Relation between left coideal and coideal.

Question

Let $H$ be a Hopf algebra. A coideal $C$ is a subset of $H$ such that $\Delta(C) \subset H \otimes C + C \otimes H$. A left coideal $C'$ is a subset of $H$ such that $\Delta(C') \subset H \otimes C'$. Since $H \otimes C \subset H \otimes C + C \otimes H$, a left coideal must be a coideal? Is this true? Are there some examples of left coideals which are not coideals (or a coideal which is not a left coideal)? Thank you very much.

Answer 1

Your definition of coideal is incomplete. A coideal must also be contained in the kernel of the counit. This means that a left or right coideal need not be a coideal. For example if $1$ is the identity in a bialgebra, the linear span of $1$ is a subcoalgebra (hence both a left and right coideal) but not a coideal.

Edited after comment by hardmath.

Answer 2

You are right in your observation. By definition a left (right) coideal is also a coideal.

However, the converse is not true: a coideal may be neither left nor right coideal. For an example, consider $k[x]$ the polynomial ring, which is a coalgebra with comultiplication given by $$ \Delta(x^n)=\Delta(x)^n=(1\otimes x+x\otimes 1)^n, \ \ \ \ \ \Delta(1)=1\otimes 1 $$ and counity $$ \epsilon(x^n)=\epsilon(x)^n=0, \ \textrm{for } \ n\geq 1, \ \ \ \ \ \epsilon(1)=1 $$ Now, consider the subspace spanned by $x$. Since $\Delta(x)=\Delta(x)=1\otimes x+x\otimes 1$, this is clearly a coideal but it is neither left nor right coideal.

(The same holds essentially for any coideal containing only primitive elements. For example consider the universal enveloping algebra $U(L)$ of a Lie algebra $L$, the images of the elements of $L$ into $U(L)$ and the subspaces of $U(L)$ generated by them).

P.S.1: Notice also that coideals and left (right) coideals are not merely subsets of the initial Hopf algebra (or: coalgebra). They also need to be subspaces. (Supposing that we are dealing with hopf algebras or coalgerbas over a field).

P.S.2: It is interesting to note that this situation is dual to the relation between ideals and left (right) ideals of algebras.