirable photocatalytic properties under visible light. [19][20][21] Especially because of the introduction of halogen in the compound system, bismuth oxyhalides (BiOX, X=F, Cl, Br and I) have received more 50 attention and showed enhanced photocatalytic activity compared with the corresponding metal oxides. [22][23][24][25] Compared with bismuth-containing oxide photocatalysts, bismuth sulfide (Bi 2 S 3 ) has drawn intensive attention for its potential applications in many areas (e.g.…”