A Hopf algebra dual to a polynomial algebra over a commutative ring
Over a field, the continuous dual \(A^0\) of an algebra \(A\) has several equivalent definitions, and has the structure of a coalgebra. Over a commutative ring \(R\), Artamonov has defined \(A^0\) as those \(f\) in \(A^*\) whose kernel contains an ideal \(I\) of \(A\) such that \(A/I\) is a finitely-generated projective \(R\)-module. When \(R\) is Noetherian, \(A^0\) has the structure of a coalgebra, and will be a bialgebra (resp. Hopf algebra) when \(A\) is a bialgebra (resp. Hopf algebra). For arbitrary commutative rings \(R\), the author shows that \(A^0\) has a coalgebra structure whenever \(A\) satisfies the following property: If \(I\) and \(J\) are ideals of \(A\) such that \(A/I\) and \(A/J\) are finitely-generated projective \(R\)-modules, then \(I\cap J\) contains an ideal \(K\) such that \(A/K\) is a finitely-generated projective \(R\)-module. This result is then used to show that if \(A=R[X]\) or \(R[X,X^{-1}]\) is a polynomial bialgebra (Hopf algebra), where \(X=(x_1,\dots,x_k)\), then \(A^0\) is a bialgebra (Hopf algebra). Finally, an application is given to linearly recursive sequences. Let \(R[x]\) be a polynomial bialgebra in one variable. \(R[x]^0\) is the bialgebra of linearly recursive sequences over \(R\), with convolution product \(*\) dual to the comultiplication in \(R[x]\). For \(f\) and \(g\) in \(R[x]^0\), the author gives an explicit recursive relation for \(f*g\) in terms of recursive relations satisfied by \(f\) and \(g\).
scientific article; zbMATH DE number 1928204