Classification of Ramification Systems for Symmetric Groups
We obtain the formula computing the number of isomorphic classes of ramification systems with characters over group $S_n$ ($n \not=6$) and their representatives.
math.SP↗
arXiv subjects
Shouchuan Zhang
Publications and source records attributed to Shouchuan Zhang.
Double Bicrosssum of Braided Lie algebras
The condition for double bicrosssum to be a braided Lie bialgebra is given. The result generalizes quantum double, bicrosssum, bicrosscosum, bisum. The quantum double of braided Lie bialgebras is constructed. The relation between double crosssum of Lie algebras and double crossproduct of Hopf algebras is given.
math.QA↗
The Factorization of Braided Hopf Algebras
We obtain the double factorization of braided bialgebras or braided Hopf algebras, give relation among integrals and semisimplicity of braided Hopf algebra and its factors.
math.RA↗
Structures and Representations of Generalized Path Algebras
It is shown that an algebra $Λ$ can be lifted with nilpotent Jacobson radical $r = r(Λ)$ and has a generalized matrix unit $\{e_{ii}\}_I$ with each $\bar e_{ii} $ in the center of $\bar Λ= Λ/r$ iff $Λ$ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, $Λ$ is a finite algebra with non-zero unity element over perfect field $k$ (e.g. a field with characteristic zero or a finite field) iff $Λ$ is isomorphic to a generalized path algebra $k (D, Ω, ρ)$ of finite directed graph with weak relations and $dim {\} Ω< \infty $; $Λ$ is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise orthogonal idempotents iff $Λ$ is isomorphic to a path algebra with relations.
math.RA↗
On pointed Hopf algebras associated to some conjugacy classes in S_n
We show that any pointed Hopf algebra with infinitesimal braiding associated to the conjugacy class of pi in S_n is infinite-dimensional, if either the order of pi is odd, or pi is a product of disjoint cycles of odd order except for exactly two transpositions.
math.QA↗
Braided m-Lie Algebras
Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of $End_F M$, where $M$ is a Yetter-Drinfeld module over $B$ with dim $B< \infty $ . In particular, generalized classical braided m-Lie algebras $sl_{q, f}(GM_G(A), F)$ and $osp_{q, t} (GM_G(A), M, F)$ of generalized matrix algebra $GM_G(A)$ are constructed and their connection with special generalized matrix Lie superalgebra $sl_{s, f}(GM_{{\bf Z}_2}(A^s), F)$ and orthosymplectic generalized matrix Lie super algebra $osp_{s, t} (GM_{{\bf Z}_2}(A^s), M^s, F)$ are established. The relationship between representations of braided m-Lie algebras and their associated algebras are established.
math.RA↗
Hopf Galois Extension in Braided Tensor Categories
The relation between crossed product and $H$-Galois extension in braided tensor category ${\cal C}$ with equivalisers and coequivalisers is established. That is, it is shown that if there exist an equivaliser and a coequivaliser for any two morphisms in ${\cal C}$, then $A = B #_σH$ is a crossed product algebra if and only if the extension $A/B$ is Galois, the canonical epic $q: A\otimes A \to A\otimes_B A$ is split and $A$ is isomorphic as left $B$-modules and right $H$-comodules to $B\otimes H$ in ${\cal C}$.
math.RA↗
Classical Yang- Baxter Equation and Low Dimensional Triangular Lie Bialgebras Over Arbitrary Field
Let $ L $ be a Lie algebra over arbitrary field $ k $ with dim $ L $ =3 and dim $ L' $ =2. All solutions of constant classical Yang- Baxter equation (CYBE) in Lie algebra $ L $ are obtained and the necessary conditions which $ (L,[\ ],Δ_{r}, r) $ is a coboundary (or triangular) Lie bialgebra are given.
math.RA↗
Integrals of Braided Hopf Algebras
The faithful quasi-dual $H^d$ and strict quasi-dual $H^{d'}$ of an infinite braided Hopf algebra $H$ are introduced and it is proved that every strict quasi-dual $H^{d'}$ is an $H$-Hopf module. The connection between the integrals and the maximal rational $H^{d}$-submodule $H^{d rat}$ of $H^{d}$ is found. That is, $H^{d rat}\cong \int ^l_{H^d} \otimes H$ is proved. The existence and uniqueness of integrals for braided Hopf algebras in the Yetter-Drinfeld category $(^B_B{\cal YD},C)$ are given.
math.QA↗
Generalized Path Algebras and Pointed Hopf Algebras
Most of pointed Hopf algebras of dimension $p^m$ with large coradical are shown to be generalized path algebras. By the theory of generalized path algebras it is obtained that the representations, homological dimensions and radicals of these Hopf algebras. The relations between the radicals of path algebras and connectivity of directed graphs are given.
math.RA↗
An Example of Double Cross Coproducts with Non-trivial Left Coaction and Right Coaction in Strictly Braided Tensor Categories
An example of double cross coproducts with both non-trivial left coaction and non-trivial right coaction in strictly braided tensor categories is given.
math.QA↗
On the Lie's Theorem for Lie Color Algebras
We show that the Lie's Theorem holds for Lie color algebras with a torsion-free abelian group $G$. We give an example to show that the torsion-free condition is necessary.
math.RA↗
Duality Theorem and Drinfeld Double in Braided Tensor Categories
Let $H$ be a finite Hopf algebra with $C_{H,H} = C_{H,H}^{-1}.$ The duality theorem is shown for $H$, i.e., $$ (R # H)# H^{\hat *} \cong R \otimes (H \bar \otimes H^{\hat *}) \hbox {as algebras in} {\cal C}.$$ Also, it is proved that the Drinfeld double $(D(H),[b])$ is a quasi-triangular Hopf algebra in ${\cal C}$.
math.RA↗
The Baer Radical Of Generalized Matrix Rings
In this paper, we introduce a new concept of generalized matrix rings and build up the general theory of radicals for g.m.rings. Meantime, we obtain $$\bar{r}_b(A)=g.m.r_b(A)=\sum\{r_b(A_{ij})\mid i, j\in I\}=r_b(A)$$
math.RA↗
The Von Neumann Regular Radical and Jacobson Radical of Crossed Products
We construct the $H$-von Neumann regular radical for $H$-module algebras and show that it is an $H$-radical property. We obtain that the Jacobson radical of twisted graded algebra is a graded ideal. For twisted $H$-module algebra $R$, we also show that $r_{j}(R#_σH)= r_{Hj}(R)#_σH$ and the Jacobson radical of $R$ is stable, when $k$ is an algebraically closed field or there exists an algebraic closure $F$ of $k$ such that $r_j(R \otimes F) = r_j(R) \otimes F$, where $H$ is a finite-dimensional, semisimple, cosemisimple, commutative or cocommutative Hopf algebra over $k$. In particular, we answer two questions J.R.Fisher asked.
math.QA↗
Homological Dimension of Crossed Products
We obtain that the global dimensions of $R$ and the crossed product $R # _σH$ are the same; meantime, their weak dimensions are also the same, when $H$ is finite-dimensional semisimple and cosemisimple Hopf algebra.
math.QA↗
The relation between the decomposition of comodules and coalgebras
T. Shudo and H. Miyamito \cite{SM78} showed that $C$ can be decomposed into a direct sum of its indecomposable subcoalgebras of $C$. Y.H. Xu \cite {XF92} showed that the decomposition was unique. He also showed that $M$ can uniquely be decomposed into a direct sum of the weak-closed indecomposable subcomodules of $M$(we call the decomposition the weak-closed indecomposable decomposition) in \cite{XSF94}. In this paper, we give the relation between the two decomposition. We show that if $M$ is a full, $W$-relational hereditary $C$-comodule, then the following conclusions hold: (1) $M$ is indecomposable iff $C$ is indecomposable; (2) $M$ is relative-irreducible iff $C$ is irreducible; (3) $M$ can be decomposed into a direct sum of its weak-closed relative-irreducible subcomodules iff $C$ can be decomposed into a direct sum of its irreducible subcoalgebras. We also obtain the relation between coradical of $C$- comodule $M$ and radical of algebra $C(M)^*$
math.QA↗
The Radicals of Hopf Module Algebras
The characterization of $H$-prime radical is given in many ways. Meantime, the relations between the radical of smash product $R # H$ and the $H$-radical of Hopf module algebra $R$ are obtained.
math.QA↗