﻿ 场代数弱局部性与商空间的研究

# 场代数弱局部性与商空间的研究On the Weak Locality of a Field Algebra and Its Quotient Space

Abstract: In this paper, we investigate the weak locality of a field algebra and give a concrete example which relates to weak locality and non-locality. The quotient space of a field algebra is obtained by its bilateral ideals. Finally, some theorems on the quotient space of a field algebra are explored.

1. 引言

$\begin{array}{l}Y:V\to glf\left(V\right)=\left\{a\left(z\right)=\underset{n\in Z}{\sum }{a}_{n}{z}^{-n-1};{a}_{n}\in EndV,n\in Z\right\}\\ \text{}a\to Y\left(a,z\right)\end{array}$

1) 真空性 $Y\left(|0〉,z\right)={I}_{V}$$Y\left(a,z\right)|0〉=a+T\left(a\right)z+\cdots \in V〚z〛$，其中 ${I}_{V}\left(\in EndV\right)$ 是恒等变换；

2) 平移不变性 $\left[T,Y\left(a,z\right)\right]=Y\left(Ta,z\right)={\partial }_{z}Y\left(a,z\right)$ (称T是V上的平移算子)；

${\left(z-w\right)}^{N}Y\left(Y\left(a,z\right)b,-w\right)c={\left(z-w\right)}^{N}{i}_{z,w}Y\left(a,z-w\right)Y\left(b,-w\right)c,N\gg 0，$

2. 预备知识

${\left(z-w\right)}^{N}\left[Y\left(a,z\right),Y\left(b,w\right)\right]=0，$

3. 主要结果及证明过程

$glf\left(V\right)=\left\{a\left(x\right)=\underset{n\in Z}{\sum }{a}_{n}{x}^{-n-1}|{a}_{n}\in EndV;\forall v\in V,{a}_{n}v=0,n>>0\right\}，$

$\begin{array}{l}n:glf\left(V\right)×glf\left(V\right)\to glf\left(V\right)\\ \text{}\left(a\left(x\right),b\left(x\right)\right)\to a{\left(x\right)}_{n}b\left(x\right)\end{array}$

$a{\left(x\right)}_{n}b\left(x\right)=\mathrm{Re}{s}_{{x}_{1}}\left\{{\left({x}_{1}-x\right)}^{n}a\left({x}_{1}\right)b\left(x\right)-{\left(-x+{x}_{1}\right)}^{n}b\left(x\right)a\left({x}_{1}\right)\right\}$

$\begin{array}{c}\mathrm{Re}{s}_{z}{\left(z-w\right)}^{N}\left[a\left(z\right),b\left(w\right)\right]=\mathrm{Re}{s}_{z}\left[\underset{i=0}{\overset{N}{\sum }}{\left(-1\right)}^{i}{z}^{N-i}{w}^{i}\underset{n,m\in Z}{\sum }\left[{a}_{n},{b}_{m}\right]{z}^{-n-1}{w}^{-m-1}\right]\\ =\mathrm{Re}{s}_{z}\left[\underset{i=0}{\overset{N}{\sum }}{\left(-1\right)}^{i}\underset{n,m\in Z}{\sum }\left[{a}_{n},{b}_{m}\right]{z}^{N-i-n-1}{w}^{i-m-1}\right]\\ =\mathrm{Re}{s}_{z}\left[\underset{i=0}{\overset{N}{\sum }}{\left(-1\right)}^{i}\underset{n,m\in Z}{\sum }\left[{a}_{N+n-i},{b}_{m+i}\right]{z}^{-n-1}{w}^{-m-1}\right]\\ =\underset{i=0}{\overset{N}{\sum }}{\left(-1\right)}^{i}\underset{m\in Z}{\sum }\left[{a}_{N-i},{b}_{m+i}\right]{w}^{-m-1}\\ =0\end{array}$

$\underset{i=0}{\overset{{N}_{2}}{\sum }}{\left(-1\right)}^{i}\left[{a}_{{N}_{2}+n-i},{b}_{m+i}\right]\ne 0,\exists n,m\in Z$

$\left[{a}_{{N}_{2}+n-k},{b}_{m+k}\right]=\left[{a}_{{N}_{2}+n-k},{b}_{{N}_{1}}\right]\ne 0,$

$\underset{i=0}{\overset{{N}_{2}}{\sum }}{\left(-1\right)}^{i}\left[{a}_{{N}_{2}+n-i},{b}_{m+i}\right]=\left[{a}_{{N}_{2}+n-k},{b}_{m+k}\right]\ne 0.$

$Y\left(a\left(x\right),z\right)=\underset{n\in Z}{\sum }a{\left(x\right)}_{n}{z}^{-n-1},$

$\begin{array}{l}\stackrel{˜}{Y}:V/I\to End\left(V/I\right)〚z,{z}^{-1}〛\\ \text{}\left[u\right]\to Y\left(\left[u\right],z\right)=\underset{n\in Ζ}{\sum }{\left[u\right]}_{n}{z}^{-n-1},n\in Z,u\in V\end{array}$

${a}_{n}u-{b}_{n}w={a}_{n}u-{b}_{n}u+{b}_{n}u-{b}_{n}w={\left(a-b\right)}_{n}u+{b}_{n}\left(u-w\right)\in I,$

2) 真空性： $Y\left(\left[|0〉\right],z\right)={I}_{V/I}$$Y\left(\left[a\right],z\right)|0〉=\left[a\right]+T\left(\left[a\right]\right)z+\cdots \in V/I〚z〛$，其中 ${I}_{v}\in EndV$ 是恒等变换。

3) 平移不变性： $\left[T,Y\left(\left[a\right],z\right)\right]=Y\left(T\left[a\right],z\right)={\partial }_{z}Y\left(\left[a\right],z\right)$，T是 $V/I$ 上的平移算子。

4) 因为 $\left(V,|0〉,Y\right)$ 是场代数，对所有的 $a,b,c\in V$，有下列结合性等式：

${\left(z-w\right)}^{N}Y\left(Y\left(a,z\right)b,-w\right)c={\left(z-w\right)}^{N}{i}_{z,w}Y\left(a,z-w\right)Y\left(b,-w\right)c,N\gg 0$

${\left(z-w\right)}^{N}Y\left(\underset{n\in Z}{\sum }{a}_{n}b{z}^{-n-1},-w\right)c={\left(z-w\right)}^{N}{i}_{z,w}\underset{m\in Z}{\sum }{a}_{m}{\left(z-w\right)}^{-m-1}\underset{k\in Z}{\sum }{b}_{k}c{\left(-w\right)}^{-k-1}$

${\left(z-w\right)}^{N}\underset{s,n\in Z}{\sum }{\left({a}_{n}b\right)}_{s}c{z}^{-n-1}{\left(-w\right)}^{-s-1}={\left(z-w\right)}^{N}{i}_{z,w}\underset{m,k\in Z}{\sum }{a}_{m}\left({b}_{k}c\right){\left(z-w\right)}^{-m-1}{\left(-w\right)}^{-k-1}.$

${\left(z-w\right)}^{N}\underset{s,n\in Z}{\sum }{\left({\left[a\right]}_{n}\left[b\right]\right)}_{s}\left[c\right]{z}^{-n-1}{\left(-w\right)}^{-s-1}={\left(z-w\right)}^{N}{i}_{z,w}\underset{m,k\in Z}{\sum }{\left[a\right]}_{m}\left({\left[b\right]}_{k}\left[c\right]\right){\left(z-w\right)}^{-m-1}{\left(-w\right)}^{-k-1}$

${\left(z-w\right)}^{N}Y\left(Y\left(\left[a\right],z\right)\left[b\right],-w\right)\left[c\right]={\left(z-w\right)}^{N}{i}_{z,w}Y\left(\left[a\right],z-w\right)Y\left(\left[b\right],-w\right)\left[c\right],N\gg 0$

$\left(V/I，\left[|0〉\right],Y\right)$ 也是一个场代数。

1) 令 $\stackrel{˜}{f}\left(\left[x\right]\right)=f\left(x\right),\forall x\in V$。映射 $\stackrel{˜}{f}$ 定义合理：若 $\left[{a}_{1}\right]=\left[{a}_{2}\right]$，即 ${a}_{1}~{a}_{2}$${a}_{1}-{a}_{2}\in I\subset \mathrm{ker}f$，有

$f\left({a}_{1}-{a}_{2}\right)=0$$f\left({a}_{1}\right)=f\left({a}_{2}\right)$$\stackrel{˜}{f}\left(\left[{a}_{1}\right]\right)=\stackrel{˜}{f}\left(\left[{a}_{2}\right]\right)$

2) $\stackrel{˜}{f}$ 是同态： $V/I\to W$。首先，它是向量空间的线性映。另外，它保持n运算：

$\stackrel{˜}{f}\left(\left[|0〉\right]\right)=|0〉,\stackrel{˜}{f}\left({\left[a\right]}_{n}\left[b\right]\right)=f\left({a}_{n}b\right)=f{\left(a\right)}_{n}f\left(b\right)=\stackrel{˜}{f}{\left(\left[a\right]\right)}_{n}\stackrel{˜}{f}\left(\left[b\right]\right).$

3) 由 $\stackrel{˜}{f}$ 的定义直接看出： $\stackrel{˜}{f}\pi =f$

4) 唯一性：若还有另外一个同态 $g:V/I\to W$，使得 $g\pi =f$。则 $\left(g\pi \right)\left(x\right)=f\left(x\right),\forall x\in V$。即 $g\left(\left[x\right]\right)=f\left(x\right),\forall \left[x\right]\in V/I$。由此可知， $g=\stackrel{˜}{f}$

5) $\stackrel{˜}{f}:V/I\to \mathrm{Im}f$ 是双射：由定义直接看出。因此， $\stackrel{˜}{f}:V/I\to \mathrm{Im}f$ 是场代数的同构映射。

$\sigma :A\to B,\text{}L\to \pi \left(L\right)=L/I.$

1) $\sigma$ 是单射：设理想 ${L}_{1},{L}_{2}\in A$，且 ${L}_{1}/I={L}_{2}/I$$\forall a\in {L}_{1}$，必有。故存在，使得。于是，从而，同理可得，则

2)是满射：设，令，现证。从而，则，同理可证。因此，L是V的理想。另外，L是V的包含I的理想，即。最后由于典范映射是满射，必有，即是满射。

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