Corrigendum to: Free Rota-Baxter systems and a Hopf algebra structure

Abstract

In this corrigendum, we give a correction that a free Rota-Baxter system is a left counital Hopf algebra but not a Hopf algebra.

Keywords:

• Gröbner-Shirshov basis
• free Rota-Baxter system
• left counital Hopf algebra

AMS 2010 Subject Classification:

• 16S15
• 13P10
• 17A50
• 16T25

This article refers to:

Free Rota-Baxter systems and a Hopf algebra structure

Dr. Xing Gao has kindly pointed out that ε is not a right counit in Lemma 3.3 of [2]. It follows that free Rota-Baxter system is not a Hopf algebra but a left counital Hopf algebra. We will correct Lemma 3.3, Theorem 3.4 and Theorem 3.8 in the paper [2].

For the definitions of left counital bialgebra and left counital Hopf algebra, see for instance [1, 3].

Lemma 3.3.

The triple $(RS(X),\Delta ,\epsilon )$ is a left counital coalgebra.

Proof.

The proof is the same as Lemma 3.3 in [2].□

Remark.

Note that ε is not a right counit on RS(X). If $R\ne S,$ then $(\text{id}\otimes \epsilon )\Delta (S(1))\ne {\beta }_r(S(1)),$ where ${\beta }_r:RS(X)\to RS(X)\otimes k,w\mapsto w\otimes 1_k.$

Theorem 3.4.

The quintuple $(RS(X),\diamond ,u,\Delta ,\epsilon )$ is a left counital bialgebra.

Proof.

By Lemmas 3.1–3.2 in [2] and Lemma 3.3, we can get the result.

A left counital bialgebra $(H,\mu ,u,\Delta ,\epsilon )$ is called a graded left counital bialgebra if there is a sequence of k-vector spaces $H^{(n)},n\ge 0,$ such that (a) $H={\oplus }_{n=0}^{\infty }{H}^{(n)};$ (b) For any $p,q\ge 0,H^{(p)}{H}^{(q)}\subseteq H^{(p+q)};$ (c) For any $n\ge 0,\Delta (H^{(n)})\subseteq \underset{p+q=n}{\oplus }{H}^{(p)}\otimes H^{(q)}.$ A graded left counital bialgebra $H={\oplus }_{n=0}^{\infty }{H}^{(n)}$ is called connected if $H^{(0)}=k$ and $\mathit{ker}\epsilon =\underset{n>0}{\oplus }{H}^{(n)}.$□

Lemma 3.7.

A connected left counital bialgebra $(H,\mu ,u,\Delta ,\epsilon )$ is a left counital Hopf algebra.

Proof.

The proof is similar to the ones in [1, 3].□

By Lemmas 3.5–3.6 in [2] and Lemma 3.7, we have the following theorem.

Theorem 3.8.

The free Rota-Baxter system $RS(X)={\oplus }_{n=0}^{\infty }{H}_{RS}^{(n)}$ is a connected left counital bialgebra. It follows that RS(X) is a left counital Hopf algebra.

Acknowledgement

We wish to express our thanks to Dr. Xing Gao and Prof. Li Guo for helpful and valuable suggestions and comments.