If a Hopf Algebra has a nontrivial, finite-dimensional right ideal, then it is finite dimensional

Let $H$ be a Hopf Algebra (over a field $K$), with comultiplication $\Delta$, counit $\varepsilon$, and antipode $S$.

A $K$-subspace V is said to be:

• A right ideal if $VH \subseteq V$
• A right coideal if $\Delta (V) \subseteq V \otimes H$ (some call this a left coideal...)

Suppose that there is a nonzero right ideal $J$ in $H$, such that $\dim_K (J)$ is finite. Then $\dim_K (H)$ is finite.

The proof breaks in two parts:

1. If $V$ is a nonzero $K$-subspace of a Hopf algebra $H$ that is both a right ideal and a right coideal, then $V=H$.
2. If $J\subseteq H$ is a finite-dimensional, nonzero right ideal of $H$, then $N:= H^* J = \{ f\cdot x = \sum x_{(1)}f(x_{(2)}) : f\in H^* , x\in J\}$ is finite dimensional, nonzero, and it is both a right ideal and a right coideal (thus $H=N$ is finite dimenisonal)

Now, the first part is easy (you prove that there is $x\in V$ such that $\varepsilon (x) = 1$, then $1_H = \varepsilon(x)1_H = \sum x_{(1)}S(x_{(2)}) \in V$).

My problem is the second part. I can prove that $N$ is a finite dimensional subspace: since $J$ is finite dimensional, we can assume it is contained in a finite dimensional subcoalgebra $D$ of $H$; then $N\subseteq D$.

Does anyone have suggestions on how to prove that $N$ is a right ideal and a right coideal?

1 Answer

This is a part of Lemma 5.3.1 in Hopf Algebra: an introduction.

Dascalescu, Sorin, Constantin Nastasescu, and Serban Raianu. Hopf   algebra: An introduction. CRC Press, 2000.