An Example of Double Cross Coproducts with Non-trivial Left Coaction and Right Coaction in Strictly Braided Tensor Categories

$U_q(sl_2)$ 的辫子余积、对极与伴随作用

Braided coproduct, antipode and adjoint action for $U_q(sl_2)$

本文为量子代数 $U_q(\mathfrak{sl}_2)$ 显式给出辫子余积、对极及伴随作用，并证明其辫子伴随作用与普通量子伴随作用相同。

Interrelations between Quantum Groups and Reflection Equation (Braided) Algebras

We show that the differential complex $Ω_{B}$ over the braided matrix algebra $BM_{q}(N)$ represents a covariant comodule with respect to the coaction of the Hopf algebra $Ω_{A}$ which is a differential extension of $GL_{q}(N)$. On the other hand, the algebra $Ω_{A}$ is a covariant braided comodule with respect to the coaction of the braided Hopf algebra $Ω_{B}$. Geometrical aspects of these results are discussed.

GL_q(N)-covariant braided differential bialgebras

We study a possibility to define the (braided) comultiplication for the GLq(N)-covariant differential complexes on some quantum spaces. We discover such `differential bialgebras' (and Hopf algebras) on the bosonic and fermionic quantum hyperplanes (with additive coproduct) and on the braided matrix algebra BMq(N) with both multiplicative and additive coproducts. The latter case is related (for N=2) to the q-Minkowski space and q-Poincare algebra.

Some Remarks on Braided Group Reconstruction and Braided Doubles

The cross coproduct braided group $Aut(C) \rcocross B$ is obtained by Tannaka-Krein reconstruction from $C^B\to C$ for a braided group $B$ in braided category $C$. We apply this construction to obtain partial solutions to two problems in braided group theory, namely the tensor problem and the braided double. We obtain $Aut(C)\rcocross Aut(C)\isom Aut(C)\lcross Aut(C)$ and higher braided group `spin chains'. The example of the braided group $B(R)\lcross B(R)\lcross...\lcross B(R)$ is described explicitly by R-matrix relations. We also obtain $Aut(C)\rcocross Aut(C)^*$ as a dual quasitriangular …

关于非酉代数余积的思考

Reflections on coproducts for non-unital algebras

本文针对非酉代数（具有非退化乘积）探讨余积与余结合性的定义问题，指出传统余积概念不适用，需借助乘子代数，并总结已有结果、补充新解答与开放问题。

Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups $U_q(g)$. They have the same FRT generators $l^\pm$ but a matrix braided-coproduct $\undΔL=L\und\tens L$ where $L=l^+Sl^-$, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices $BM_q(2)$; it is a braided-commutative bialgebra in a braided category…