# Article

 Title: A representation of the coalgebra of derivations for smooth spaces (English) Author: Fischer, Gerald Language: English Journal: Proceedings of the 18th Winter School "Geometry and Physics" Volume: Issue: 1998 Year: Pages: 135-141 . Category: math . Summary: Let $K$ be a field. The generalized Leibniz rule for higher derivations suggests the definition of a coalgebra ${\cal D}^k_K$ 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 ${\cal D}^k_K$ 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. (English) MSC: 16W25 MSC: 16W30 MSC: 16W60 MSC: 32C38 idZBL: Zbl 0962.16027 idMR: MR1692264 . Date available: 2009-07-13T21:41:18Z Last updated: 2012-09-18 Stable URL: http://hdl.handle.net/10338.dmlcz/701632 .

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