The coradical of a Jordan (alternative) coalgebra and the quasiregular radical of its dual algebra (Q881133)

The author proves the Jordan and alternative analogues of the theorem that if \(C\) is an associative coalgebra over a field, with coradical \(C_0\), then the orthogonal compliment of \(C_0\) in the convolution algebra \(C^*\) is the Jacobson radical of \(C^*\). The author credits the associative result to \textit{R. G. Heyneman} and \textit{D. E. Radford} [J. Algebra 28, 215--246 (1974; Zbl 0291.16008)]. This result was also proved by \textit{L. A. Grünenfelder} [Math. Z. 116, 166--182 (1970; Zbl 0201.02502)]. For \(C\) a Jordan or alternative coalgebra (i.e., \(C^*\) is a Jordan or alternative algebra), the radical of \(C^*\) is defined by the analogue of the description of the Jacobson radical of an associative algebra in terms of quasiregularity. The proof is given in the Jordan case. The proof in the alternative case is similar.