Derivations

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 23 October 2009

Derivations

Let $𝔽$ be a field. A vector space over $𝔽$ is an abelian group $V$ with a map $$\begin{array}{rcl}𝔽\times V& ⟶& V\\ \left(c,v\right)& ⟼& cv\end{array}$$ such that

1. if $c_1,c_2\in 𝔽$ and $v\in V$ then $\left(c_1+c_2\right)v=c_{1}v+c_{2}v$,
2. if $c\in 𝔽$ and $v_1,v_2\in V$ then $c\left(v_1+v_2\right)=c{v}_1+c{v}_2$,
3. if $c_1,c_2\in 𝔽$ and $v\in V$ then $c_1\left(c_{2}v\right)=\left(c_1{c}_2\right)v$, and
4. if $v\in V$ then $1·v=v$.

Let $𝔽$ be a field. Let $V,W$ be vector spaces over $𝔽$. An $𝔽$-linear map from $V$ to $W$ is a function $\varphi :V\to W$ such that

1. $\varphi$ is a group homomorphism,
2. if $c\in 𝔽$ and $v\in V$ then $\varphi \left(cv\right)=c\varphi \left(v\right)$.

Let $𝔽$ be a field. An algebra is a vector space $A$ over $𝔽$ with an operation $$\begin{array}{rcl}A\times A& ⟶& A\\ \left(a_1,a_2\right)& ⟼& a_1{a}_2\end{array}$$ such that $A$ is a ring and scalar multiplication is the composition of the map $$\begin{array}{rcl}𝔽& ⟶& A\\ \xi & ⟼& \xi ·1\end{array}$$ and the multiplication in $A$.

Let $𝔽$ be a field. Let $A$ be an $𝔽$-algebra. A derivation of $A$ is an $𝔽$-linear map $d:A\to A$ such that $$\text{if}{a}_1,a_2\in A\text{then}d\left(a_1{a}_2\right)=a_{1}d\left(a_2\right)+d\left(a_1\right)a_2\text{.}$$

1. There is a unique derivation ${\displaystyle \frac{d}{dx}}$ of $𝔽\left[x\right]$ such that ${\displaystyle \frac{dx}{dx}=1}$.
2. If $p\in 𝔽\left[x\right]$ then $$\frac{dp}{dx}=\left(\text{coefficient of }y\text{ in }p\left(x+y\right)\right)\text{.}$$
3. If $p\in 𝔽\left[x\right]$ then $$p=\sum _{k\in {\mathbb{Z}}_{\ge 0}}{\left.\left({\left(\frac{d}{dx}\right)}^{k}p\right)\right|}_{x=0}{x}^k$$
4. There is a unique extension of ${\displaystyle \frac{d}{dx}}$ to a derivation of $𝔽\left(x\right)$.
5. There is a unique extension of ${\displaystyle \frac{d}{dx}}$ to a derivation of $𝔽[[x]]$.
6. There is a unique extension of ${\displaystyle \frac{d}{dx}}$ to a derivation of $𝔽((x))$.
7. If $p\in 𝔽[[x]]$ then $$\frac{dp}{dx}=\left(\text{coefficient of }y\text{ in }p\left(x+y\right)\right)\text{.}$$
8. If $p\in 𝔽[[x]]$ then $$p=\sum _{k\in {\mathbb{Z}}_{\ge 0}}{\left.\left({\left(\frac{d}{dx}\right)}^{k}p\right)\right|}_{x=0}{x}^k$$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)