Finite-dimensional Jordan algebras admitting the structure of a Jordan bialgebra

We define the concepts of a triangular and a quasitriangular Jordan bialgebras. It is proved that a finite-dimensional Jordan algebra J over an algebraically closed field Φ admits the structure of a quasitriangular Jordan bialgebra with nonzero comultiplication, provided that J is not a direct sum of fields, algebras H(Φ₂) and H(Φ₃), null extensions of Φ, and of algebras with zero multiplication. Supported by RFFR grant No. 98-01-01142. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 40–67, January–February, 1999.     

The structure of Hopf algebras

This survey considers work published between 1970 and 1990 on the algebraic aspects of Hopf theory. There are detailed discussions of the properties of antipodes, group and primitive elements, integrals, crossed products, Galois theory, Lie coalgebras, the category of Hopf algebras, quantum groups, etc.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 29, pp. 3–63, 1991.     

Construction of Rota-Baxter algebras via Hopf module algebras

We propose the notion of Hopf module algebra and show that the projection onto the subspace of coinvariants is an idempotent Rota-Baxter operator of weight-1. We also provide a construction of Hopf module algebras by using Yetter-Drinfeld module algebras. As an application,we prove that the positive part of a quantum group admits idempotent Rota-Baxter algebra structures.     

Hopf bifurcation of a predator-prey system with stage structure and harvesting

In this paper, a two-species predator-prey system with stage structure and harvesting is investigated. The existence of Hopf bifurcations of the system is given. And the stability and directions of Hopf bifurcations are determined by applying the normal form theory and the center manifold theorem.     

The module structure of integral designs

The existence problem for t-designs with prescribed parameters is solved by allowing positive and negative integral multiplicities for the blocks.     

Hopf structure in nambu-lien-algebras

We define and study the Hopf structures in the ternary (and n-ary) Nambu-Lie algebra. The fundamental concepts of 3-coalgebra, 3-bialgebra, and Hopf 3-algebra are introduced. Some examples of the Hopf structures and analyzed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 1, pp. 87–93, January, 1998.     

Free Rota-Baxter systems and a Hopf algebra structure

In this paper, we give a linear basis of a free Rota-Baxter system on a set by using the Gröbner-Shirshov bases method and then we obtain a Hopf algebra structure on a free Rota-Baxter system.     

On the growth of algebras with bialgebra action

Ifk is a field,A ak-algebra, andB ak-bialgebra which acts onA, we study the rate of growth ofA under its algebra structure together with the action ofB. We then briefly place our results in the more general context of a vector spaceA on which anoperad acts, and sketch an application to a (still open) question on finitely generated subalgebras of free associative algebras. Dedicated to the memory of Shimshon Amitsur 1991Mathematics Subject Classifications: Primary: 11N45, 16P90, 16W30; secondary: 08B20, 08C15, 11P99, 13N99, 17A01. Work done while the author held NSF contract DMS 93-03379. An erratum to this article is available at .     

Stability and Hopf bifurcation analysis of an eco-epidemic model with a stage structure

In this paper, an eco-epidemiological model with a stage structure is considered. The asymptotical stability of the five equilibria, the existence of stability switches about positive equilibrium, is investigated. It is found that Hopf bifurcation occurs when the delay τ passes though a critical value. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

The Structure of Hopf Algebras with a Weak Projection

We analyze the structure of a Hopf algebra H that has a Hopf subalgebra H " and a left H "-module coalgebra projection onto H ". In this situation H H " Q for Q = H / H "⁺ H, and the Hopf algebra structure on H can be recovered from suitable structures on Q, among others an in general nonassociative multiplication. The construction of H from H " and Q generalizes Radford biproduct, double crossproducts, and certain bicrossproducts. Further examples are Hopf algebras with a triangular decomposition, like all quantized enveloping algebras. In an appendix, we improve a standard criterion for a bicrossproduct A B of two Hopf algebras to be a Hopf algebra, and we show that in this case the antipode of the bicrossproduct is bijective if the antipodes of the factors are.     

弱模余代数的结构定理

设H为弱Hopf代数,C为弱右H-模余代数,令C=C/C·ker L.利用Smash余积来研究弱模余代数上的结构定理,并给出了C与C×H作为余代数同构的条件.     

The structure theorem for weak module coalgebras

Let H be a weak Hopf algebra, let C be a weak right H-module coalgebra, and let $ \bar C = {C \mathord{\left/ {\vphantom {C C}} \right. \kern-0em} C} \cdot Ker \sqcap ^L $ . We prove a structure theorem for weak module coalgebras, namely, C is isomorphic as a weak right H-module coalgebra to a weak smash coproduct $ \bar C $ × H defined on a k-space $$ \{ \Sigma c_{(0)} \otimes h_2 \varepsilon (c_{( - 1)} h_1 )|c \in C,h \in H\} $$ if there exists a weak right H-module coalgebra map ?: C → H.     

The structure theorem for comodule algebras over Hopf algebroids

We obtain the structure theorem for $(H, \mathcal{H})$ -Hopf bimodules over Hopf algebroids, where H is the total algebra of the Hopf algebroid  $\mathcal{H}$ . Based on this theorem, we investigate the structure theorem for comodule algebras over Hopf algebroids.     

一类具有时滞的病菌-免疫力模型的Hopf分歧

考虑病菌的一种信息交流机制,建立一类病菌与免疫系统竞争的时滞传染病模型.分析正平衡点的存在性、渐近稳定性、Hopf分歧的存在性及方向.运用计算机数值模拟验证所得理论结果,为传染病的控制和预防提供了理论基础和数值依据.