The exactness, reflexivity and semisimplicity of convolution product algebras (Q2721714)

Let \(A\) be an algebra over a field \(k\), \(C\) a coalgebra over \(k\). Then one can define a convolution algebra \(\Hom(C,A)\). Dually, there is a convolution coalgebra \(A*C\), which is isomorphic to \(A^o\otimes C\). The authors prove that the natural isomorphisms \(\Hom(C,\Hom(C,A))\cong\Hom(C\otimes C,A)\) and \(\Hom(A*C,A)\cong\Hom(A^o,\Hom(C,A))\) are algebra isomorphisms. Let \(C\) and \(D\) be coalgebras, and \(A\) and \(B\) algebras. Then \((A*C)\otimes(B*D)\cong(A\otimes B)*(C\otimes D)\) as coalgebras. Let \(C\) be finite-dimensional. If \(A\) is a reflexive (weak reflexive, proper, resp.) algebra then so is \(\Hom(C,A)\). Conversely, if \(\Hom(C,A)\) is proper then so is \(A\). Assume that \(k\) is algebraically closed. If \(C\) and \(D\) are cosemisimple coalgebras and \((C\otimes D)^*\cong C^*\otimes D^*\) as algebras, then \(C\otimes D\) is cosemisimple. If \(D\) is a bialgebra and \(C\otimes D\) is cosemisimple, then \(C\) is also cosemisimple. This paper also proves that if \(A\) and \(C\) are bialgebras then so is \(A*C\), and \(A*C\cong A^o\otimes C\) as bialgebras. If \(A\) and \(C\) are Hopf algebras then so is \(A*C\).