Weyl algebra#Properties of the Weyl algebra

{{see also|Canonical commutation relation}}

The Weyl algebra arises naturally in the context of quantum mechanics and the process of canonical quantization. Consider a classical phase space with canonical coordinates $(q\_1,\; p\_1,\; \backslash dots,\; q\_n,\; p\_n)$. These coordinates satisfy the Poisson bracket relations:$$$$

\{q_i, q_j\} = 0, \quad \{p_i, p_j\} = 0, \quad \{q_i, p_j\} = \delta_{ij}.

In canonical quantization, one seeks to construct a Hilbert space of states and represent the classical observables (functions on phase space) as self-adjoint operators on this space. The canonical commutation relations are imposed:$$$$

[\hat{q}_i, \hat{q}_j] = 0, \quad [\hat{p}_i, \hat{p}_j] = 0, \quad [\hat{q}_i, \hat{p}_j] = i\hbar \delta_{ij},

where $[\backslash cdot,\; \backslash cdot]$ denotes the commutator. Here, $\backslash hat\{q\}\_i$ and $\backslash hat\{p\}\_i$ are the operators corresponding to $q\_i$ and $p\_i$ respectively. Erwin Schrödinger proposed in 1926 the following:{{sfn | Landsman | 2007 | p=428}}

• $\backslash hat\{q\_j\}$ with multiplication by $x\_j$.
• $\backslash hat\{p\}\_j$ with $-i\backslash hbar\; \backslash partial\_\{x\_j\}$.

With this identification, the canonical commutation relation holds.

The Weyl algebras have different constructions, with different levels of abstraction.

The Weyl algebra $A\_n$ can be concretely constructed as a representation.

In the differential operator representation, similar to Schrödinger's canonical quantization, let $q\_j$ be represented by multiplication on the left by $x\_j$, and let $p\_j$ be represented by differentiation on the left by $\backslash partial\_\{x\_j\}$.

In the matrix representation, similar to the matrix mechanics, $$

A_1

is represented by{{sfn|Coutinho|1997|pp=598–599}}$$$$

P=\begin{bmatrix}

0 & 1 & 0 & 0 & \cdots \\

0 & 0 & 2 & 0 & \cdots \\

0 & 0 & 0 & 3 & \cdots \\

\vdots & \vdots & \vdots & \vdots & \ddots

\end{bmatrix}, \quad Q=\begin{bmatrix}

0 & 0 & 0 & 0 & \ldots \\

1 & 0 & 0 & 0 & \cdots \\

0 & 1 & 0 & 0 & \cdots \\

\vdots & \vdots & \vdots & \vdots & \ddots

\end{bmatrix}

$A\_n$ can be constructed as a quotient of a free algebra in terms of generators and relations.

One construction starts with an abstract vector space V (of dimension 2n) equipped with a symplectic form ω. Define the Weyl algebra W(V) to be

: $W(V)\; :=\; T(V)\; /\; (\backslash !(\; v\; \backslash otimes\; u\; -\; u\; \backslash otimes\; v\; -\; \backslash omega(v,u),\; \backslash text\{\; for\; \}\; v,u\; \backslash in\; V\; )\backslash !),$

where T(V) is the tensor algebra on V, and the notation $(\backslash !(\; )\backslash !)$ means "the ideal generated by".

In other words, W(V) is the algebra generated by V subject only to the relation {{math|vu − uv {{=}} ω(v, u)}}. Then, W(V) is isomorphic to A_n via the choice of a Darboux basis for {{mvar|ω}}.

$A\_n$ is also a quotient of the universal enveloping algebra of the Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the central element of the Heisenberg algebra (namely [q, p]) equal to the unit of the universal enveloping algebra (called 1 above).

The algebra W(V) is a quantization of the symmetric algebra Sym(V). If V is over a field of characteristic zero, then W(V) is naturally isomorphic to the underlying vector space of the symmetric algebra Sym(V) equipped with a deformed product – called the Groenewold–Moyal product (considering the symmetric algebra to be polynomial functions on V^∗, where the variables span the vector space V, and replacing iħ in the Moyal product formula with 1).

The isomorphism is given by the symmetrization map from Sym(V) to W(V)

: $a\_1\; \backslash cdots\; a\_n\; \backslash mapsto\; \backslash frac\{1\}\{n!\}\; \backslash sum\_\{\backslash sigma\; \backslash in\; S\_n\}\; a\_\{\backslash sigma(1)\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; a\_\{\backslash sigma(n)\}~.$

If one prefers to have the iħ and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by q_i and iħ∂_qi (as per quantum mechanics usage).

Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the Moyal quantization (if for the latter one restricts to polynomial functions), but the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication.

Stated in another way, let the Moyal star product be denoted $f\; \backslash star\; g$, then the Weyl algebra is isomorphic to $(\backslash mathbb\; C[x\_1,\; \backslash dots,\; x\_n],\; \backslash star)$.{{sfn|Coutinho|1997|pp=602–603}}

In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra, which is also referred to as the orthogonal Clifford algebra.{{sfn|Lounesto|Ablamowicz|2004|p=xvi}}{{sfn|Micali|Boudet|Helmstetter|1992|pp=83-96}}

The Weyl algebra is also referred to as the symplectic Clifford algebra.{{sfn | Lounesto | Ablamowicz | 2004|p=xvi}}{{sfn | Micali | Boudet | Helmstetter | 1992 | pp=83-96}}{{sfn | Helmstetter | Micali | 2008 | p=xii}} Weyl algebras represent for symplectic bilinear forms the same structure that Clifford algebras represent for non-degenerate symmetric bilinear forms.{{sfn | Helmstetter | Micali | 2008 | p=xii}}

The Weyl algebra can be constructed as a D-module.{{sfn | Coutinho | 1997 | pp=600–601}} Specifically, the Weyl algebra corresponding to the polynomial ring $R[x\_1,\; ...,\; x\_n]$ with its usual partial differential structure is precisely equal to Grothendieck's ring of differential operations $D\_\{\backslash mathbb\{A\}^n\_R\; /\; R\}$.{{sfn | Coutinho | 1997 | pp=600–601}}

More generally, let $X$ be a smooth scheme over a ring $R$. Locally, $X\; \backslash to\; R$ factors as an étale cover over some $\backslash mathbb\{A\}^n\_R$ equipped with the standard projection.{{Cite web |title=Section 41.13 (039P): Étale and smooth morphisms—The Stacks project |url=https://stacks.math.columbia.edu/tag/039P |access-date=2024-09-29 |website=stacks.math.columbia.edu}} Because "étale" means "(flat and) possessing null cotangent sheaf",{{Cite web |title=etale morphism of schemes in nLab |url=https://ncatlab.org/nlab/show/etale+morphism+of+schemes |access-date=2024-09-29 |website=ncatlab.org}} this means that every D-module over such a scheme can be thought of locally as a module over the $n^\backslash text\{th\}$ Weyl algebra.

Let $R$ be a commutative algebra over a subring $S$. The ring of differential operators $D\_\{R/S\}$ (notated $D\_R$ when $S$ is clear from context) is inductively defined as a graded subalgebra of $\backslash operatorname\{End\}\_\{S\}(R)$:

• $D^0\_R=R$
• $$

D^k_R=\left\{d \in \operatorname{End}_{S}(R):[d, a] \in D^{k-1}_R \text { for all } a \in R\right\} .

Let $D\_R$ be the union of all $D^k\_R$ for $k\; \backslash geq\; 0$. This is a subalgebra of $\backslash operatorname\{End\}\_\{S\}(R)$.

In the case $R\; =\; S[x\_1,\; ...,\; x\_n]$, the ring of differential operators of order $\backslash leq\; n$ presents similarly as in the special case $S\; =\; \backslash mathbb\{C\}$ but for the added consideration of "divided power operators"; these are operators corresponding to those in the complex case which stabilize $\backslash mathbb\{Z\}[x\_1,\; ...,\; x\_n]$, but which cannot be written as integral combinations of higher-order operators, i.e. do not inhabit $D\_\{\backslash mathbb\{A\}^n\_\backslash mathbb\{Z\}\; /\; \backslash mathbb\{Z\}\}$. One such example is the operator $\backslash partial\_\{x\_1\}^\{[p]\}\; :\; x\_1^N\; \backslash mapsto\; \{N\; \backslash choose\; p\}\; x\_1^\{N-p\}$.

Explicitly, a presentation is given by

:$D\_\{S[x\_1,\; \backslash dots,\; x\_\backslash ell]/S\}^n\; =\; S\; \backslash langle\; x\_1,\; \backslash dots,\; x\_\backslash ell,\; \backslash \{\backslash partial\_\{x\_i\},\; \backslash partial\_\{x\_i\}^\{[2]\},\; \backslash dots,\; \backslash partial\_\{x\_i\}^\{[n]\}\backslash \}\_\{1\; \backslash leq\; i\; \backslash leq\; \backslash ell\}\; \backslash rangle$

with the relations

:$[x\_i,\; x\_j]\; =\; [\backslash partial\_\{x\_i\}^\{[k]\},\; \backslash partial\_\{x\_j\}^\{[m]\}]\; =\; 0$

:

:$\backslash partial\_\{x\_i\}^\{[k]\}\; \backslash partial\_\{x\_i\}^\{[m]\}\; =\; \{k+m\; \backslash choose\; k\}\; \backslash partial\_\{x\_i\}^\{[k+m]\}\; ~~~~~\backslash text\{when\; \}k+m\; \backslash leq\; n$

where $\backslash partial\_\{x\_i\}^\{[0]\}\; =\; 1$ by convention. The Weyl algebra then consists of the limit of these algebras as $n\; \backslash to\; \backslash infty$.{{Cite journal |last=Grothendieck |first=Alexander |date=1964 |title=Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, Première partie |url=http://www.numdam.org/item/PMIHES_1964__20__5_0/ |journal=Publications Mathématiques de l'IHÉS |language=en |volume=20 |pages=5–259 |issn=1618-1913}}{{Pg|location=Ch. IV.16.II}}

When $S$ is a field of characteristic 0, then $D^1\_R$ is generated, as an $R$-module, by 1 and the $S$-derivations of $R$. Moreover, $D\_R$ is generated as a ring by the $R$-subalgebra $D^1\_R$. In particular, if $S\; =\; \backslash mathbb\{C\}$ and $R=\backslash mathbb\{C\}[x\_1,\; ...,\; x\_n]$, then $D^1\_R=R+\; \backslash sum\_i\; R\; \backslash partial\_\{x\_i\}$. As mentioned, $A\_n\; =\; D\_R$.{{sfn | Coutinho | 1995 | pp=20-24}}

Many properties of $$

A_1

apply to $$

A_n

with essentially similar proofs, since the different dimensions commute.

{{Main|General Leibniz rule}}

{{Math theorem

| name = Theorem

| note = general Leibniz rule

| math_statement = $$$$

p^k q^m = \sum_{l=0}^k \binom{k}{l} \frac{m!}{(m-l)!} q^{m-l} p^{k-l} = q^mp^k + mk q^{m-1}p^{k-1} + \cdots

}}

{{Math proof|title=Proof|proof=

Under the $$

p \mapsto x, q \mapsto \partial_x

representation, this equation is obtained by the general Leibniz rule. Since the general Leibniz rule is provable by algebraic manipulation, it holds for $$

A_1

as well.

}}In particular, $[q,\; q^m\; p^n]\; =\; -nq^mp^\{n-1\}$ and $[p,\; q^mp^n]\; =\; mq^\{m-1\}p^n$.

{{Math theorem

| math_statement = The center of Weyl algebra $A\_n$ is the underlying field of constants $F$.

| name = Corollary

}}

{{Math proof|title=Proof|proof=

If the commutator of $f$ with either of $p,\; q$ is zero, then by the previous statement, $f$ has no monomial $p^nq^m$ with $n\; >\; 0$ or $m\; >\; 0$.

}}

{{Math theorem

| name = Theorem

| note =

| math_statement = $$

A_n

has a basis $$

\{q^m p^n : m, n \geq 0\}

.{{Sfn|Coutinho|1995|p=9|loc=Proposition 2.1}}

}}

{{Math proof|title=Proof|proof=

By repeating the commutator relations, any monomial can be equated to a linear sum of these. It remains to check that these are linearly independent. This can be checked in the differential operator representation. For any linear sum $$

\sum_{m, n} c_{m,n} x^m \partial_x^n

with nonzero coefficients, group it in descending order: $$

p_N(x) \partial_x^N + p_{N-1}(x) \partial_x^{N-1} + \cdots + p_M(x) \partial_x^M

, where $$

p_M

is a nonzero polynomial. This operator applied to $$

x^M

results in $$

M! p_M(x) \neq 0

.

}}

This allows $$

A_1

to be a graded algebra, where the degree of $$

\sum_{m, n} c_{m,n} q^m p^n

is $$

\max (m + n)

among its nonzero monomials. The degree is similarly defined for $$

A_n

.

{{Math theorem

| name = Theorem

| math_statement = For $A\_n$:{{sfn | Coutinho | 1995 | pp=14-15}}

\deg(g + h) \leq \max(\deg(g), \deg(h))

\deg([g, h]) \leq \deg(g) + \deg(h) - 2

\deg(g h) = \deg(g) + \deg(h)

}}

{{Math proof|title=Proof|proof=

We prove it for $A\_1$, as the $A\_n$ case is similar.

The first relation is by definition. The second relation is by the general Leibniz rule. For the third relation, note that $\backslash deg(g\; h)\; \backslash leq\; \backslash deg(g)\; +\; \backslash deg(h)$, so it is sufficient to check that $gh$ contains at least one nonzero monomial that has degree $\backslash deg(g)\; +\; \backslash deg(h)$. To find such a monomial, pick the one in $g$ with the highest degree. If there are multiple such monomials, pick the one with the highest power in $q$. Similarly for $h$. These two monomials, when multiplied together, create a unique monomial among all monomials of $gh$, and so it remains nonzero.

}}

{{Math theorem

| name = Theorem

| math_statement = $A\_n$ is a simple domain.{{sfn | Coutinho | 1995 | p=16}}

}}

That is, it has no two-sided nontrivial ideals and has no zero divisors.

{{Math proof|title=Proof|proof=

Because $\backslash deg(gh)\; =\; \backslash deg(g)\; +\; \backslash deg(h)$, it has no zero divisors.

Suppose for contradiction that $I$ is a nonzero two-sided ideal of $A\_1$, with $I\; \backslash neq\; A\_1$. Pick a nonzero element $f\; \backslash in\; I$ with the lowest degree.

If $f$ contains some nonzero monomial of form $xx^m\backslash partial^n\; =\; x^\{m+1\}\; \backslash partial^n$, then

[\partial, f] = \partial f - f \partial

contains a nonzero monomial of form

\partial x^{m+1} \partial^n - x^{m+1} \partial^n \partial = (m+1) x^m \partial^n.

Thus $[\backslash partial,\; f]$ is nonzero, and has degree $\backslash leq\; \backslash deg(f)-1$. As $I$ is a two-sided ideal, we have $[\backslash partial,\; f]\; \backslash in\; I$, which contradicts the minimality of $\backslash deg(f)$.

Similarly, if $f$ contains some nonzero monomial of form $x^m\backslash partial^n\backslash partial$, then $[x,\; f]\; =\; xf\; -\; fx$ is nonzero with lower degree.

}}

{{See|Derivation (differential algebra)}}{{Math theorem|

| math_statement = The derivations of $A\_n$ are in bijection with the elements of $A\_n$ up to an additive scalar.{{sfn | Dirac | 1926 | pp=415–417}}}}

That is, any derivation $D$ is equal to $[\backslash cdot,\; f]$ for some $f\; \backslash in\; A\_n$; any $f\backslash in\; A\_n$ yields a derivation $[\backslash cdot,\; f]$; if $f,\; f\text{'}\; \backslash in\; A\_n$ satisfies $[\backslash cdot,\; f]\; =\; [\backslash cdot,\; f\text{'}]$, then $f\; -\; f\text{'}\; \backslash in\; F$.

The proof is similar to computing the potential function for a conservative polynomial vector field on the plane.{{sfn | Coutinho | 1997 | p=597}}

{{Collapse top|title=Proof}}

Since the commutator is a derivation in both of its entries, $[\backslash cdot,\; f]$ is a derivation for any $f\backslash in\; A\_n$. Uniqueness up to additive scalar is because the center of $A\_n$ is the ring of scalars.

It remains to prove that any derivation is an inner derivation by induction on $n$.

Base case: Let $D:\; A\_1\; \backslash to\; A\_1$ be a linear map that is a derivation. We construct an element $r$ such that $[p,\; r]\; =\; D(p),\; [q,r]\; =\; D(q)$. Since both $D$ and $[\backslash cdot,\; r]$ are derivations, these two relations generate $[g,\; r]\; =\; D(g)$ for all $g\backslash in\; A\_1$.

Since $[p,\; q^mp^n]\; =\; mq^\{m-1\}p^n$, there exists an element $f\; =\; \backslash sum\_\{m,n\}\; c\_\{m,n\}\; q^m\; p^n$ such that $$$$

[p, f] = \sum_{m,n} m c_{m,n} q^m p^n = D(p)

\begin{aligned}

0 &\stackrel{[p, q] = 1}{=} D([p, q]) \\

&\stackrel{D \text{ is a derivation}}{=} [p, D(q)] + [D(p), q] \\

&\stackrel{[p,f] = D(p)}{=} [p, D(q)] + [[p,f], q] \\

&\stackrel{\text{Jacobi identity}}{=} [p, D(q) - [q, f]]

\end{aligned}

Thus, $D(q)\; =\; g(p)\; +\; [q,\; f]$ for some polynomial $g$. Now, since $[q,\; q^m\; p^n]\; =\; -nq^mp^\{n-1\}$, there exists some polynomial $h(p)$ such that $[q,\; h(p)]\; =\; g(p)$. Since $[p,\; h(p)]\; =\; 0$, $r\; =\; f\; +\; h(p)$ is the desired element.

For the induction step, similarly to the above calculation, there exists some element $r\; \backslash in\; A\_n$ such that $[q\_1,\; r]\; =\; D(q\_1),\; [p\_1,\; r]\; =\; D(p\_1)$.

Similar to the above calculation, $$$$

[x, D(y) - [y, r]] = 0

for all $x\; \backslash in\; \backslash \{p\_1,\; q\_1\backslash \},\; y\; \backslash in\; \backslash \{p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash \}$. Since $[x,\; D(y)\; -\; [y,\; r]]$ is a derivation in both $x$ and $y$, $[x,\; D(y)\; -\; [y,\; r]]\; =\; 0$ for all $x\backslash in\; \backslash langle\; p\_1,\; q\_1\backslash rangle$ and all $y\; \backslash in\; \backslash langle\; p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash rangle$. Here, $\backslash langle\; \backslash rangle$ means the subalgebra generated by the elements.

Thus, $\backslash forall\; y\; \backslash in\; \backslash langle\; p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash rangle$, $$$$

D(y) - [y, r] \in \langle p_2, \dots, p_n, q_2, \dots, q_n\rangle

Since $D\; -\; [\backslash cdot,\; r]$ is also a derivation, by induction, there exists $r\text{'}\; \backslash in\; \backslash langle\; p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash rangle$ such that $D(y)\; -\; [y,\; r]\; =\; [y,\; r\text{'}]$ for all $y\; \backslash in\; \backslash langle\; p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash rangle$.

Since $p\_1,\; q\_1$ commutes with $\backslash langle\; p\_2,\; \backslash dots,\; p\_n,\; q\_2,\; \backslash dots,\; q\_n\backslash rangle$, we have $D(y)\; =\; [y,\; r\; +\; r\text{'}]$ for all $y\; \backslash in\; \backslash \{p\_1,\; \backslash dots,\; p\_n,\; q\_1,\; \backslash dots,\; q\_n\backslash \}$, and so for all of $A\_n$.

{{Collapse bottom}}

{{further|Stone–von Neumann theorem}}

In the case that the ground field {{mvar|F}} has characteristic zero, the nth Weyl algebra is a simple Noetherian domain.{{sfn | Coutinho | 1995 | p=70}} It has global dimension n, in contrast to the ring it deforms, Sym(V), which has global dimension 2n.

It has no finite-dimensional representations. Although this follows from simplicity, it can be more directly shown by taking the trace of σ(q) and σ(Y) for some finite-dimensional representation σ (where {{nowrap|1=[q,p] = 1}}).

: $\backslash mathrm\{tr\}([\backslash sigma(q),\backslash sigma(Y)])=\backslash mathrm\{tr\}(1)~.$

Since the trace of a commutator is zero, and the trace of the identity is the dimension of the representation, the representation must be zero dimensional.

In fact, there are stronger statements than the absence of finite-dimensional representations. To any finitely generated A_n-module M, there is a corresponding subvariety Char(M) of {{nowrap|V × V^∗}} called the 'characteristic variety'{{what|date=August 2016}} whose size roughly corresponds to the size{{what|date=August 2016}} of M (a finite-dimensional module would have zero-dimensional characteristic variety). Then Bernstein's inequality states that for M non-zero,

: $\backslash dim(\backslash operatorname\{char\}(M))\backslash geq\; n$

An even stronger statement is Gabber's theorem, which states that Char(M) is a co-isotropic subvariety of {{nowrap|V × V^∗}} for the natural symplectic form.

The situation is considerably different in the case of a Weyl algebra over a field of characteristic {{nowrap|p > 0}}.

In this case, for any element D of the Weyl algebra, the element D^p is central, and so the Weyl algebra has a very large center. In fact, it is a finitely generated module over its center; even more so, it is an Azumaya algebra over its center. As a consequence, there are many finite-dimensional representations which are all built out of simple representations of dimension p.

The ideals and automorphisms of $A\_1$ have been well-studied.{{sfn|Berest|Wilson|2000|pp=127–147}}{{sfn|Cannings|Holland|1994|pp=116–141}} The moduli space for its right ideal is known.{{sfn|Lebruyn|1995|pp=32–48}} However, the case for $A\_n$ is considerably harder and is related to the Jacobian conjecture.{{sfn|Coutinho|1995|loc=section 4.4}}

For more details about this quantization in the case n = 1 (and an extension using the Fourier transform to a class of integrable functions larger than the polynomial functions), see Wigner–Weyl transform.

Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.

Weyl algebras also generalize in the case of algebraic varieties. Consider a polynomial ring

: $R\; =\; \backslash frac\{\backslash mathbb\{C\}[x\_1,\backslash ldots,x\_n]\}\{I\}.$

Then a differential operator is defined as a composition of $\backslash mathbb\{C\}$-linear derivations of $R$. This can be described explicitly as the quotient ring

: $\backslash text\{Diff\}(R)\; =\; \backslash frac\{\backslash \{\; D\; \backslash in\; A\_n\backslash colon\; D(I)\; \backslash subseteq\; I\; \backslash \}\}\{\; I\backslash cdot\; A\_n\}.$

• Jacobian conjecture
• Dixmier conjecture

== Notes ==

{{reflist}}

• {{cite book | last=Coutinho | first=S. C. | title=A Primer of Algebraic D-Modules | publisher=Cambridge University Press | publication-place=Cambridge [England] ; New York, NY, USA | date=1995| isbn=978-0-521-55119-9|doi=10.1017/cbo9780511623653}}
• {{cite journal | last=Coutinho | first=S. C. | title=The Many Avatars of a Simple Algebra | journal=The American Mathematical Monthly | volume=104 | issue=7 | date=1997 | issn=0002-9890 | doi=10.1080/00029890.1997.11990687 | pages=593–604}}
• {{cite journal | last=Dirac | first=P. A. M. | authorlink=Paul Dirac | title=On Quantum Algebra | journal=Mathematical Proceedings of the Cambridge Philosophical Society | volume=23 | issue=4 | date=1926 | issn=0305-0041 | doi=10.1017/S0305004100015231 | pages=412–418}}
• {{cite book | last=Helmstetter | first=J. | last2=Micali | first2=A. | title=Quadratic Mappings and Clifford Algebras | publisher=Birkhäuser | publication-place=Basel ; Boston | date=2008 | isbn=978-3-7643-8605-4 | oclc=175285188}}
• {{cite book | last=Landsman | first=N.P. | title=Philosophy of Physics | chapter=BETWEEN CLASSICAL AND QUANTUM | publisher=Elsevier | date=2007 | isbn=978-0-444-51560-5 | doi=10.1016/b978-044451560-5/50008-7}}
• {{cite book | last=Lounesto | first=P. | last2=Ablamowicz | first2=R. | title=Clifford Algebras | publisher=Springer Science & Business Media | publication-place=Boston | date=2004 | isbn=0-8176-3525-4}}
• {{cite book | last=Micali | first=A. | last2=Boudet | first2=R. | last3=Helmstetter | first3=J. | title=Clifford Algebras and their Applications in Mathematical Physics | publisher=Springer Science & Business Media | publication-place=Dordrecht | date=1992 | isbn=0-7923-1623-1}}
• {{cite journal |first1=M. Rausch |last1=de Traubenberg |first2=M. J. |last2=Slupinski |first3=A. |last3=Tanasa |title=Finite-dimensional Lie subalgebras of the Weyl algebra |journal=J. Lie Theory |year=2006 |volume=16 |pages=427–454 |arxiv=math/0504224}}
• {{cite book |first=Will |last=Traves |chapter=Differential Operations on Grassmann Varieties |editor-last=Campbell |editor-first=H. |editor2-last=Helminck |editor2-first=A. |editor3-last=Kraft |editor3-first=H. |editor4-last=Wehlau |editor4-first=D. |title=Symmetry and Spaces |publisher=Birkhäuse |series=Progress in Mathematics |volume=278 |year=2010 |isbn=978-0-8176-4875-6 |pages=197–207 |doi=10.1007/978-0-8176-4875-6_10 }}
• {{cite book |author-link=Tsit Yuen Lam |author=Tsit Yuen Lam |title=A first course in noncommutative rings |publisher=Springer |edition=2nd |year=2001 |isbn=978-0-387-95325-0 |pages=6 |volume=131 |series=Graduate Texts in Mathematics}}
• {{cite journal |last=Berest |first=Yuri |last2=Wilson |first2=George |date=September 1, 2000 |title=Automorphisms and ideals of the Weyl algebra |journal=Mathematische Annalen |volume=318 |issue=1 |pages=127–147 |doi=10.1007/s002080000115 |issn=0025-5831|arxiv=math/0102190 }}
• {{cite journal |last=Cannings |first=R.C. |last2=Holland |first2=M.P. |year=1994 |title=Right Ideals of Rings of Differential Operators |journal=Journal of Algebra |publisher=Elsevier BV |volume=167 |issue=1 |pages=116–141 |doi=10.1006/jabr.1994.1179 |issn=0021-8693}}
• {{cite journal |last=Lebruyn |first=L. |year=1995 |title=Moduli Spaces for Right Ideals of the Weyl Algebra |journal=Journal of Algebra |publisher=Elsevier BV |volume=172 |issue=1 |pages=32–48 |doi=10.1006/jabr.1995.1046 |issn=0021-8693|hdl=10067/123950151162165141 |hdl-access=free }}

Category:Algebras

Category:Differential operators

Category:Ring theory