Differential algebra

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

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Differential ring

A differential ring is a ring R equipped with one or more derivations.

\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 may not be commutative, so the somewhat standard d(xy) = xdy + ydx form of the product rule in commutative settings may be false. In index-free notation, 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).

Differential field

A differential field is a field K, together with a derivation. The theory of differential fields, DF, is given by the usual field axioms along with two extra axioms involving the derivation. As above, the derivation must obey the product rule, or Leibniz rule over the elements of the field, in order to be worthy of being called a derivation. That is, for any two elements u, v of the field, one has

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

since multiplication on the field is commutative. The derivation must also be distributive over addition in the field:

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

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

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: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.

Examples

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

Any field pure can be interpretted 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 addition 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 et 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+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).

Graded derivations

If we have a graded algebra A, and D is an homogeneous linear map of grade d = \left| D \right| on A then D is an homogeneous derivation if D(ab)=D(a)b+\epsilon^{|a||D|}aD(b), \epsilon = \pm1 acting on homogeneous elements of A. A graded derivation is sum of homogeneous derivations with the same \epsilon.

If the commutator factor \epsilon = 1, this definition reduces to the usual case.

If \epsilon = -1, however, we have D(ab)=D(a)b+(-1)^{|a|}aD(b), for odd \left| D \right|. They are called antiderivations.

Examples of anti-derivations include the exterior derivative and the interior product acting on differential forms.

Graded derivations of superalgebras (i.e. \mathbb{Z}_2 -graded algebras) are often called superderivations.

See also

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 Verlang (1996).
  • A. Magid, Lectures on Differential Galois Theory, American Math. Soc., 1994

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