# 微分代数

## 微分环

$\partial:R \to R$

$\partial(r_1 r_2)=(\partial r_1) r_2 + r_1 (\partial r_2),$

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

## 微分域

$\partial(uv) = u \,\partial v + v\, \partial u,\,$

$\partial (u + v) = \partial u + \partial v\ .\,$

## 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 $k \in K$$x \in A$

$\partial (kx) = k \partial x.\,$

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

$\partial (xy) = (\partial x) y + x(\partial y),\,$

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

## 李代数上的导子

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

## 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)。

$\partial(u) = u$

## 伪微分算子环

$R((\xi^{-1})) = \left\{ \sum_{n<\infty} r_n \xi^n | r_n \in R \right\}.$

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

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

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

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

## 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