Differential algebra
Encyclopedia
In mathematics
, differential rings, differential fields, and differential algebras are rings
, fields
, and algebras
equipped with a derivation
, which is a unary
function that is linear and satisfies the Leibniz product law. 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 differentiation with respect to t.
such that each derivation satisfies the Leibniz product rule
for every
. 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 
is multiplication on the ring, the product rule is the identity
where
means the function which maps a pair 
to the pair 
.
, or Leibniz rule over the elements of the field, to be worthy of being called a derivation. That is, for any two elements u, v of the field, one has
since multiplication on the field is commutative. The derivation must also be distributive over addition in the field:
If K is a differential field then the field of constants
and 
one has
In index-free notation, if
is the ring morphism 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
and 
one has
and
is a linear map 
satisfying the Leibniz rule:
For any
, ad(a) is a derivation on 
, which follows from the Jacobi identity
. Any such derivation is called an inner derivation.
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
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 derivative
s, Lie derivative
s, 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.
s on them.
This is the ring
Multiplication on this ring is defined as
Here
is the binomial coefficient
. Note the identities
which makes use of the identity
and
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, differential rings, differential fields, and differential algebras are rings
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, fields
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, and algebras
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
equipped with a derivation
Derivation (abstract algebra)
In abstract algebra, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field K, a K-derivation is a K-linear map D: A → A that satisfies Leibniz's law: D = b + a.More...
, which is a unary
Unary operation
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation on A....
function that is linear and satisfies the Leibniz product law. 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 differentiation with respect to t.
Differential ring
A differential ring is a ring R equipped with one or more derivations, that is additive homomorphismssuch that each derivation satisfies the Leibniz product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
for every
. 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 is multiplication on the ring, the product rule is the identitywhere
means the function which maps a pair to the pair .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 ruleProduct rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
, or Leibniz rule over the elements of the field, to be worthy of being called a derivation. That is, for any two elements u, v of the field, one has
since multiplication on the field is commutative. The derivation must also be distributive over addition in the field:
If K is a differential field then the field of constants
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 and one hasIn index-free notation, if
is the ring morphism defining scalar multiplication on the algebra, one hasAs above, the derivation must obey the Leibniz rule over the algebra multiplication, and must be linear over addition. Thus, for all
and one hasand
Derivation on a Lie algebra
A derivation on a Lie algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
is a linear map satisfying the Leibniz rule:For any
, ad(a) is a derivation on , which follows from the Jacobi identityJacobi identity
In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...
. Any such derivation is called an inner derivation.
Examples
If 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
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
Differentially closed field
In mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by...
. 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
Differential Galois theory
In mathematics, differential Galois theory studies the Galois groups of differential equations.Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of...
.
Naturally occurring examples of derivations are partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
s, Lie derivative
Lie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
s, the Pincherle derivative, and the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
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 operatorPseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory....
s on them.
This is the ring
Multiplication on this ring is defined as
Here
is the binomial coefficientBinomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written \tbinom nk , and it is the coefficient of the x k term in...
. Note the identities
which makes use of the identity
and
See also
- Differential Galois theoryDifferential Galois theoryIn mathematics, differential Galois theory studies the Galois groups of differential equations.Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of...
- Kähler differentialKähler differentialIn mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.-Presentation:The idea was introduced by Erich Kähler in the 1930s...
- Differentially closed fieldDifferentially closed fieldIn mathematics, a differential field K is differentially closed if every finite system of differential equations with a solution in some differential field extending K already has a solution in K. This concept was introduced by...
- A D-moduleD-moduleIn mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations...
is an algebraic structure with several differential operators acting on it. - A differential graded algebraDifferential graded algebraIn mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure.- Definition :...
is a differential algebra with an additional grading. - Arithmetic derivativeArithmetic derivativeIn number theory, the arithmetic derivative, or number derivative, is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis.-Definition:...
- Differential calculus over commutative algebrasDifferential calculus over commutative algebrasIn mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms...
- Difference algebraDifference algebraDifference algebra is analogous to differential algebra but concerned with difference equations rather than differential equations.-References:*Alexander Levin , , Springer, ISBN 9781402069468...
- Differential algebraic geometryDifferential algebraic geometryDifferential algebraic geometry is an area of differential algebra that adapts concepts and methods from algebraic geometry and applies them to systems of differential equations/algebraic differential equations....
- Picard–Vessiot theoryPicard–Vessiot theoryIn differential algebra, Picard–Vessiot theory is the study of the differential field extension generated by the solutions of a linear differential equation, using the differential Galois group of the field extension. A major goal is to describe when the differential equation can be solved by...
External links
- David Marker's home page has several online surveys discussing differential fields.