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 in one variable over the complex numbers, formula_1, where the derivation is 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 in 1950.
Differential ring.
A "differential ring" is a ring "R" equipped with one or more "derivations", that are homomorphisms of additive groups
such that each derivation ∂ satisfies the Leibniz product rule
for every formula_4. Note that the ring could be noncommutative, so the somewhat standard d("xy") = "x"d"y" + "y"d"x" form of the product rule in commutative settings may be false. If formula_5 is multiplication on the ring, the product rule is the identity
where formula_7 means the function which maps a pair formula_8 to the pair formula_9.
Note that a differential ring is a (not necessarily graded) formula_10-differential algebra.
Differential field.
A differential field is a commutative field "K" equipped with derivations.
The well-known formula for differentiating fractions
follows from the product rule. Indeed, we must have
By the product rule, we then have
Solving with respect to formula_14, we obtain the sought identity.
If "K" is a differential field then "the field of constants" of "K" is formula_15
A differential algebra over a field "K" is a "K"-algebra "A" wherein the derivation(s) commutes with the scalar multiplication. That is, for all formula_16 and formula_17 one has
If formula_19 is the ring homomorphism to the center of A defining scalar multiplication on the algebra, one has
As above, the derivation must obey the Leibniz rule over the algebra multiplication, and must be linear over addition. Thus, for all formula_21 and formula_22 one has
and
Derivation on a Lie algebra.
A derivation on a Lie algebra formula_25 is a linear map formula_26 satisfying the Leibniz rule:
For any formula_28, ad("a") is a derivation on formula_25, 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 formula_30, the rationals are always a subfield of the field of constants of formula_30.
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 ∂("u"2) = "u" ∂("u") + ∂("u")"u" = 2"u"∂("u").
The differential field Q("t") fails to have a solution to the differential equation
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 set of formal infinite sums
where formula_34 means that the sum runs on all integers that are not greater than a fixed (finite) value.
This set is made a ring with the multiplication defined by linearly extending the following formula for "monomials":
where formula_36 is the binomial coefficient. (If formula_37 the sum is finite, as the terms with formula_38 are all equal to zero.)
In particular, one has
for , , and , and using the identity
formula_40