Differential algebra

In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natural example of a differential field is the field of rational functions C(t) in one variable, over the complex numbers, where the derivation is the differentiation with respect to t.

Differential algebra refers also to the area of mathematics consisting in the study of these algebraic objects and their use for an algebraic study of the differential equations. Differential algebra was introduced by Joseph Ritt.^[1]

Differential ring

A differential ring is a ring R equipped with one or more derivations, that is additive homomorphisms

\partial:R \to R\,

such that each derivation ∂ satisfies the Leibniz product rule

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

for every $r_1, r_2 \in R$ . Note that the ring could be noncommutative, so the somewhat standard d(xy) = xdy + ydx form of the product rule in commutative settings may be false. If $M:R \times R \to R$ is multiplication on the ring, the product rule is the identity

\partial \circ M = M \circ (\partial \times \operatorname{id}) + M \circ (\operatorname{id} \times \partial).

where $f\times g$ means the function which maps a pair $(x,y)$ to the pair $(f(x),g(y))$ .

Differential field

A differential field is a commutative field K equipped with derivations.

The well-known formula for differentiating fractions

\partial\left(\frac u v\right) = \frac{\partial(u)\,v - u\,\partial(v)}{v^2}

follows from the product rule. Indeed, we must have

\partial\left(\frac u v \times v\right) = \partial(u)

By the product rule, we then have

\partial\left(\frac u v\right) \, v + \frac u v \, \partial (v) = \partial(u)

Solving with respect to $\partial (u/v)$ , we obtain the sought identity.

If K is a differential field then the field of constants of K is $k = \{u \in K : \partial(u) = 0\}.$

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 + x(\partial y)

and

\partial (ax+by) = a\,\partial x + 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)] + [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. This derivation extends to the universal enveloping algebra of the Lie algebra.

Examples

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

Any ring is a differential ring with respect to the trivial derivation which maps any ring element to zero.

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 an algebra.

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+n-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

↑ Ritt, Joseph Fels (1950). Differential Algebra. New York: AMS Colloquium Publications (volume 33).

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.

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