Differential algebra

A differential algebra over a field K is a K-algebra A wherein the derivation(s) commutes with the field. That is, for all $k \in K$ and $x \in A$ one has

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

In index-free notation, if $\eta \colon K\to A$ is the ring morphism defining scalar multiplication on the algebra, one has

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

As above, the derivation must obey the Leibniz rule over the algebra multiplication, and must be linear over addition. Thus, for all $a,b \in K$ and $x,y \in A$ one has

$\partial (xy) = (\partial x) y %2B x(\partial y)$

and

$\partial (ax%2Bby) = a\,\partial x %2B b\,\partial y.$

Derivation on a Lie algebra

A derivation on a Lie algebra $\mathfrak{g}$ is a linear map $D \colon \mathfrak{g} \to \mathfrak{g}$ satisfying the Leibniz rule:

$D([a,b]) = [a,D(b)] %2B [D(a),b]$

For any $a \in \mathfrak{g}$ , ad(a) is a derivation on $\mathfrak{g}$ , which follows from the Jacobi identity. Any such derivation is called an inner derivation.

Examples

If $A$ is unital, then ∂(1) = 0 since ∂(1) = ∂(1 × 1) = ∂(1) + ∂(1). For example, in a differential field of characteristic zero the rationals are always a subfield of the constant field.

Any field pure can be interpreted as a constant differential field.

The field Q(t) has a unique structure as a differential field, determined by setting ∂(t) = 1: the field axioms along with the axioms for derivations ensure that the derivation is differentiation with respect to t. For example, by commutativity of multiplication and the Leibniz law one has that ∂(u²) = u ∂(u) + ∂(u)u= 2u∂(u).

The differential field Q(t) fails to have a solution to the differential equation

$\partial(u) = u$

but expands to a larger differential field including the function e^t which does have a solution to this equation. A differential field with solutions to all systems of differential equations is called a differentially closed field. Such fields exist, although they do not appear as natural algebraic or geometric objects. All differential fields (of bounded cardinality) embed into a large differentially closed field. Differential fields are the objects of study in differential Galois theory.

Naturally occurring examples of derivations are partial derivatives, Lie derivatives, the Pincherle derivative, and the commutator with respect to an element of the algebra. All these examples are tightly related, with the concept of derivation as the major unifying theme.

Ring of pseudo-differential operators

Differential rings and differential algebras are often studied by means of the ring of pseudo-differential operators on them.

This is the ring

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

Multiplication on this ring is defined as

$(r\xi^m)(s\xi^n) = \sum_{k=0}^m r (\partial^k s) {m \choose k} \xi^{m%2Bn-k}.$

Here ${m \choose k}$ is the binomial coefficient. Note the identities

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

which makes use of the identity

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

and

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

References

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

External links

David Marker's home page has several online surveys discussing differential fields.

Differential algebra

Contents

Differential ring

Differential field