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By Euler L.

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T ˝ id/ ı † ı . A; /cop is a Hopf algebra with antipode T . 14. i) ) ii), iii): Suppose that S is invertible. A; /cop . Á. a/// D Á. 1/ / D 22 Chapter 1. 1/ D Á. a// for all a 2 A. A; /cop be a Hopf algebra with antipode T . a// D a for all a 2 A. A; /. 16. A; /, we have S 2 D idA . Proof. A; /cop . By . 17. B; B / be Hopf algebras and F W A ! B a unital and counital morphism of bialgebras. Then F ı SA D SB ı F , that is, F is a morphism of Hopf algebras. Proof. A; B/. F ı SA /: Therefore F is invertible with respect to the convolution, and its inverse is SB ıF D F ı SA .

A; /cop . Á. a/// D Á. 1/ / D 22 Chapter 1. 1/ D Á. a// for all a 2 A. A; /cop be a Hopf algebra with antipode T . a// D a for all a 2 A. A; /. 16. A; /, we have S 2 D idA . Proof. A; /cop . By . 17. B; B / be Hopf algebras and F W A ! B a unital and counital morphism of bialgebras. Then F ı SA D SB ı F , that is, F is a morphism of Hopf algebras. Proof. A; B/. F ı SA /: Therefore F is invertible with respect to the convolution, and its inverse is SB ıF D F ı SA . 9, Hopf algebras can be characterized as those bialgebras for which the identity map is invertible with respect to the convolution product.

A; A / be a coalgebra. Then the dual space A0 is an algebra with respect to the multiplication mA0 W A0 ˝ A0 ,! A /0 ! A; A /. ii) Let A be a finite-dimensional algebra. mA /0 ! A0 ; A0 / coincides with evaluation at the unit of A. A; A / be a finite-dimensional bialgebra. Then A0 , equipped with the multiplication and comultiplication defined above, is a bialgebra. 4. SA /0 . The Š natural isomorphism ÃA W A ! A00 of vector spaces is an isomorphism of bialgebras or Hopf algebras, respectively.

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An observation on the sums of divisors by Euler L.


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