Given a commutative unital ring and a coassociative -coalgebra with comultiplication and counit , a -submodule is
a right coideal if ,
a left coideal if ,
a coideal if and .
Here, denotes the image of (this distinction is not necessary when is a flat -module, for example when is a field).
In other words, a left coideal is simply a left subcomodule in with coaction being the comultiplication and a coideal is a subbicomodule in , where the comultiplication plays simultaneously the roles of left and right -coactions on itself.
If is a coideal, then the quotient -module is equipped with the canonical structure of a coassociative -coalgebra, the quotient co(al)gebra; the comultiplication is induced via choosing an arbitrary representative in each class, namely
or, in Sweedler notation, where .
Last revised on January 10, 2019 at 07:38:16. See the history of this page for a list of all contributions to it.