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],...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.展开更多
Let 0 →I → A →A/I →0 be a short exact sequence of C^＊-algebras with A unital. Suppose that I has tracial topological rank no more than one and A/I belongs to a class of certain C^＊-algebras. We show that A has t...Let 0 →I → A →A/I →0 be a short exact sequence of C^＊-algebras with A unital. Suppose that I has tracial topological rank no more than one and A/I belongs to a class of certain C^＊-algebras. We show that A has trazial topological rank no more than one if the extension is quasidiagonal, and A has the property （P1） if the extension is tracially quasidiagonal.展开更多
文摘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.
基金supported by National Natural Science Foundation of China (Grant No. 11071188)
文摘Let 0 →I → A →A/I →0 be a short exact sequence of C^＊-algebras with A unital. Suppose that I has tracial topological rank no more than one and A/I belongs to a class of certain C^＊-algebras. We show that A has trazial topological rank no more than one if the extension is quasidiagonal, and A has the property （P1） if the extension is tracially quasidiagonal.
