arXiv subjects
Publications and source records attributed to Xiao-Wu Chen.
We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra $sl_2^c$.
Using certain pairings of couples, we obtain a large class of two-sided non-degenerated graded Hopf pairings for quantum symmetric algebras.
Let $A$ be a $(G, χ)$-Hopf algebra with bijection antipode and let $M$ be a $G$-graded $A$-bimodule. We prove that there exists an isomorphism \mathrm{HH}^*_{\rm gr}(A, M)\cong{\rm Ext}^*_{A{-}{\rm gr}} (\K, {^{ad}(M)}), where $\K$ is viewed as the trivial graded $A$-module via the counit of $A$, $^{ad} M$ is the adjoint $A$-module associated to the graded $A$-bimodule $M$ and $\mathrm{HH}_{\rm gr}$ denotes the $G$-graded Hochschild cohomology. As an application, we deduce that the cohomology of color Lie algebra $L$ is isomorphic to the graded Hochschild cohomology of the universal enveloping algebra $U(L)$, solving a question of M. Scheunert.
We introduce the graded bialgebra deformations, which explain Andruskiewitsch-Schneider's liftings method. We also relate this graded bialgebra deformation with the corresponding graded bialgebra cohomology groups, which is the graded version of the one due to Gerstenhaber-Schack.
Let $K$ be a field of characteristic 0 containing all roots of unity. We classify all the Hopf structures on monomial $K$-coalgebras, or, in dual version, on monomial $K$-algebras.
We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical $C_0$ of an arbitrary coalgebra $C$, we give an alternative definition of the Gabriel quiver of $C$, and then show that it coincides with the known $\operatorname {Ext}$ quiver of $C$ and the link quiver of $C$. The dual Gabriel theorem for a coalgebra with separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of $C_1$ of any coalgebra $C$, which can be regarded as a generalization of the first part of the well-known Taft-Wilson Theorem for pointed coalgebras. As applications, we give a characterization of locally finite coalgebras via their Gabriel quivers, and a property of the Gabriel quiver of a quasi-coFrobenius coalgebra.
The aim of this paper is to construct comodules of $U_q(sl_2)$ and modules of $SL_q(2)$ via quiver, where $q$ is not a root of unity. By embedding the quantized algebra $U_q(sl_2)$ into the path coalgebra $k\mathcal{D}^c$, where $\mathcal{D}$ is the Gabriel quiver of $U_q(sl_2)$ as a coalgebra, we obtain a basis of $U_q(sl_2)$ in terms of combinations of paths in the quiver $\mathcal{D}$; this special basis enable us to describe the category of $U_q(sl_2)$-comodules by certain representations of $\mathcal{D}$; and this description further permits us to construct a class of modules of the quantum special linear group $SL_q(2)$, from certain representations of $\mathcal{D}$, via the duality between $U_q(sl_2)$ and $SL_q(2)$.