Let H be a commutative, noetherian, semisimple and cosemisimple Hopf algebra with a bijective antipode over a field k. Then the semisimplicity of YD（H） is considered, where YD （H） means the disjoint union of the c...Let H be a commutative, noetherian, semisimple and cosemisimple Hopf algebra with a bijective antipode over a field k. Then the semisimplicity of YD（H） is considered, where YD （H） means the disjoint union of the category of generalized Yetter-Drinfeld modules nYD^H（ α, β） for any α, β E Aut Hopf（H）. First, the fact that YD（H） is closed under Mor is proved. Secondly, based on the properties of finitely generated projective modules and semisimplicity of H, YD（H） satisfies the exact condition. Thus each object in YD（H） can be decomposed into simple ones since H is noetherian and cosemisimple. Finally, it is proved that YD （H） is a sernisimple category.展开更多
Let H be a Hopf algebra and HYD the Yetter- Drinfeld category over H. First, the enveloping algebra of generalized H-Hom-Lie algebra L, i.e., Hom-Lie algebra L H in the category HYD, is constructed. Secondly, it is o...Let H be a Hopf algebra and HYD the Yetter- Drinfeld category over H. First, the enveloping algebra of generalized H-Hom-Lie algebra L, i.e., Hom-Lie algebra L H in the category HYD, is constructed. Secondly, it is obtained that U（L） = T（ L）/L where I is the Hom-ideal of T（L） generated by {ll＇-l_（（-1））·l＇l_0-[l,l＇]｜l,l＇∈L}, and u： L,T（L）/I is the canonical map. Finally, as the applications of the result, the enveloping algebras of generalized H-Lie algebras, i.e., the Lie algebras in the category MyDn and the Hom-Lie algebras in the category of left H-comodules are presented, respectively.展开更多
Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of ...Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of the crossed product in the classic Hopf algebra theory, first, let A be a k-linear category and H be a Hopf algebra, and the two crossed products A#_σH and A#＇_σH are isomorphic under some conditions. Then, let A#_σH be a crossed product category for a finite dimensional and semisimple Hopf algebra H. If V is a left A#σH-module and WC V is a submodule such that W has a complement as a left A-module, then W has a complement as a A#_σH-module.展开更多
Although E Maddy （1997） says on naturalism： ＂This is not, in itself, a philosophy of mathematics [...]＂ （161）, already by its name, or by those whose interest has called on it （Quine, Putnam et al.） ... it an...Although E Maddy （1997） says on naturalism： ＂This is not, in itself, a philosophy of mathematics [...]＂ （161）, already by its name, or by those whose interest has called on it （Quine, Putnam et al.） ... it anyhow reveals desire to be it. Insofar as otherwise, the semantic potential of the word leaves far behind it （after all scarce） results it achieved from the relation of an exact （mathematical） expression and （overly rich） intuitive reality of Being. We plead here already from the perspective of the slogan ＂One and All＂ of the first philosopher： Tales, when by the number （which one forebodes） one could go to such an extent into areas of reality （Pythagoras）, or when （especially in the human sphere） is being over again actual final cause of Aristotle the philosophy and the mathematics to accomplish far more fruitful encounter with the Being. Alain Badiou （1988） has already pointed that： ＂Mathematics is ontology,＂ and the category theory in mathematics, having covered by itself other fields of this science, continues to find applications in a series of＂non-traditional＂ domains of reality. In that correlation the philosophy can express its （primary） needs for truth, justice, beauty, ... as well as for the overall development in the sense of purpose--also because of an undreamed power of the technological development （of hardwares and softwares） today. Namely, the naturalism in mathematics, which developed an abundant reflection on the place （importance of） the mathematical idiom in sciences--in the balance of criticism--has come rather to meager provisions, such as： ＂preestablished harmony of thinking,＂ ＂ontic commitment,＂ （Quine 1960） ＂the hygiene of mind,＂ （Maddy 1996） ＂success argument,＂ （Putnam 1975） ＂pragmatic argument,＂ （Resnik 1981） etc., which only are few places from the encounter of an exact expression such as is mathematical one and the reality of natuural sciences. Instead of philosophy of mathematics to radicalize its claims from the perspective of that （powerful） mathematical idiom and the excessive reality of Being and man＇s place in it--this time, in the spirit of biocosmology （neo-Aristotelism）.展开更多
With the purpose of providing a categorical treatment of weak multiplier bialgebras （introduced by BShm, Gomez-Torrecillas and Ldpez-Centella in 2015）, an ap- propriate notion of morphism for these algebraic objects...With the purpose of providing a categorical treatment of weak multiplier bialgebras （introduced by BShm, Gomez-Torrecillas and Ldpez-Centella in 2015）, an ap- propriate notion of morphism for these algebraic objects is proposed. This allows us to define a category wmb of （regular） weak multiplier bialgebras （with a right full comultipli- cation）, containing as a full subcategory the category wba of weak bialgebras defined by BShm, Gomez-Torrecillas and Lopez-Centella in 2014. We present a great source of ex- amples of these morphisms proving that, under some assumption, a functor between small categories induces a morphism of this kind between the natural weak multiplier bialgebra structures carried by the linear spans of the arrow sets of the categories. We explore the notion of elements of group-like type in a weak multiplier bialgebra, proposing a definition in the line of the one by the aforementioned authors for weak bialgebras. We show a big number of its properties and provide more general versions of many results known in the context of weak bialgebras. In particular, in analogy with the classical bialgebra setting （where the set of group-like elements is a monoid）, we prove that the set of these elements possesses a structure of category.展开更多
基金The National Natural Science Foundation of China(No.11371088)the Fundamental Research Funds for the Central Universities(No.3207013906)the Natural Science Foundation of Jiangsu Province(No.BK2012736)
文摘Let H be a commutative, noetherian, semisimple and cosemisimple Hopf algebra with a bijective antipode over a field k. Then the semisimplicity of YD（H） is considered, where YD （H） means the disjoint union of the category of generalized Yetter-Drinfeld modules nYD^H（ α, β） for any α, β E Aut Hopf（H）. First, the fact that YD（H） is closed under Mor is proved. Secondly, based on the properties of finitely generated projective modules and semisimplicity of H, YD（H） satisfies the exact condition. Thus each object in YD（H） can be decomposed into simple ones since H is noetherian and cosemisimple. Finally, it is proved that YD （H） is a sernisimple category.
基金The National Natural Science Foundation of China(No.11371088)the Excellent Young Talents Fund of Anhui Province(No.2013SQRL092ZD)+2 种基金the Natural Science Foundation of Higher Education Institutions of Anhui Province(No.KJ2015A294)China Postdoctoral Science Foundation(No.2015M571725)the Excellent Young Talents Fund of Chuzhou University(No.2013RC001)
文摘Let H be a Hopf algebra and HYD the Yetter- Drinfeld category over H. First, the enveloping algebra of generalized H-Hom-Lie algebra L, i.e., Hom-Lie algebra L H in the category HYD, is constructed. Secondly, it is obtained that U（L） = T（ L）/L where I is the Hom-ideal of T（L） generated by {ll＇-l_（（-1））·l＇l_0-[l,l＇]｜l,l＇∈L}, and u： L,T（L）/I is the canonical map. Finally, as the applications of the result, the enveloping algebras of generalized H-Lie algebras, i.e., the Lie algebras in the category MyDn and the Hom-Lie algebras in the category of left H-comodules are presented, respectively.
基金The National Natural Science Foundation of China(No.11371088)the Natural Science Foundation of Jiangsu Province(No.BK2012736)+1 种基金the Fundamental Research Funds for the Central Universitiesthe Research Innovation Program for College Graduates of Jiangsu Province(No.KYLX15_0109)
文摘Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of the crossed product in the classic Hopf algebra theory, first, let A be a k-linear category and H be a Hopf algebra, and the two crossed products A#_σH and A#＇_σH are isomorphic under some conditions. Then, let A#_σH be a crossed product category for a finite dimensional and semisimple Hopf algebra H. If V is a left A#σH-module and WC V is a submodule such that W has a complement as a left A-module, then W has a complement as a A#_σH-module.
文摘Although E Maddy （1997） says on naturalism： ＂This is not, in itself, a philosophy of mathematics [...]＂ （161）, already by its name, or by those whose interest has called on it （Quine, Putnam et al.） ... it anyhow reveals desire to be it. Insofar as otherwise, the semantic potential of the word leaves far behind it （after all scarce） results it achieved from the relation of an exact （mathematical） expression and （overly rich） intuitive reality of Being. We plead here already from the perspective of the slogan ＂One and All＂ of the first philosopher： Tales, when by the number （which one forebodes） one could go to such an extent into areas of reality （Pythagoras）, or when （especially in the human sphere） is being over again actual final cause of Aristotle the philosophy and the mathematics to accomplish far more fruitful encounter with the Being. Alain Badiou （1988） has already pointed that： ＂Mathematics is ontology,＂ and the category theory in mathematics, having covered by itself other fields of this science, continues to find applications in a series of＂non-traditional＂ domains of reality. In that correlation the philosophy can express its （primary） needs for truth, justice, beauty, ... as well as for the overall development in the sense of purpose--also because of an undreamed power of the technological development （of hardwares and softwares） today. Namely, the naturalism in mathematics, which developed an abundant reflection on the place （importance of） the mathematical idiom in sciences--in the balance of criticism--has come rather to meager provisions, such as： ＂preestablished harmony of thinking,＂ ＂ontic commitment,＂ （Quine 1960） ＂the hygiene of mind,＂ （Maddy 1996） ＂success argument,＂ （Putnam 1975） ＂pragmatic argument,＂ （Resnik 1981） etc., which only are few places from the encounter of an exact expression such as is mathematical one and the reality of natuural sciences. Instead of philosophy of mathematics to radicalize its claims from the perspective of that （powerful） mathematical idiom and the excessive reality of Being and man＇s place in it--this time, in the spirit of biocosmology （neo-Aristotelism）.
文摘With the purpose of providing a categorical treatment of weak multiplier bialgebras （introduced by BShm, Gomez-Torrecillas and Ldpez-Centella in 2015）, an ap- propriate notion of morphism for these algebraic objects is proposed. This allows us to define a category wmb of （regular） weak multiplier bialgebras （with a right full comultipli- cation）, containing as a full subcategory the category wba of weak bialgebras defined by BShm, Gomez-Torrecillas and Lopez-Centella in 2014. We present a great source of ex- amples of these morphisms proving that, under some assumption, a functor between small categories induces a morphism of this kind between the natural weak multiplier bialgebra structures carried by the linear spans of the arrow sets of the categories. We explore the notion of elements of group-like type in a weak multiplier bialgebra, proposing a definition in the line of the one by the aforementioned authors for weak bialgebras. We show a big number of its properties and provide more general versions of many results known in the context of weak bialgebras. In particular, in analogy with the classical bialgebra setting （where the set of group-like elements is a monoid）, we prove that the set of these elements possesses a structure of category.
