Given a commutative unital ring kk and a coassociative kk-coalgebra C=(C,Δ,ϵ)C = (C,\Delta,\epsilon) with comultiplication Δ\Delta and counit ϵ:Ck\epsilon\colon C\to k, a kk-submodule ICI\subseteq C is

  • a left coideal if Δ(I)IC¯\Delta(I)\subseteq \overline{I\otimes C},

  • a right coideal if Δ(I)CI¯\Delta(I)\subseteq \overline{C\otimes I},

  • a coideal if Δ(I)IC¯+CI¯\Delta(I)\subseteq \overline{I\otimes C}+\overline{C\otimes I} and ϵ(I)=0\epsilon(I)=0.

Here, IC¯\overline{I \otimes C} denotes the image of ICCCI \otimes C \to C \otimes C (this distinction is not necessary when CC is a flat kk-module, for example when kk is a field).

In other words, a left coideal is simply a left subcomodule in CC with coaction being the comultiplication and a coideal is a subbicomodule in CC, where the comultiplication plays simultaneously the roles of left and right CC-coactions on itself.

If ICI\subseteq C is a coideal, then the quotient kk-module C/IC/I is equipped with the canonical structure of a coassociative kk-coalgebra, the quotient co(al)gebra; the comultiplication is induced via choosing an arbitrary representative in each class, namely

Δ C/I(c+I):=Δ C(c)+IC+CIC/IC/I, \Delta_{C/I}(c+I):= \Delta_C(c) + I\otimes C + C\otimes I \in C/I \otimes C/I,

or, in Sweedler notation, Δ C/I(c)=(c (1)+I)(c (2)+I)\Delta_{C/I}(c) = \sum (c_{(1)}+I)\otimes (c_{(2)}+I) where Δ C(c)=c (1)c (2)\Delta_C(c)= \sum c_{(1)}\otimes c_{(2)}.

Revised on January 28, 2016 10:40:17 by Martin B? (