### ${\partial}_{\psi}$- difference calculus Bernoulli-Taylor formula

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Introduction: This article will present just one example of a general construction known as the Bernstein-Gelfand-Gelfand (BGG) resolution. It was the motivating example from two lectures on the BGG resolution given at the 19th Czech Winter School on Geometry and Physics held in Srní in January 1999. This article may be seen as a technical example to go with a more elementary introduction which will appear elsewhere [M. Eastwood, Notices Am. Math. Soc. 46, No. 11, 1368-1376 (1999)]. In fact, there...

Summary: The Ado theorem is a fundamental fact, which has a reputation of being a `strange theorem'. We give its natural proof.

An $n$-ary Poisson bracket (or generalized Poisson bracket) on the manifold $M$ is a skew-symmetric $n$-linear bracket $\{,\cdots ,\}$ of functions which is a derivation in each argument and satisfies the generalized Jacobi identity of order $n$, i.e., $$\sum _{\sigma \in {S}_{2n-1}}(sign\sigma )\{\{{f}_{{\sigma}_{1}},\cdots ,{f}_{{\sigma}_{n}}\},{f}_{{\sigma}_{n+1}},\cdots ,{f}_{{\sigma}_{2n-1}}\}=0,$$${S}_{2n...}$

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 ${d}_{0},...,{d}_{k}$, 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 ${\mathcal{D}}_{K}^{k}$ 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.

Let $X$ be a reduced $n$-dimensional complex space, for which the set of singularities consists of finitely many points. If ${X}^{\text{'}}\subseteq X$ denotes the set of smooth points, the author considers a holomorphic vector bundle $E\to {X}^{\text{'}}\setminus A$, equipped with a Hermitian metric $h$, where $A$ represents a closed analytic subset of complex codimension at least two. The results, surveyed in this paper, provide criteria for holomorphic extension of $E$ across $A$, or across the singular points of $X$ if $A=\u2300$. The approach taken here is via the metric...