Weyl algebra

Template:Short description In abstract algebra, the Weyl algebras are abstracted from the ring of differential operators with polynomial coefficients. They are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics.

In the simplest case, these are differential operators. Let $F$ be a field, and let $F[x]$ be the ring of polynomials in one variable with coefficients in $F$. Then the corresponding Weyl algebra consists of differential operators of form

$f_m(x)\underset{x}{\overset{m}{\partial }}+f_{m-1}(x)\underset{x}{\overset{m-1}{\partial }}+\cdots +f_1(x){\partial }_x+f_0(x)$

This is the first Weyl algebra $A_1$. The n-th Weyl algebra $A_n$ are constructed similarly.

Alternatively, $A_1$ can be constructed as the quotient of the free algebra on two generators, q and p, by the ideal generated by $([p,q]-1)$. Similarly, $A_n$ is obtained by quotienting the free algebra on 2n generators by the ideal generated by$$([p_i,q_j]-{\delta }_{i,j}),\mathrm{\forall }i,j=1,\dots ,n$$where ${\delta }_{i,j}$ is the Kronecker delta.

More generally, let $(R,\mathrm{\Delta })$ be a partial differential ring with commuting derivatives $\mathrm{\Delta }=\{{\partial }_1,\dots ,{\partial }_m\}$. The Weyl algebra associated to $(R,\mathrm{\Delta })$ is the noncommutative ring $R[{\partial }_1,\dots ,{\partial }_m]$ satisfying the relations ${\partial }_{i}r=r{\partial }_i+{\partial }_i(r)$ for all $r\in R$. The previous case is the special case where $R=F[x_1,\dots ,x_n]$ and $\mathrm{\Delta }=\{{\partial }_{x_1},\dots ,{\partial }_{x_n}\}$ where $F$ is a field.

This article discusses only the case of $A_n$ with underlying field $F$ characteristic zero, unless otherwise stated.

The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension.

Motivation

Template:See also 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 $(q_1,p_1,\dots ,q_n,p_n)$. These coordinates satisfy the Poisson bracket relations:$$\{q_i,q_j\}=0,\{p_i,p_j\}=0,\{q_i,p_j\}={\delta }_{ij}.$$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:$$[{\hat{q}}_i,{\hat{q}}_j]=0,[{\hat{p}}_i,{\hat{p}}_j]=0,[{\hat{q}}_i,{\hat{p}}_j]=i\hslash {\delta }_{ij},$$where $[\cdot ,\cdot ]$ denotes the commutator. Here, ${\hat{q}}_i$ and ${\hat{p}}_i$ are the operators corresponding to $q_i$ and $p_i$ respectively. Erwin Schrödinger proposed in 1926 the following:Template:Sfn

• $\widehat{q_j}$ with multiplication by $x_j$.
• ${\hat{p}}_j$ with $-i\hslash {\partial }_{x_j}$.

With this identification, the canonical commutation relation holds.

Constructions

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

Representation

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 ${\partial }_{x_j}$.

In the matrix representation, similar to the matrix mechanics, $A_1$ is represented byTemplate:Sfn$$P=\left[\begin{array}{ccccc}0& 1& 0& 0& \cdots \\ 0& 0& 2& 0& \cdots \\ 0& 0& 0& 3& \cdots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right],Q=\left[\begin{array}{ccccc}0& 0& 0& 0& \dots \\ 1& 0& 0& 0& \cdots \\ 0& 1& 0& 0& \cdots \\ ⋮& ⋮& ⋮& ⋮& \ddots \end{array}\right]$$

Generator

$A_n$ 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

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

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

In other words, W(V) is the algebra generated by V subject only to the relation Template:Math. Then, W(V) is isomorphic to A_n via the choice of a Darboux basis for Template:Mvar.

$A_n$ 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).

Quantization

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)

$a_1\cdots a_n\mapsto \frac{1}{n!}\sum _{\sigma \in S_n}{a}_{\sigma (1)}\otimes \cdots \otimes a_{\sigma (n)}.$

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 $f\star g$, then the Weyl algebra is isomorphic to $(ℂ[x_1,\dots ,x_n],\star )$.Template:Sfn

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.Template:SfnTemplate:Sfn

The Weyl algebra is also referred to as the symplectic Clifford algebra.Template:SfnTemplate:SfnTemplate:Sfn Weyl algebras represent for symplectic bilinear forms the same structure that Clifford algebras represent for non-degenerate symmetric bilinear forms.Template:Sfn

D-module

The Weyl algebra can be constructed as a D-module.Template:Sfn 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 $D_{{𝔸}_R^n/R}$.Template:Sfn

More generally, let $X$ be a smooth scheme over a ring $R$. Locally, $X\to R$ factors as an étale cover over some ${𝔸}_R^n$ equipped with the standard projection. Because "étale" means "(flat and) possessing null cotangent sheaf", this means that every D-module over such a scheme can be thought of locally as a module over the $n^{\text{th}}$ 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 ${\mathrm{End}}_S(R)$:

• $D_R^0=R$
• $D_R^k=\{d\in {\mathrm{End}}_S(R):[d,a]\in D_R^{k-1}\text{ for all }a\in R\}.$

Let $D_R$ be the union of all $D_R^k$ for $k\ge 0$. This is a subalgebra of ${\mathrm{End}}_S(R)$.

In the case $R=S[x_1,...,x_n]$, the ring of differential operators of order $\le n$ presents similarly as in the special case $S=ℂ$ but for the added consideration of "divided power operators"; these are operators corresponding to those in the complex case which stabilize $\mathbb{Z}[x_1,...,x_n]$, but which cannot be written as integral combinations of higher-order operators, i.e. do not inhabit $D_{{𝔸}_{\mathbb{Z}}^n/\mathbb{Z}}$. One such example is the operator $\underset{x_1}{\overset{[p]}{\partial }}:x_1^N\mapsto (\genfrac{}{}{0ex}{}{N}{p})x_1^{N-p}$.

Explicitly, a presentation is given by

$D_{S[x_1,\dots ,x_{\ell }]/S}^n=S⟨x_1,\dots ,x_{\ell },\{{\partial }_{x_i},\underset{x_i}{\overset{[2]}{\partial }},\dots ,\underset{x_i}{\overset{[n]}{\partial }}{\}}_{1\le i\le \ell }⟩$

with the relations

$[x_i,x_j]=[\underset{x_i}{\overset{[k]}{\partial }},\underset{x_j}{\overset{[m]}{\partial }}]=0$

$[\underset{x_i}{\overset{[k]}{\partial }},x_j]=\{\begin{array}{cc}\underset{x_i}{\overset{[k-1]}{\partial }}& \text{if }i=j\\ 0& \text{if }i\ne j\end{array}$

$\underset{x_i}{\overset{[k]}{\partial }}\underset{x_i}{\overset{[m]}{\partial }}=(\genfrac{}{}{0ex}{}{k+m}{k})\underset{x_i}{\overset{[k+m]}{\partial }}\text{when }k+m\le n$

where $\underset{x_i}{\overset{[0]}{\partial }}=1$ by convention. The Weyl algebra then consists of the limit of these algebras as $n\to \mathrm{\infty }$.Template:Pg

When $S$ is a field of characteristic 0, then $D_R^1$ 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 $D_R^1$. In particular, if $S=ℂ$ and $R=ℂ[x_1,...,x_n]$, then $D_R^1=R+\sum _{i}R{\partial }_{x_i}$. As mentioned, $A_n=D_R$.Template:Sfn

Properties of A_n

Many properties of $A_1$ apply to $A_n$ with essentially similar proofs, since the different dimensions commute.

General Leibniz rule

Template:Math proofIn particular, $[q,q^m{p}^n]=-n{q}^m{p}^{n-1}$ and $[p,q^m{p}^n]=m{q}^{m-1}{p}^n$. Template:Math theorem

Degree

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 $A_n$.

Template:Math theorem That is, it has no two-sided nontrivial ideals and has no zero divisors. Template:Math proof

Derivation

Template:SeeTemplate:Math theorem That is, any derivation $D$ is equal to $[\cdot ,f]$ for some $f\in A_n$; any $f\in A_n$ yields a derivation $[\cdot ,f]$; if $f,f^{\prime }\in A_n$ satisfies $[\cdot ,f]=[\cdot ,f^{\prime }]$, then $f-f^{\prime }\in F$.

The proof is similar to computing the potential function for a conservative polynomial vector field on the plane.Template:Sfn

Template:Collapse top Since the commutator is a derivation in both of its entries, $[\cdot ,f]$ is a derivation for any $f\in A_n$. 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 $D:A_1\to A_1$ 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 $[\cdot ,r]$ are derivations, these two relations generate $[g,r]=D(g)$ for all $g\in A_1$.

Since $[p,q^m{p}^n]=m{q}^{m-1}{p}^n$, there exists an element $f=\sum _{m,n}{c}_{m,n}{q}^m{p}^n$ such that $$[p,f]=\sum _{m,n}m{c}_{m,n}{q}^m{p}^n=D(p)$$

$$\begin{array}{cc}0& \stackrel{[p,q]=1}{=}D([p,q])\\ & \stackrel{D\text{ is a derivation}}{=}[p,D(q)]+[D(p),q]\\ & \stackrel{[p,f]=D(p)}{=}[p,D(q)]+[[p,f],q]\\ & \stackrel{\text{Jacobi identity}}{=}[p,D(q)-[q,f]]\end{array}$$

Thus, $D(q)=g(p)+[q,f]$ for some polynomial $g$. Now, since $[q,q^m{p}^n]=-n{q}^m{p}^{n-1}$, 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 $r\in A_n$ 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 $x\in \{p_1,q_1\},y\in \{p_2,\dots ,p_n,q_2,\dots ,q_n\}$. Since $[x,D(y)-[y,r]]$ is a derivation in both $x$ and $y$, $[x,D(y)-[y,r]]=0$ for all $x\in ⟨p_1,q_1⟩$ and all $y\in ⟨p_2,\dots ,p_n,q_2,\dots ,q_n⟩$. Here, $⟨⟩$ means the subalgebra generated by the elements.

Thus, $\mathrm{\forall }y\in ⟨p_2,\dots ,p_n,q_2,\dots ,q_n⟩$, $$D(y)-[y,r]\in ⟨p_2,\dots ,p_n,q_2,\dots ,q_n⟩$$

Since $D-[\cdot ,r]$ is also a derivation, by induction, there exists $r^{\prime }\in ⟨p_2,\dots ,p_n,q_2,\dots ,q_n⟩$ such that $D(y)-[y,r]=[y,r^{\prime }]$ for all $y\in ⟨p_2,\dots ,p_n,q_2,\dots ,q_n⟩$.

Since $p_1,q_1$ commutes with $⟨p_2,\dots ,p_n,q_2,\dots ,q_n⟩$, we have $D(y)=[y,r+r^{\prime }]$ for all $y\in \{p_1,\dots ,p_n,q_1,\dots ,q_n\}$, and so for all of $A_n$. Template:Collapse bottom

Representation theory

Zero characteristic

In the case that the ground field Template:Mvar has characteristic zero, the nth Weyl algebra is a simple Noetherian domain.Template:Sfn 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 Template:Nowrap).

$\mathrm{t}\mathrm{r}([\sigma (q),\sigma (Y)])=\mathrm{t}\mathrm{r}(1).$

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 Template:Nowrap called the 'characteristic variety'Template:What whose size roughly corresponds to the sizeTemplate:What of M (a finite-dimensional module would have zero-dimensional characteristic variety). Then Bernstein's inequality states that for M non-zero,

$\dim (\mathrm{char}(M))\ge n$

An even stronger statement is Gabber's theorem, which states that Char(M) is a co-isotropic subvariety of Template:Nowrap for the natural symplectic form.

Positive characteristic

The situation is considerably different in the case of a Weyl algebra over a field of characteristic Template:Nowrap.

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.

Generalizations

The ideals and automorphisms of $A_1$ have been well-studied.Template:SfnTemplate:Sfn The moduli space for its right ideal is known.Template:Sfn However, the case for $A_n$ is considerably harder and is related to the Jacobian conjecture.Template:Sfn

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.

Affine varieties

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

$R=\frac{ℂ[x_1,\dots ,x_n]}{I}.$

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

$\text{Diff}(R)=\frac{\{D\in A_n:D(I)\subseteq I\}}{I\cdot A_n}.$

See also

• Jacobian conjecture
• Dixmier conjecture

References

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