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Acta Mathematica Sinica,English Series 1996年第3期12卷 244-248页

作者： Hao Zhifeng, Department of Applied Mathematics South China University of Technology Guangzhou, 510641 ChinaTong Wenting, Department of Mathematics Nanjing University Nanjing, 210008 China Department of Applied Mathematics South China University of Technology Guangzhou China Department of Mathematics Nanjing University Nanjing China

In this paper, we obtain the following main theorem for a free quadratic bialgebra J: (a) For p≠0, J is a pointed cosemisimple coalgebra. For p=0, J is a hyperalgebra. (b) For p≠0 and q≠0, J has antipode S iff p·q+2=0 and S(x)=x. Forp=0 or q=0, J has antipode and S(x)=-x. (c) All left J*-modules are rational. Also, we give some applications in homological theory and algebraic K-theory.

关键词： Free quadratic bialgebra antipode Cosemisimple Rational module

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Galois linear maps and their construction

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Journal of Southeast University(English Edition) 2019年第4期35卷 522-526页

作者： Gu Yue Wang Wei Wang Shuanhong School of Mathematics Southeast UniversityNanjing 211189China Nanjing Research Institute of Electronic Engineering Nanjing 210007China

The condition of an algebra to be a Hopf algebra or a Hopf(co)quasigroup can be determined by the properties of Galois linear *** a bialgebra H,if it is unital and associative as an algebra and counital coassociative as a coalgebra,then the Galois linear maps T1 and T2 can be *** such a bialgebra H,it is a Hopf algebra if and only if T1 is ***,T1^-1 is a right H-module map and a left H-comodule map(similar to T2).On the other hand,for a unital algebra(no need to be associative),and a counital coassociative coalgebra A,if the coproduct and counit are both algebra morphisms,then the sufficient and necessary condition of A to be a Hopf quasigroup is that T1 is bijective,and T1^-1 is left compatible with ΔT1-11^r and right compatible with mT1-1^l at the same time(The properties are similar to T2).Furthermore,as a corollary,the quasigroups case is also considered.

关键词： Galois linear map antipode Hopf algebra Hopf(co)quasigroup

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The tensor algebras of Yetter-Drinfeld module

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上海师范大学学报（自然科学版） 2014年第5期43卷 523-528页

作者： Yanhua Wang School of Mathematics Shanghai University of Finance and Economics

The antipode of a Yetter-Drinfeld Hopf algebra is an anti-algebra and anti-coalgebra map is proved. It is also proved that the tensor algebra of Yetter-Drinfeld Hopf module is a Yetter-Drinfeld Hopf algebra.

关键词： Hopf algebra antipode comodule Yetter-Drinfeld Hopf algebras

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A Short Note on the Conjugation in the Leibniz-Hopf Algebra

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Journal of Mathematics and System Science 2014年第1期4卷 34-36页

作者： Neset Deniz Turgay Independent Scholar 126/7 Sok. No. 10/1 Daire 3 Evka 3 Bornova- Izmir 35050 Turkey

The Leibniz-Hopf algebra is the free associative Z - algebra with one generator in each positive degree and coproduct is given by the Cartan formula. It has been also known as the ＇ring ofnoncommutative symmetric functions＇ [1], and to be isomorphic to the Solomon Descent algebra [ 12]. This Hopf algebra has links with algebra,topology and combinatorics. In this article we consider another approach of proof for the antipode formula in the Leibniz-Hopf algebra by using some properties of words in [2].

关键词： Hopfalgebra Leibniz-Hopf algebra antipode Steenrod algebra.

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Some Notes on the Dual Leibniz-Hopf algebra

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Journal of Mathematics and System Science 2013年第11期3卷 573-576页

作者： Neset Deniz Turgay

The Leibniz-Hopf algebra is the free associative algebra with one generator in each positive degree and coproduct given by the Cartan formula. Quasi-symmetric functions are a generalisation of symmetric functions [7],and the algebra of quasi-symmetric functions appear as the dual of the Leibniz-Hopf algebra. The Leibniz-Hopf algebra and its dual are word Hopf algebras and play an important role in combinatorics, algebra and topology. We give some properties of words and consider an another view of proof for the antipode in the dual Leibniz-Hopf algebra.

关键词： Hopf algebra Leibniz-Hopf algebra antipode quasi-symmetric functions.

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