基于Fréchet-Grassmann代数的超空间上超光滑函数的定义与刻画
Definition and characterization of supersmooth functions on superspace based on Fréchet-Grassmann algebra
摘要：
在由可数无限Grassmann生成元构成的Fréchet-Grassmann（FG-）代数${\fR}$的基础上，引入了超空间${\fR}^{m|n}$ .在定义了${\euc}^m$上光滑函数到${\fR}^{m|0}$上光滑函数的Grassmann延拓之后，我们在${\fR}^{m|n}$上引入了一类超光滑函数.本文在Fréchet空间上刻画了Gâteaux（但不一定是Fréchet）可微范畴中的超光滑函数. $G^{\infty}$ -函数的这种类型的参数主要是在Banach-Grassmann（BG-）代数上完成的，但是我们发现当我们处理诸如Dirac，Weyl或Pauli方程的PDE系统时，在FG-代数内工作是相当自然的。在该应用中，我们需要证明（超）Hamilton方程的解是超光滑的。初始数据。虽然我们在以前的作品中采取了这一观点，但处理得相当不够。因此，我们在此重新看待这个问题，给予肯定的回答。给出了${\fR}^{m|n}$上超光滑函数的局部或整体反函数定理。
英文摘要：
Preparing the Fréchet-Grassmann (FG-)algebra ${\fR}$ composed with countably infinite Grassmann generators, we introduce the superspace ${\fR}^{m|n}$. After defining Grassmann continuation of smooth functions on ${\euc}^m$ to those on ${\fR}^{m|0}$, we introduce a class of functions on ${\fR}^{m|n}$ which are called supersmooth. In this paper, we characterize such supersmooth functions in Gâteaux (but not necessarily Fréchet) differentiable category on Fréchet but not on Banach space. This type of arguments for $G^{\infty}$-functions is mainly done on the Banach-Grassmann (BG-)algebra, but we find it rather natural to work within FG-algebra when we treat systems of PDE such as Dirac, Weyl or Pauli equations. In that application, we need to prove that the solution of the (super) Hamilton equation is supersmooth w.r.t. initial data. Though we took this point of view in our previous works, but is managed rather insufficiently. Therefore, we re-treat this subject here to answer affirmatively. We give also local or global inverse function theorems for supersmooth functions on ${\fR}^{m|n}$.
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