# On some formula in connected cocommutative Hopf algebras over a field of characteristic 0

Colloquium Mathematicae (2000)

• Volume: 83, Issue: 2, page 271-279
• ISSN: 0010-1354

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## Abstract

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Let H be a cocommutative connected Hopf algebra, where K is a field of characteristic zero. Let ${H}^{+}=Ker$ and ${h}^{+}=h-\left(h\right)$ for $h\in H$. We prove that ${d}_{h}={\sum }_{r=1}^{\infty }\left({\left(-1\right)}^{r+1}/r\right)\sum {h}_{1}^{+}...{h}_{r}^{+}$ is primitive, where $\sum {h}_{1}\otimes ...\otimes {h}_{r}={\Delta }_{r-1}\left(h\right)$.

## How to cite

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Wiśniewski, Piotr. "On some formula in connected cocommutative Hopf algebras over a field of characteristic 0." Colloquium Mathematicae 83.2 (2000): 271-279. <http://eudml.org/doc/210786>.

@article{Wiśniewski2000,
abstract = {Let H be a cocommutative connected Hopf algebra, where K is a field of characteristic zero. Let $H^\{+\} = Ker$ and $h^\{+\} = h - (h)$ for $h ∈ H$. We prove that $d_h = ∑_\{r=1\}^∞ ((-1)^\{r+1\}/r) ∑ h_1^\{+\}...h_r^\{+\}$ is primitive, where $∑ h_1 ⊗ ... ⊗ h_r=Δ_\{r-1\}(h)$.},
author = {Wiśniewski, Piotr},
journal = {Colloquium Mathematicae},
keywords = {Hopf algebras; derivations; primitive elements; enveloping algebras; counits},
language = {eng},
number = {2},
pages = {271-279},
title = {On some formula in connected cocommutative Hopf algebras over a field of characteristic 0},
url = {http://eudml.org/doc/210786},
volume = {83},
year = {2000},
}

TY - JOUR
AU - Wiśniewski, Piotr
TI - On some formula in connected cocommutative Hopf algebras over a field of characteristic 0
JO - Colloquium Mathematicae
PY - 2000
VL - 83
IS - 2
SP - 271
EP - 279
AB - Let H be a cocommutative connected Hopf algebra, where K is a field of characteristic zero. Let $H^{+} = Ker$ and $h^{+} = h - (h)$ for $h ∈ H$. We prove that $d_h = ∑_{r=1}^∞ ((-1)^{r+1}/r) ∑ h_1^{+}...h_r^{+}$ is primitive, where $∑ h_1 ⊗ ... ⊗ h_r=Δ_{r-1}(h)$.
LA - eng
KW - Hopf algebras; derivations; primitive elements; enveloping algebras; counits
UR - http://eudml.org/doc/210786
ER -

## References

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1. [1] S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conf. Ser. in Math. 82, Amer. Math. Soc., Providence, RI, 1993.
2. [2] S. A. Saymeh, On Hasse-Schmidt higher derivations, Osaka J. Math. 23 (1986), 503-508. Zbl0609.13017
3. [3] M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969.
4. [4] N. Ya. Vilenkin, Combinatorics, Academic Press, New York, 1971.

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