For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\Delta(f)(a,b) = f(ab)$ is well-defined.) What is the corresponding construction for a non-unital algebra. The coproduct part still works, but we have no counit, since this should arise as the dual of the unit. So are non-counital coalgebras studied in the literature? If so what are some references?

## 1 Answer

The finite dual of a non-unital algebra has been introduced in Semiperfect and coreflexive coalgebras, S. Dăscălescu, M. C. Iovanov, Forum Math. 27 (2015), No. 5, 2587--2608. See also: arXiv:1512.09344 [math.RT].

For the (more general) notion of noncounital corings (and some relevant examples) you can also see ch.4, sect. 26 of Corings and comodules, T. Brzezinski, R. Wisbauer

Infinitesimal Hopf algebras(they form part of the structure of the latter). $\endgroup$