# 微分代数

## 微分环

${\displaystyle \partial :R\to R}$

${\displaystyle \partial (r_{1}r_{2})=(\partial r_{1})r_{2}+r_{1}(\partial r_{2}),}$

${\displaystyle \partial \circ M=M\circ (\partial \otimes \operatorname {id} )+M\circ (\operatorname {id} \otimes \partial ).\,}$

## 微分域

${\displaystyle \partial (uv)=u\,\partial v+v\,\partial u,\,}$

${\displaystyle \partial (u+v)=\partial u+\partial v\ .\,}$

## 微分代数

K 上一个微分代数是一个 K-代数 A，其中的导子与域可交换。即对所有 ${\displaystyle k\in K}$${\displaystyle x\in A}$

${\displaystyle \partial (kx)=k\partial x.\,}$

${\displaystyle \partial \circ M\circ (\eta \times \operatorname {Id} )=M\circ (\eta \times \partial ).\,}$

${\displaystyle \partial (xy)=(\partial x)y+x(\partial y),\,}$

${\displaystyle \partial (ax+by)=a\,\partial x+b\,\partial y.\,}$

## 李代数上的导子

${\displaystyle D([a,b])=[a,D(b)]+[D(a),b]\,}$

## 例

Q(t) 具有惟一的结构成为一个微分域，由令 ∂(t) = 1 确定：域公理与导子的公理奇异保证导子是关于 t 的导数。例如，由乘法与莱布尼兹法则的交换性有 ∂(u2) = u ∂(u) + ∂(u)u= 2u∂(u)。

${\displaystyle \partial (u)=u}$

## 伪微分算子环

${\displaystyle R((\xi ^{-1}))=\left\{\sum _{n<\infty }r_{n}\xi ^{n}|r_{n}\in R\right\}.}$

${\displaystyle (r\xi ^{m})(s\xi ^{n})=\sum _{k=0}^{m}r(\partial ^{k}s){m \choose k}\xi ^{m+n-k}.}$

${\displaystyle \xi ^{-1}r=\sum _{n=0}^{\infty }(-1)^{n}(\partial ^{n}r)\xi ^{-1-n}}$

${\displaystyle {-1 \choose n}=(-1)^{n}}$

${\displaystyle r\xi ^{-1}=\sum _{n=0}^{\infty }\xi ^{-1-n}(\partial ^{n}r).}$

## 参考文献

• Buium, Differential Algebra and Diophantine Geometry, Hermann (1994).
• I. Kaplansky, Differential Algebra, Hermann (1957).
• E. Kolchin, Differential Algebra and Algebraic Groups, 1973
• D. Marker, Model theory of differential fields, Model theory of fields, Lecture notes in Logic 5, D. Marker, M. Messmer and A. Pillay, Springer Verlag (1996).
• A. Magid, Lectures on Differential Galois Theory, American Math. Soc., 1994