Differential algebra

A derivation ${\textstyle \partial }$ on a ring ${\textstyle {\mathcal {R}}}$ is a function ${\displaystyle \partial :R\to R\,}$ such that

$${\displaystyle \partial (r_{1}+r_{2})=\partial r_{1}+\partial r_{2}}$$

and

${\displaystyle \partial (r_{1}r_{2})=(\partial r_{1})r_{2}+r_{1}(\partial r_{2})\quad }$ (Leibniz product rule),

for every ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ in ${\displaystyle R.}$

A derivation is linear over the integers since these identities imply ${\displaystyle \partial (0)=\partial (1)=0}$ and ${\displaystyle \partial (-r)=-\partial (r).}$

A differential ring is a commutative ring ${\displaystyle R}$ equipped with one or more derivations that commute pairwise; that is,

$${\displaystyle \partial _{1}(\partial _{2}(r))=\partial _{2}(\partial _{1}(r))}$$

for every pair of derivations and every ${\displaystyle r\in R.}$[1]^: 58–59 When there is only one derivation one talks often of an ordinary differential ring; otherwise, one talks of a partial differential ring.

A differential field is differentiable ring that is also a field. A differential algebra ${\displaystyle A}$ over a differential field ${\displaystyle K}$ is a differential ring that contains ${\displaystyle K}$ as a subring such that the restriction to ${\displaystyle K}$ of the derivations of ${\displaystyle A}$ equal the derivations of ${\displaystyle K.}$ (A more general definition is given below, which covers the case where ${\displaystyle K}$ is not a field, and is essentially equivalent when ${\displaystyle K}$ is a field.)

A Witt algebra is a differential ring that contains the field ${\displaystyle \mathbb {Q} }$ of the rational numbers. Equivalently, this is a differential algebra over ${\displaystyle \mathbb {Q} ,}$ since ${\displaystyle \mathbb {Q} }$ can be considered as a differential field on which every derivation is the zero function.

The constants of a differential ring are the elements ${\displaystyle r}$ such that ${\displaystyle \partial r=0}$ for every derivation ${\displaystyle \partial .}$ The constants of a differential ring form a subring and the constants of a differentiable field form a subfield.[1]^: 58–60 This meaning of "constant" generalizes the concept of a constant function, and must not be confused with the common meaning of a constant.

In the following identities, ${\displaystyle \delta }$ is a derivation of a differential ring ${\displaystyle R.}$[6]^: 76

• If ${\displaystyle r\in R}$ and ${\displaystyle c}$ is a constant in ${\displaystyle R}$ (that is, ${\displaystyle \delta c=0}$), then
  $${\displaystyle \delta (cr)=c\delta (r).}$$

  
• If ${\displaystyle r\in R}$ and ${\displaystyle u}$ is a unit in ${\displaystyle R,}$ then
  $${\displaystyle \delta \left({\frac {r}{u}}\right)={\frac {\delta (r)u-r\delta (u)}{u^{2}}}}$$

  
• If ${\displaystyle n}$ is a nonnegative integer and ${\displaystyle r\in R}$ then
  $${\displaystyle \delta (r^{n})=nr^{n-1}\delta (r)}$$

  
• If ${\displaystyle u_{1},\ldots ,u_{n}}$ are units in ${\displaystyle R,}$ and ${\displaystyle e_{1},\ldots ,e_{n}}$ are integers, one has the logarithmic derivative identity:
  $${\displaystyle {\frac {\delta (u_{1}^{e_{1}}\ldots u_{n}^{e_{n}})}{u_{1}^{e_{1}}\ldots u_{n}^{e_{n}}}}=e_{1}{\frac {\delta (u_{1})}{u_{1}}}+\dots +e_{n}{\frac {\delta (u_{n})}{u_{n}}}.}$$

  

A derivation operator or higher-order derivation is the composition of several derivations. As the derivations of a differential ring are supposed to commute, the order of the derivations does not matter, and a derivation operator may be written as

$${\displaystyle \delta _{1}^{e_{1}}\circ \cdots \circ \delta _{n}^{e_{n}},}$$

where ${\displaystyle \delta _{1},\ldots ,\delta _{n}}$ are the derivations under consideration, ${\displaystyle e_{1},\ldots ,e_{n}}$ are nonnegative integers, and the exponent of a derivation denotes the number of times this derivation is composed in the operator.

The sum ${\displaystyle o=e_{1}+\cdots +e_{n}}$ is called the order of derivation. If ${\displaystyle o=1}$ the derivation operator is one of the original derivations. If ${\displaystyle o=0}$, one has the identity function, which is generally considered as the unique derivation operator of order zero. With these conventions, the derivation operators form a free commutative monoid on the set of derivations under consideration.

A derivative of an element ${\displaystyle x}$ of a differential ring is the application of a derivation operator to ${\displaystyle x,}$ that is, with the above notation, ${\displaystyle \delta _{1}^{e_{1}}\circ \cdots \circ \delta _{n}^{e_{n}}(x).}$ A proper derivative is a derivative of positive order.[1]^: 58–59

A differential ideal ${\displaystyle I}$ of a differential ring ${\displaystyle R}$ is an ideal of the ring ${\displaystyle R}$ that is closed (stable) under the derivations of the ring; that is, ${\textstyle \partial x\in I,}$ for every derivation ${\displaystyle \partial }$ and every ${\displaystyle x\in I.}$. A differential ideal is said proper if it is not the whole ring. For avoiding confusion, an ideal that is not a differential ideal is sometimes called an algebraic ideal.

The radical of a differential ideal is the same as its radical as an algebraic ideal, that is, the set of the ring elements that have a power in the ideal. The radical of a differential ideal is also a differential ideal. A radical or perfect differential ideal is a differential ideal that equals its radical.[7]^: 3–4 A prime differential ideal is a differential ideal that is prime in the usual sense; that is, if a product belongs to the ideal, at least one of the factors belongs to the ideal. A prime differential ideal is always a radical differential ideal.

A discovery of Ritt is that, although the classical theory of algebraic ideals does not work for differential ideals, a large part of it can be extended to radical differential ideals, and this makes them fundamental in differential algebra.

The intersection of any family of differential ideals is a differential ideal, and the intersection of any family of radical differential ideals is a radical differential ideal.[1]^: 61–62 It follows that, given a subset ${\displaystyle S}$ of a differential ring, there are three ideals generated by it, which are the intersections of, respectively, all algebraic ideals, all differential ideals, and all radical differential ideals that contain it.[1]^: 61–62[8]^: 21

The algebraic ideal generated by ${\displaystyle S}$ is the set of the finite linear combinations of elements of ${\displaystyle S,}$ and is commonly denoted as ${\displaystyle (S)}$ or ${\displaystyle \langle S\rangle .}$

The differential ideal generated by ${\displaystyle S}$ is the set of the finite linear combinations of elements of ${\displaystyle S}$ and of the derivatives of any order of these elements; it is commonly denoted as ${\displaystyle [S].}$ When ${\displaystyle S}$ is finite, ${\displaystyle [S]}$ is generally not finitely generated as an algebraic ideal.

The radical differential ideal generated by ${\displaystyle S}$ is commonly denoted as ${\displaystyle \{S\}.}$ There is no known way to characterize its element in a similar way as for the two other cases.

The ranking of derivatives is a total order and an admisible order, defined as:[1]^: 75–76[17]^: 1141[10]^: 10

${\textstyle \forall p\in \Theta Y,\ \forall \theta _{\mu }\in \Theta$ :\theta _{\mu }p>p.}

${\textstyle \forall p,q\in \Theta Y,\ \forall \theta _{\mu }\in \Theta :p\geq q\Rightarrow \theta _{\mu }p\geq \theta _{\mu }q.}$

Each derivative has an integer tuple, and a monomial order ranks the derivative by ranking the derivative's integer tuple. The integer tuple identifies the differential indeterminate, the derivative's multi-index and may identify the derivative's order. Types of ranking include:[18]^: 83

• Orderly ranking: ${\displaystyle \forall y_{i},y_{j}\in Y,\ \forall \theta _{\mu },\theta _{\nu }\in \Theta \$ :\ \operatorname {ord} (\theta _{\mu })\geq \operatorname {ord} (\theta _{\nu })\Rightarrow \theta _{\mu }y_{i}\geq \theta _{\nu }y_{j}}
• Elimination ranking: ${\displaystyle \forall y_{i},y_{j}\in Y,\ \forall \theta _{\mu },\theta _{\nu }\in \Theta \$ :\ y_{i}\geq y_{j}\Rightarrow \theta _{\mu }y_{i}\geq \theta _{\nu }y_{j}}

In this example, the integer tuple identifies the differential indeterminate and derivative's multi-index, and lexicographic monomial order, ${\textstyle \geq _{\text{lex}}}$, determines the derivative's rank.[19]^: 4

${\displaystyle \eta (\delta _{1}^{e_{1}}\circ \cdots \circ \delta _{n}^{e_{n}}(y_{j}))=(j,e_{1},\ldots ,e_{n})}$.

${\displaystyle \eta (\theta _{\mu }y_{j})\geq _{\text{lex}}\eta (\theta _{\nu }y_{k})\Rightarrow \theta _{\mu }y_{j}\geq \theta _{\nu }y_{k}.}$

This is the standard polynomial form: ${\displaystyle p=a_{d}\cdot u_{p}^{d}+a_{d-1}\cdot u_{p}^{d-1}+\cdots +a_{1}\cdot u_{p}+a_{0}}$.[1]^: 75–76[19]^: 4

• Leader or leading derivative is the polynomial's highest ranked derivative: ${\displaystyle u_{p}}$.
• Coefficients ${\displaystyle a_{d},\ldots ,a_{0}}$ do not contain the leading derivative ${\textstyle u_{p}}$.
• Degree of polynomial is the leading derivative's greatest exponent: ${\displaystyle \deg _{u_{p}}(p)=d}$.
• Initial is the coefficient: ${\displaystyle I_{p}=a_{d}}$.
• Rank is the leading derivative raised to the polynomial's degree: ${\displaystyle u_{p}^{d}}$.
• Separant is the derivative: ${\displaystyle S_{p}={\frac {\partial p}{\partial u_{p}}}}$.

Separant set is ${\displaystyle S_{A}=\{S_{p}\mid p\in A\}}$, initial set is ${\displaystyle I_{A}=\{I_{p}\mid p\in A\}}$ and combined set is ${\textstyle H_{A}=S_{A}\cup I_{A}}$.[12]^: 159

Partially reduced (partial normal form) polynomial ${\textstyle q}$ with respect to polynomial ${\textstyle p}$ indicates these polynomials are non-ground field elements, ${\textstyle p,q\in {\mathcal {K}}\{Y\}\setminus {\mathcal {K}}}$, and ${\displaystyle q}$ contains no proper derivative of ${\displaystyle u_{p}}$.[1]^: 75[18]^: 84[12]^: 159

Partially reduced polynomial ${\textstyle q}$ with respect to polynomial ${\textstyle p}$ becomes reduced (normal form) polynomial ${\textstyle q}$ with respect to ${\textstyle p}$ if the degree of ${\textstyle u_{p}}$ in ${\textstyle q}$ is less than the degree of ${\textstyle u_{p}}$ in ${\textstyle p}$.[1]^: 75[18]^: 84[12]^: 159

An autoreduced polynomial set has every polynomial reduced with respect to every other polynomial of the set. Every autoreduced set is finite. An autoreduced set is triangular meaning each polynomial element has a distinct leading derivative.[7]^: 6[1]^: 75

Ritt's reduction algorithm identifies integers ${\textstyle i_{A_{k}},s_{A_{k}}}$ and transforms a differential polynomial ${\textstyle f}$ using pseudodivision to a lower or equally ranked remainder polynomial ${\textstyle f_{red}}$ that is reduced with respect to the autoreduced polynomial set ${\textstyle A}$. The algorithm's first step partially reduces the input polynomial and the algorithm's second step fully reduces the polynomial. The formula for reduction is:[1]^: 75

${\displaystyle f_{\text{red}}\equiv \prod _{A_{k}\in A}I_{A_{k}}^{i_{A_{k}}}\cdot S_{A_{k}}^{i_{A_{k}}}\cdot f,{\pmod {[A]}}{\text{ with }}i_{A_{k}},s_{A_{k}}\in \mathbb {N} .}$

Set ${\textstyle A}$ is a differential chain if the rank of the leading derivatives is ${\textstyle u_{A_{1}}<\dots <u_{A_{m}}}$ and ${\textstyle \forall i,\ A_{i}}$ is reduced with respect to ${\textstyle A_{i+1}}$[11]^: 294

Autoreduced sets ${\textstyle A}$ and ${\textstyle B}$ each contain ranked polynomial elements. This procedure ranks two autoreduced sets by comparing pairs of identically indexed polynomials from both autoreduced sets.[1]^: 81

• ${\displaystyle A_{1}<\cdots <A_{m}\in A}$ and ${\displaystyle B_{1}<\cdots <B_{n}\in B}$ and ${\displaystyle i,j,k\in \mathbb {N} }$.
• ${\displaystyle {\text{rank }}A<{\text{rank }}B}$ if there is a ${\displaystyle k\leq \operatorname {minimum} (m,n)}$ such that ${\displaystyle A_{i}=B_{i}}$ for ${\textstyle 1\leq i<k}$ and ${\displaystyle A_{k}<B_{k}}$.
• ${\displaystyle \operatorname {rank} A<\operatorname {rank} B}$ if ${\displaystyle n<m}$ and ${\displaystyle A_{i}=B_{i}}$ for ${\displaystyle 1\leq i\leq n}$.
• ${\displaystyle \operatorname {rank} A=\operatorname {rank} B}$ if ${\displaystyle n=m}$ and ${\displaystyle A_{i}=B_{i}}$ for ${\displaystyle 1\leq i\leq n}$.

A characteristic set ${\textstyle C}$ is the lowest ranked autoreduced subset among all the ideal's autoreduced subsets whose subset polynomial separants are non-members of the ideal ${\textstyle {\mathcal {I}}}$.[1]^: 82

The delta polynomial applies to polynomial pair ${\textstyle p,q}$ whose leaders share a common derivative, ${\textstyle \theta _{\alpha }u_{p}=\theta _{\beta }u_{q}}$. The least common derivative operator for the polynomial pair's leading derivatives is ${\textstyle \theta _{pq}}$, and the delta polynomial is:[1]^: 136[12]^: 160

${\displaystyle \operatorname {\Delta -poly} (p,q)=S_{q}\cdot {\frac {\theta _{pq}p}{\theta _{p}}}-S_{p}\cdot {\frac {\theta _{pq}q}{\theta _{q}}}}$

A coherent set is a polynomial set that reduces its delta polynomial pairs to zero.[1]^: 136[12]^: 160

A regular system ${\textstyle \Omega }$ contains a autoreduced and coherent set of differential equations ${\textstyle A}$ and a inequation set ${\textstyle H_{\Omega }\supseteq H_{A}}$ with set ${\textstyle H_{\Omega }}$ reduced with respect to the equation set.[12]^: 160

Regular differential ideal ${\textstyle {\mathcal {I}}_{\text{dif}}}$ and regular algebraic ideal ${\textstyle {\mathcal {I}}_{\text{alg}}}$ are saturation ideals that arise from a regular system.[12]^: 160 Lazard's lemma states that the regular differential and regular algebraic ideals are radical ideals.[20]

• Regular differential ideal: ${\textstyle {\mathcal {I}}_{\text{dif}}=[A]:H_{\Omega }^{\infty }.}$
• Regular algebraic ideal: ${\textstyle {\mathcal {I}}_{\text{dif}}=(A):H_{\Omega }^{\infty }.}$

The Rosenfeld–Gröbner algorithm decomposes the radical differential ideal as a finite intersection of regular radical differential ideals. These regular differential radical ideals, represented by characteristic sets, are not necessarily prime ideals and the representation is not necessarily minimal.[12]^: 158

The membership problem is to determine if a differential polynomial ${\textstyle p}$ is a member of an ideal generated from a set of differential polynomials ${\textstyle S}$. The Rosenfeld–Gröbner algorithm generates sets of Gröbner bases. The algorithm determines that a polynomial is a member of the ideal if and only if the partially reduced remainder polynomial is a member of the algebraic ideal generated by the Gröbner bases.[12]^: 164

The Rosenfeld–Gröbner algorithm facilitates creating Taylor series expansions of solutions to the differential equations.[21]

Example 1: ${\textstyle (\operatorname {Mer} (\operatorname {f} (y),\partial _{y})}$ is the differential meromorphic function field with a single standard derivation.

Example 2: ${\textstyle (\mathbb {C} \{y\},(1+3\cdot y+y^{2})\cdot \partial _{y})}$ is a differential field with a linear differential operator as the derivation.

Define ${\textstyle E^{a}(p(y))=p(y+a)}$ as shift operator ${\textstyle E^{a}}$ for polynomial ${\textstyle p(y)}$.

A shift-invariant operator ${\textstyle T}$ commutes with the shift operator: ${\textstyle E^{a}\circ T=T\circ E^{a}}$.

The Pincherle derivative, a derivation of shift-invariant operator ${\textstyle T}$, is ${\textstyle T^{\prime }=T\circ y-y\circ T}$.[22]^: 694

Ring of integers is ${\displaystyle (\mathbb {Z} .\delta )}$, and every integer is a constant.

• The derivation of 1 is zero. ${\textstyle \delta (1)=\delta (1\cdot 1)=\delta (1)\cdot 1+1\cdot \delta (1)=2\cdot \delta (1)\Rightarrow \delta (1)=0}$.
• Also, ${\displaystyle \delta (m+1)=\delta (m)+\delta (1)=\delta (m)\Rightarrow \delta (m+1)=\delta (m)}$.
• By induction, ${\displaystyle \delta (1)=0\ \wedge \ \delta (m+1)=\delta (m)\Rightarrow \forall \ m\in \mathbb {Z} ,\ \delta (m)=0}$.

Field of rational numbers is ${\displaystyle (\mathbb {Q} .\delta )}$, and every rational number is a constant.

• Every rational number is a quotient of integers.

  ${\displaystyle \forall r\in \mathbb {Q} ,\ \exists \ a\in \mathbb {Z} ,\ b\in \mathbb {Z} /\{0\},\ r={\frac {a}{b}}}$

• Apply the derivation formula for quotients recognizing that derivations of integers are zero:

  ${\displaystyle \delta (r)=\delta \left({\frac {a}{b}}\right)={\frac {\delta (a)\cdot b-a\cdot \delta (b)}{b^{2}}}=0}$.

Constants form the subring of constants ${\textstyle (\mathbb {C} ,\partial _{y})\subset (\mathbb {C} \{y\},\partial _{y})}$.[1]^: 60

Element ${\textstyle \exp(y)}$ simply generates differential ideal ${\textstyle [\exp(y)]}$ in the differential ring ${\textstyle (\mathbb {C} \{y,\exp(y)\},\partial _{y})}$.[7]^: 4

Any ring with identity is a ${\textstyle \operatorname {{\mathcal {Z}}-} }$algebra.[23]^: 343 Thus a differential ring is a ${\textstyle \operatorname {{\mathcal {Z}}-} }$algebra.

If ring ${\textstyle {\mathcal {R}}}$ is a subring of the center of unital ring ${\textstyle {\mathcal {M}}}$, then ${\textstyle {\mathcal {M}}}$ is an ${\textstyle \operatorname {{\mathcal {R}}-} }$algebra.[23]^: 343 Thus, a differential ring is an algebra over its differential subring. This is the natural structure of an algebra over its subring.[1]^: 75

Ring ${\textstyle (\mathbb {Q} \{y,z\},\partial _{y})}$ has irreducible polynomials, ${\textstyle p}$ (normal, squarefree) and ${\textstyle q}$ (special, ideal generator).

${\textstyle \partial _{y}(y)=1,\ \partial _{y}(z)=1+z^{2},\ z=\tan(y)}$

${\textstyle p(y)=1+y^{2},\ \partial _{y}(p)=2\cdot y,\ \gcd(p,\partial _{y}(p))=1}$

${\textstyle q(z)=1+z^{2},\ \partial _{y}(q)=2\cdot z\cdot (1+z^{2}),\ \gcd(q,\partial _{y}(q))=q}$

Ring ${\textstyle (\mathbb {Q} \{y_{1},y_{2}\},\delta )}$ has derivatives ${\textstyle \delta (y_{1})=y_{1}^{\prime }}$ and ${\textstyle \delta (y_{2})=y_{2}^{\prime }}$

• Map each derivative to an integer tuple: ${\textstyle \eta (\delta ^{(i_{2})}(y_{i_{1}}))=(i_{1},i_{2})}$.
• Rank derivatives and integer tuples: ${\textstyle y_{2}^{\prime \prime }\ (2,2)>y_{2}^{\prime }\ (2,1)>y_{2}\ (2,0)>y_{1}^{\prime \prime }\ (1,2)>y_{1}^{\prime }\ (1,1)>y_{1}\ (1,0)}$.

The leading derivatives, and initials are:

${\textstyle p={\color {Blue}(y_{1}+y_{1}^{\prime })}\cdot ({\color {Red}y_{2}^{\prime \prime }})^{2}+3\cdot y_{1}^{2}\cdot {\color {Red}y_{2}^{\prime \prime }}+(y_{1}^{\prime })^{2}}$

${\textstyle q={\color {Blue}(y_{1}+3\cdot y_{1}^{\prime })}\cdot {\color {Red}y_{2}^{\prime \prime }}+y_{1}\cdot y_{2}^{\prime }+(y_{1}^{\prime })^{2}}$

${\textstyle r={\color {Blue}(y_{1}+3)}\cdot ({\color {Red}y_{1}^{\prime \prime }})^{2}+y_{1}^{2}\cdot {\color {Red}y_{1}^{\prime \prime }}+2\cdot y_{1}}$

${\textstyle S_{p}=2\cdot (y_{1}+y_{1}^{\prime })\cdot y_{2}^{\prime \prime }+3\cdot y_{1}^{2}}$.

${\textstyle S_{q}=y_{1}+3\cdot y_{1}^{\prime }}$

${\textstyle S_{r}=2\cdot (y_{1}+3)\cdot y_{1}^{\prime \prime }+y_{1}^{2}}$

• Autoreduced sets are ${\textstyle \{p,r\}}$ and ${\textstyle \{q,r\}}$. Each set is triangular with a distinct polynomial leading derivative.
• The non-autoreduced set ${\textstyle \{p,q\}}$ contains only partially reduced ${\textstyle p}$ with respect to ${\textstyle q}$; this set is non-triangular because the polynomials have the same leading derivative.

Symbolic integration uses algorithms involving polynomials and their derivatives such as Hermite reduction, Czichowski algorithm, Lazard-Rioboo-Trager algorithm, Horowitz-Ostrogradsky algorithm, squarefree factorization and splitting factorization to special and normal polynomials.[6]^{: 41, 51, 53, 102, 299, 309}

Differential algebra can determine if a set of differential polynomial equations has a solution. A total order ranking may identify algebraic constraints. An elimination ranking may determine if one or a selected group of independent variables can express the differential equations. Using triangular decomposition and elimination order, it may be possible to solve the differential equations one differential indeterminate at a time in a step-wise method. Another approach is to create a class of differential equations with a known solution form; matching a differential equation to its class identifies the equation's solution. Methods are available to facilitate the numerical integration of a differential-algebraic system of equations.[10]^: 41–47

In a study of non-linear dynamical systems with chaos, researchers used differential elimination to reduce differential equations to ordinary differential equations involving a single state variable. They were successful in most cases, and this facilitated developing approximate solutions, efficiently evaluating chaos, and constructing Lypapunov functions.[24] Researchers have applied differential elimination to understanding cellular biology, compartmental biochemical models, parameter estimation and quasi-steady state approximation (QSSA) for biochemical reactions.[25][26] Using differential Gröbner bases, researchers have investigated non-classical symmetry properties of non-linear differential equations.[27] Other applications include control theory, model theory, and algebraic geometry.[28][9][8] Differential algebra also applies to differential-difference equations.[17]

A ${\textstyle \operatorname {\mathbb {Z} -graded} }$ vector space ${\textstyle V_{\bullet }}$ is a collection of vector spaces ${\textstyle V_{m}}$ with integer degree ${\textstyle |v|=m}$ for ${\textstyle v\in V_{m}}$. A direct sum can represent this graded vector space:[29]^: 48

${\displaystyle V_{\bullet }=\bigoplus _{m\in \mathbb {Z} }V_{m}}$

A differential graded vector space or chain complex, is a graded vector space ${\textstyle V_{\bullet }}$ with a differential map or boundary map ${\textstyle d_{m}:V_{m}\to V_{m-1}}$ with ${\displaystyle d_{m}\circ d_{m+1}=0}$ .[29]^: 50–51

A cochain complex is a graded vector space ${\textstyle V^{\bullet }}$ with a differential map or coboundary map ${\textstyle d_{m}:V_{m}\to V_{m+1}}$ with ${\displaystyle d_{m+1}\circ d_{m}=0}$.[29]^: 50–51

A differential graded algebra is a graded algebra ${\textstyle A}$ with a linear derivation ${\textstyle d:A\to A}$ with ${\displaystyle d\circ d=0}$ that follows the graded Leibniz product rule.[29]^: 58–59

• Graded Leibniz product rule: ${\displaystyle \forall a,b\in A,\ d(a\cdot b)=d(a)\cdot b+(-1)^{|a|}\cdot a\cdot d(b)}$ with ${\displaystyle |a|}$ the degree of vector ${\displaystyle a}$.

A Lie algebra is a finite-dimensional real or complex vector space ${\textstyle {\mathcal {g}}}$ with a bilinear bracket operator ${\textstyle [,]:{\mathcal {g}}\times {\mathcal {g}}\to {\mathcal {g}}}$ with Skew symmetry and the Jacobi identity property.[30]^: 49

• Skew symmetry: ${\displaystyle [X,Y]=-[Y,X]}$
• Jacobi identity property: ${\displaystyle [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0}$

for all ${\displaystyle X,Y,Z\in {\mathcal {g}}}$.

The adjoint operator, ${\textstyle \operatorname {ad_{X}} (Y)=[Y,X]}$ is a derivation of the bracket because the adjoint's effect on the binary bracket operation is analogous to the derivation's effect on the binary product operation. This is the inner derivation determined by ${\textstyle X}$.[30]^: 51[31]^: 9

${\displaystyle \operatorname {ad} _{X}([Y,Z])=[\operatorname {ad} _{X}(Y),Z]+[Y,\operatorname {ad} _{X}(Z)]}$

The universal enveloping algebra ${\textstyle U({\mathcal {g}})}$ of Lie algebra ${\textstyle {\mathcal {g}}}$ is a maximal associative algebra with identity, generated by Lie algebra elements ${\textstyle {\mathcal {g}}}$ and containing products defined by the bracket operation. Maximal means that a linear homomorphism maps the universal algebra to any other algebra that otherwise has these properties. The adjoint operator is a derivation following the Leibniz product rule.[30]^: 247

• Product in ${\displaystyle U({\mathcal {g}})}$ : ${\displaystyle X\cdot Y-Y\cdot X=[X,Y]}$
• Leibniz product rule: ${\displaystyle \operatorname {ad} _{X}(Y\cdot Z)=\operatorname {ad} _{X}(Y)\cdot Z+Y\cdot \operatorname {ad} _{X}(Z)}$

for all ${\displaystyle X,Y,Z\in U({\mathcal {g}})}$.

The Weyl algebra is an algebra ${\textstyle A_{n}(K)}$ over a ring ${\textstyle K[p_{1},q_{1},\dots ,p_{n},q_{n}]}$ with a specific noncommutative product: [32]^: 7–8

${\displaystyle p_{i}\cdot q_{i}-q_{i}\cdot p_{i}=1,\$ :\ i\in \{1,\dots ,n\}} .

All other indeterminate products are commutative for ${\textstyle i,j\in \{1,\dots ,n\}}$:

${\displaystyle p_{i}\cdot q_{j}-q_{j}\cdot p_{i}=0{\text{ if }}i\neq j,\ p_{i}\cdot p_{j}-p_{j}\cdot p_{i}=0,\ q_{i}\cdot q_{j}-q_{j}\cdot q_{i}=0}$.

A Weyl algebra can represent the derivations for a commutative ring's polynomials ${\textstyle f\in K[y_{1},\ldots ,y_{n}]}$. The Weyl algebra's elements are endomorphisms, the elements ${\textstyle p_{1},\ldots ,p_{n}}$ function as standard derivations, and map compositions generate linear differential operators. D-module is a related approach for understanding differential operators. The endomorphisms are:[32]^: 7–8

${\displaystyle q_{j}(y_{k})=y_{j}\cdot y_{k},\ q_{j}(c)=c\cdot y_{j}{\text{ with }}c\in K,\ p_{j}(y_{j})=1,\ p_{j}(y_{k})=0{\text{ if }}j\neq k,\ p_{j}(c)=0{\text{ with }}c\in K}$

The associative, possibly noncommutative ring ${\textstyle A}$ has derivation ${\textstyle d:A\to A}$.[33]^: 268

The pseudo-differential operator ring ${\textstyle A((\partial ^{-1}))}$ is a left ${\textstyle \operatorname {A-module} }$ containing ring elements ${\textstyle L}$:[33]^: 268[23]^: 337[34]

${\displaystyle a_{i}\in A,\ i,i_{\min }\in \mathbb {N} ,\ |i_{\min }|>0\$ :\ L=\sum _{i\geq i_{\min }}^{n}a_{i}\cdot \partial ^{i}}

The derivative operator is ${\textstyle d(a)=\partial \circ a-a\circ \partial }$.[33]^: 268

The binomial coefficient is ${\displaystyle {\Bigl (}{i \atop k}{\Bigr )}}$.

Pseudo-differential operator multiplication is:[33]^: 268

${\displaystyle \sum _{i\geq i_{\min }}^{n}a_{i}\cdot \partial ^{i}\cdot \sum _{j\geq j_{\min }}^{m}b_{i}\cdot \partial ^{j}=\sum _{i,j;k\geq 0}{\Bigl (}{i \atop k}{\Bigr )}\cdot a_{i}\cdot d^{k}(b_{j})\cdot \partial ^{i+j-k}}$