# A representation of the coalgebra of derivations for smooth spaces

• Proceedings of the 18th Winter School "Geometry and Physics", Publisher: Circolo Matematico di Palermo(Palermo), page 135-141

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## Abstract

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Let $K$ be a field. The generalized Leibniz rule for higher derivations suggests the definition of a coalgebra ${𝒟}_{K}^{k}$ for any positive integer $k$. This is spanned over $K$ by ${d}_{0},...,{d}_{k}$, and has comultiplication $\Delta$ and counit $\epsilon$ defined by $\Delta \left({d}_{i}\right)={\sum }_{j=0}^{i}{d}_{j}\otimes {d}_{i-j}$ and $\epsilon \left({d}_{i}\right)={\delta }_{0,i}$ (Kronecker’s delta) for any $i$. This note presents a representation of the coalgebra ${𝒟}_{K}^{k}$ by using smooth spaces and a procedure of microlocalization. The author gives an interpretation of this result following the principles of the quantum theory of geometric spaces.

## How to cite

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Fischer, Gerald. "A representation of the coalgebra of derivations for smooth spaces." Proceedings of the 18th Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1999. 135-141. <http://eudml.org/doc/221879>.

@inProceedings{Fischer1999,
abstract = {Let $K$ be a field. The generalized Leibniz rule for higher derivations suggests the definition of a coalgebra $\{\mathcal \{D\}\}^k_K$ for any positive integer $k$. This is spanned over $K$ by $d_0,\ldots ,d_k$, and has comultiplication $\Delta$ and counit $\varepsilon$ defined by $\Delta (d_i)=\sum _\{j=0\}^id_j\otimes d_\{i-j\}$ and $\varepsilon (d_i)=\delta _\{0,i\}$ (Kronecker’s delta) for any $i$. This note presents a representation of the coalgebra $\{\mathcal \{D\}\}^k_K$ by using smooth spaces and a procedure of microlocalization. The author gives an interpretation of this result following the principles of the quantum theory of geometric spaces.},
author = {Fischer, Gerald},
booktitle = {Proceedings of the 18th Winter School "Geometry and Physics"},
keywords = {Winter school; Proceedings; Geometry; Physics; Srní (Czech Republic)},
location = {Palermo},
pages = {135-141},
publisher = {Circolo Matematico di Palermo},
title = {A representation of the coalgebra of derivations for smooth spaces},
url = {http://eudml.org/doc/221879},
year = {1999},
}

TY - CLSWK
AU - Fischer, Gerald
TI - A representation of the coalgebra of derivations for smooth spaces
T2 - Proceedings of the 18th Winter School "Geometry and Physics"
PY - 1999
CY - Palermo
PB - Circolo Matematico di Palermo
SP - 135
EP - 141
AB - Let $K$ be a field. The generalized Leibniz rule for higher derivations suggests the definition of a coalgebra ${\mathcal {D}}^k_K$ for any positive integer $k$. This is spanned over $K$ by $d_0,\ldots ,d_k$, and has comultiplication $\Delta$ and counit $\varepsilon$ defined by $\Delta (d_i)=\sum _{j=0}^id_j\otimes d_{i-j}$ and $\varepsilon (d_i)=\delta _{0,i}$ (Kronecker’s delta) for any $i$. This note presents a representation of the coalgebra ${\mathcal {D}}^k_K$ by using smooth spaces and a procedure of microlocalization. The author gives an interpretation of this result following the principles of the quantum theory of geometric spaces.
KW - Winter school; Proceedings; Geometry; Physics; Srní (Czech Republic)
UR - http://eudml.org/doc/221879
ER -

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