Weyl Algebra

Oh, you want to delve into the Weyl algebra, do you? Fine. Don’t expect a warm welcome. This isn’t a tea party. It’s an exploration of structures that are, frankly, more interesting than most conversations I’m forced to endure. Just try not to break anything.

Differential algebra

In the realm of abstract algebra , the Weyl algebras emerge as abstract echoes of differential operators that have been coaxed into behaving nicely, specifically those with polynomial coefficients. They bear the name of Hermann Weyl , a man who apparently found the intricacies of the Heisenberg uncertainty principle in quantum mechanics sufficiently compelling to warrant such an abstraction. A quaint notion, if you ask me.

Consider the most rudimentary form. Let there be a field , which we’ll call $F$, and the ring of polynomials in a single variable, $F[x]$. The corresponding Weyl algebra, the first of its kind, $A_1$, is composed of differential operators of this particular flavor:

$$f_{m}(x)\partial {x}^{m}+f{m-1}(x)\partial {x}^{m-1}+\cdots +f{1}(x)\partial {x}+f{0}(x)$$

where each $f_{i}(x)$ is, as previously stated, an element of $F[x]$. It’s a rather straightforward construction, really. The $n$-th Weyl algebra, denoted $A_n$, follows a similar, albeit scaled-up, blueprint.

Alternatively, $A_1$ can be conjured into existence by taking the free algebra on two generators, let’s call them $q$ and $p$, and then crushing it down by introducing the ideal generated by the relation $([p,q]-1)$. This $[p,q]$ business is, of course, the commutator , a fundamental source of discord in these algebraic worlds. For $A_n$, we simply expand this to the free algebra on $2n$ generators and impose the relations $([p_i,q_j]-\delta_{i,j})$ for all $i,j$ from 1 to $n$, where $\delta_{i,j}$ is the ever-present Kronecker delta . It’s a way of forcing a specific kind of non-commutativity.

More generally, if you’re presented with a partial differential ring , $(R, \Delta)$, where $\Delta = {\partial_1, \ldots, \partial_m}$ is a set of commuting derivatives, the associated Weyl algebra is the noncommutative ring $R[\partial_1, \ldots, \partial_m]$. This ring adheres to the rule $\partial_i r = r\partial_i + \partial_i(r)$ for any $r \in R$. The initial case we discussed is a specific instance where $R = F[x_1, \ldots, x_n]$ and $\Delta = {\partial_{x_1}, \ldots, \partial_{x_n}}$, with $F$ being a field, naturally.

This article, for the most part, will confine itself to the $A_n$ constructed over a field $F$ of characteristic zero , unless some peculiar circumstance dictates otherwise. It’s worth noting that the Weyl algebra stands as a peculiar example of a simple ring that stubbornly refuses to be a matrix ring over a division ring . It’s also a non-commutative specimen of a domain and a prime illustration of an Ore extension .

Motivation

The genesis of the Weyl algebra is intimately tied to the abstract machinations of quantum mechanics and the rather ambitious process of canonical quantization . Imagine a classical phase space , a rather abstract landscape defined by canonical coordinates $(q_1, p_1, \ldots, q_n, p_n)$. These coordinates are bound by the Poisson bracket relations:

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

Now, in the grand scheme of canonical quantization, the objective is to construct a Hilbert space of states and then map these classical observables—essentially functions on this phase space—to self-adjoint operators inhabiting this Hilbert space. The crucial step is to impose the canonical commutation relations:

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

where $[\cdot, \cdot]$ denotes the commutator and $\hat{q}_i, \hat{p}_i$ are the operators that represent the classical $q_i$ and $p_i$. It was Erwin Schrödinger who, back in 1926, proposed a rather elegant identification:

• $\hat{q}_j$ is to be represented by simple multiplication by $x_j$.
• $\hat{p}j$ is to be represented by the derivative $-i\hbar \partial{x_j}$.

With this mapping, the canonical commutation relations, the very bedrock of quantum mechanics, are satisfied. It’s a rather neat trick, turning abstract concepts into tangible operators.

Constructions

The Weyl algebras, in their essence, can be constructed through various means, each offering a different vantage point, from the concrete to the profoundly abstract.

Representation

The Weyl algebra $A_n$ can be materialized as a concrete representation . In the differential operator representation, much like Schrödinger’s approach to canonical quantization, we assign:

• $q_j$ is mapped to multiplication by $x_j$.
• $p_j$ is mapped to differentiation by $\partial_{x_j}$.

Then, the commutation relation $[q_i, p_j] = \delta_{ij}$ translates directly to $[\hat{x}_i, \hat{p}j] = \delta{ij}$ in this operational setting.

In the realm of matrix representations, reminiscent of matrix mechanics , $A_1$ can be embodied by the following infinite matrices:

$$P = \begin{bmatrix} 0 & 1 & 0 & 0 & \cdots \ 0 & 0 & 2 & 0 & \cdots \ 0 & 0 & 0 & 3 & \cdots \ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix}, \quad Q = \begin{bmatrix} 0 & 0 & 0 & 0 & \cdots \ 1 & 0 & 0 & 0 & \cdots \ 0 & 1 & 0 & 0 & \cdots \ \vdots & \vdots & \vdots & \vdots & \ddots \end{bmatrix}$$

These matrices, $P$ and $Q$, when subjected to the commutator $[P, Q]$, yield the identity matrix, thereby satisfying the fundamental relation of $A_1$.

Generator

$A_n$ can also be constructed as a quotient of a free algebra, defined by its generators and the relations that bind them. One method involves starting with an abstract vector space $V$ of dimension $2n$, endowed with a symplectic form $\omega$. The Weyl algebra $W(V)$ is then defined as:

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

Here, $T(V)$ signifies the tensor algebra on $V$, and the notation $(( \cdot ))$ denotes the ideal generated by the enclosed elements. In simpler terms, $W(V)$ is the algebra generated by $V$ with the sole constraint that $vu - uv = \omega(v, u)$. When $V$ is equipped with a Darboux basis for $\omega$, $W(V)$ becomes isomorphic to $A_n$.

Furthermore, $A_n$ can be viewed as a quotient of the universal enveloping algebra of the Heisenberg algebra . The Heisenberg algebra, itself the Lie algebra of the Heisenberg group , has a central element, $[q, p]$. By setting this central element equal to 1, the unit element in the universal enveloping algebra, we arrive at $A_n$.

Quantization

The algebra $W(V)$, as defined earlier, is a quantization of the symmetric algebra $\text{Sym}(V)$. When $V$ is over a field of characteristic zero, $W(V)$ bears a natural isomorphism to $\text{Sym}(V)$ itself, but with a subtly altered product – the Groenewold–Moyal product . This perspective views the symmetric algebra as polynomial functions on $V^*$, where the variables span $V$, and the Moyal product is obtained by replacing $i\hbar$ with 1.

The isomorphism is realized through a symmetrization map from $\text{Sym}(V)$ 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 retain the $i\hbar$ and work over the complex numbers, the Weyl algebra could be defined from the outset using generators $q_i$ and $i\hbar\partial_{q_i}$, aligning more directly with the conventions of quantum mechanics .

Thus, the Weyl algebra serves as a quantization of the symmetric algebra. It’s closely related to Moyal quantization , particularly when restricted to polynomial functions. The key distinction lies in their formulation: the Weyl algebra is presented via generators and relations (as differential operators), while Moyal quantization is defined through a deformed multiplication.

To put it another way, if we denote the Moyal star product by $f \star g$, the Weyl algebra is isomorphic to $(\mathbb{C}[x_1, \dots, x_n], \star)$.

For comparison, the analogous quantization to the Weyl algebra in the context of exterior algebras is the Clifford algebra , sometimes referred to as the orthogonal Clifford algebra. The Weyl algebra itself is also known as the symplectic Clifford algebra. It essentially mirrors the role that Clifford algebras play for non-degenerate symmetric bilinear forms, but for symplectic bilinear forms .

D-module

The Weyl algebra can also be understood as a D-module . Specifically, the Weyl algebra associated with the polynomial ring $R[x_1, \dots, x_n]$, equipped with its standard differential structure, precisely corresponds to Grothendieck’s ring of differential operations, $D_{\mathbb{A}^n_R/R}$.

More broadly, consider a smooth scheme $X$ over a ring $R$. Locally, the map $X \to R$ can be seen as an étale cover over some $\mathbb{A}^n_R$, with the standard projection. Since “étale” implies flatness and a null cotangent sheaf, this means that any D-module over such a scheme can be locally viewed 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 $D_{R/S}$ (or simply $D_R$ when $S$ is implicit) is inductively defined as a graded subalgebra of $\text{End}_S(R)$:

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

The entire ring $D_R$ is the union of all $D_R^k$ for $k \geq 0$, forming a subalgebra of $\text{End}_S(R)$.

In the specific case where $R = S[x_1, \dots, x_n]$, the ring of differential operators of order $\leq n$ presents similarly to the characteristic zero case, but with the addition of “divided power operators.” These are operators that correspond to those in the complex case but cannot be expressed as integral combinations of higher-order operators. An example is the operator $\partial_{x_1}^{[p]}$, which maps $x_1^N$ to $\binom{N}{p}x_1^{N-p}$.

Explicitly, a presentation is given by:

$$D_{S[x_1, \dots, x_\ell]/S}^n = S\langle x_1, \dots, x_\ell, {\partial_{x_i}, \partial_{x_i}^{[2]}, \dots, \partial_{x_i}^{[n]}}_{1 \leq i \leq \ell}\rangle$$

subject to the relations:

$$[x_i, x_j] = [\partial_{x_i}^{[k]}, \partial_{x_j}^{[m]}] = 0$$ $$[\partial_{x_i}^{[k]}, x_j] = \begin{cases} \partial_{x_i}^{[k-1]} & \text{if } i=j \ 0 & \text{if } i \neq j \end{cases}$$ $$\partial_{x_i}^{[k]}\partial_{x_i}^{[m]} = \binom{k+m}{k}\partial_{x_i}^{[k+m]} \quad \text{when } k+m \leq n$$

where $\partial_{x_i}^{[0]} = 1$ by convention. The Weyl algebra then emerges as the limit of these algebras as $n \to \infty$.

When $S$ is a field of characteristic 0, $D_R^1$ is generated as an $R$-module by 1 and the $S$-derivations of $R$. Furthermore, $D_R$ is generated as a ring by $D_R^1$. Specifically, if $S=\mathbb{C}$ and $R=\mathbb{C}[x_1, \dots, x_n]$, then $D_R^1 = R + \sum_i R\partial_{x_i}$. As previously noted, $A_n = D_R$.

Properties of $A_n$

Many of the properties observed in $A_1$ extend to $A_n$ with minimal fuss, as the higher dimensions commute in a rather predictable fashion.

General Leibniz rule

• Theorem (general Leibniz rule): $$p^k q^m = \sum_{l=0}^{k} \binom{k}{l} \frac{m!}{(m-l)!} q^{m-l} p^{k-l} = q^m p^k + mk q^{m-1} p^{k-1} + \cdots$$

  Proof: When viewed through the lens of the $p \mapsto \partial_x, q \mapsto x$ representation, this identity arises directly from the general Leibniz rule. Since this rule can be proven through sheer algebraic manipulation, it holds sway over $A_1$ as well.

  Specifically, we find that: $$[q, q^m p^n] = -nq^m p^{n-1}$$ $$[p, q^m p^n] = mq^{m-1} p^n$$

• Corollary: The center of the Weyl algebra $A_n$ is simply its underlying field of scalars, $F$.

  Proof: If the commutator of an element $f$ with either $p$ or $q$ is zero, then, based on the preceding statement, $f$ cannot contain any monomial of the form $p^n q^m$ where $n > 0$ or $m > 0$. This leaves only scalar multiples of the identity, thus placing $f$ within the center $F$.

Degree

• Theorem: $A_n$ possesses a basis given by ${q^m p^n : m, n \geq 0}$.

  Proof: Through repeated application of the commutator relations, any monomial can be expressed as a linear combination of these basis elements. The challenge then becomes demonstrating their linear independence. This can be verified in the differential operator representation. For any linear sum $\sum_{m,n} c_{m,n} x^m \partial_x^n$ with non-zero coefficients, we can group it by descending order:

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

  where $p_M(x)$ is a non-zero polynomial. Applying this operator to $x^M$ yields $M! p_M(x)$, which is demonstrably non-zero.

  This basis structure allows $A_1$ to be a graded algebra , where the degree of $\sum_{m,n} c_{m,n} q^m p^n$ is determined by the maximum value of $m+n$ among its non-zero monomials. A similar definition of degree applies to $A_n$.

• Theorem: For $A_n$:

  • $\deg(g+h) \leq \max(\deg(g), \deg(h))$
  • $\deg([g,h]) \leq \deg(g) + \deg(h) - 2$
  • $\deg(gh) = \deg(g) + \deg(h)$

  Proof: We’ll prove this for $A_1$, as the $A_n$ case follows a similar logic. The first relation is inherent in the definition of degree. The second relation stems directly from the general Leibniz rule. For the third relation, we know that $\deg(gh) \leq \deg(g) + \deg(h)$. Thus, it suffices to show that $gh$ contains at least one non-zero monomial with the degree $\deg(g) + \deg(h)$. To find such a monomial, we select the term with the highest degree in $g$. If there are multiple such terms, we choose the one with the highest power of $q$. We do the same for $h$. The product of these two selected monomials yields a unique monomial within $gh$, ensuring it remains non-zero.

• Theorem: $A_n$ is a simple domain . That is, it possesses no non-trivial two-sided ideals and no zero divisors .

  Proof: The property $\deg(gh) = \deg(g) + \deg(h)$ directly implies that there are no zero divisors.

  Now, suppose, for the sake of contradiction, that $I$ is a non-zero two-sided ideal of $A_1$, with $I \neq A_1$. Let’s pick a non-zero element $f \in I$ that has the minimum possible degree.

  If $f$ contains a non-zero monomial of the form $x \cdot x^m \partial^n = x^{m+1} \partial^n$, then the commutator $[\partial, f] = \partial f - f\partial$ will contain a non-zero monomial of the form $\partial x^{m+1} \partial^n - x^{m+1} \partial^n \partial = (m+1)x^m \partial^n$. Thus, $[\partial, f]$ is non-zero and has a degree strictly less than $\deg(f)$. Since $I$ is a two-sided ideal, $[\partial, f] \in I$, which contradicts the assumption that $f$ had the minimal degree.

  Similarly, if $f$ contains a non-zero monomial of the form $x^m \partial^n \partial$, then the commutator $[x, f] = xf - fx$ will be non-zero and have a lower degree, again leading to a contradiction. These arguments, when applied across all possible monomials, demonstrate that no such non-zero ideal $I$ can exist.

Derivation

• Further information: Derivation (differential algebra)

• Theorem: The derivations of $A_n$ are in a one-to-one correspondence with the elements of $A_n$, up to an additive scalar. In other words, any derivation $D$ can be expressed as $[ \cdot, f ]$ for some $f \in A_n$. Conversely, any $f \in A_n$ yields a derivation $[ \cdot, f ]$. If $f, f’ \in A_n$ yield the same derivation, i.e., $[ \cdot, f ] = [ \cdot, f’ ]$, then $f - f’$ must be an element of the scalar field $F$.

  The proof for this theorem mirrors the process of finding a potential function for a conservative polynomial vector field in the plane.

  Proof: Since the commutator is a derivation in both of its arguments, $[ \cdot, f ]$ is indeed a derivation for any $f \in A_n$. The uniqueness up to an additive scalar is guaranteed by the fact that the center of $A_n$ is precisely the field of scalars $F$.

  The core of the proof involves demonstrating that any derivation is an inner derivation, achieved through induction on $n$.

  Base case (n=1): Let $D: A_1 \to A_1$ be a linear map that is a derivation. We aim to construct an element $r$ such that $[p, r] = D(p)$ and $[q, r] = D(q)$. Since both $D$ and $[ \cdot, r ]$ are derivations, these relations imply that $[g, r] = D(g)$ for all $g \in A_1$. Given that $[p, q^m p^n] = mq^{m-1}p^n$, there exists an element $f = \sum_{m,n} c_{m,n} q^m p^n$ such that $[p, f] = \sum_{m,n} mc_{m,n} q^m p^n = D(p)$. The Jacobi identity, combined with the derivation property of $D$, leads to: $$0 = D([p, q]) = [p, D(q)] + [D(p), q] = [p, D(q)] + [[p, f], q]$$ Rearranging, we get: $$[p, D(q) - [q, f]] = 0$$ This implies that $D(q) - [q, f]$ must commute with $p$. Since $[p, q^m p^n] = -nq^{m-1}p^n$, we can find a polynomial $h(p)$ such that $[q, h(p)] = D(q) - [q, f]$. Crucially, $[p, h(p)] = 0$. Thus, $r = f + h(p)$ serves as the desired element.

  Inductive step: For the inductive step, we similarly establish the existence of an element $r \in A_n$ such that $[q_1, r] = D(q_1)$ and $[p_1, r] = D(p_1)$. The commutation relations imply that $[x, D(y) - [y, r]] = 0$ for all $x \in {p_1, q_1}$ and $y \in {p_2, \dots, p_n, q_2, \dots, q_n}$. Since $[x, D(y) - [y, r]]$ is a derivation in both $x$ and $y$, it follows that $[x, D(y) - [y, r]] = 0$ for all $x \in \langle p_1, q_1 \rangle$ and all $y \in \langle p_2, \dots, p_n, q_2, \dots, q_n \rangle$. Here, $\langle \cdot \rangle$ denotes the subalgebra generated by the elements. Therefore, for any $y \in \langle p_2, \dots, p_n, q_2, \dots, q_n \rangle$, we have $D(y) - [y, r] \in \langle p_2, \dots, p_n, q_2, \dots, q_n \rangle$. Since $D - [ \cdot, r ]$ is also a derivation, by induction, there exists an $r’ \in \langle p_2, \dots, p_n, q_2, \dots, q_n \rangle$ such that $D(y) - [y, r] = [y, r’]$ for all $y \in \langle p_2, \dots, p_n, q_2, \dots, q_n \rangle$. As $p_1, q_1$ commute with $\langle p_2, \dots, p_n, q_2, \dots, q_n \rangle$, we find that $D(y) = [y, r + r’]$ for all $y \in {p_1, \dots, p_n, q_1, \dots, q_n}$, and consequently, for all of $A_n$.

Representation theory

• Further information: Stone–von Neumann theorem

Zero characteristic

When the ground field $F$ is of characteristic zero , the $n^{\text{th}}$ Weyl algebra $A_n$ is a simple , Noetherian domain . Its global dimension is $n$, a notable contrast to its deformed counterpart, $\text{Sym}(V)$, which has a global dimension of $2n$.

$A_n$ possesses no finite-dimensional representations. While this is a consequence of its simplicity, it can be more directly demonstrated by considering the trace of $\sigma(q)$ and $\sigma(p)$ for a hypothetical finite-dimensional representation $\sigma$ (where $[q, p] = 1$).

$$\text{tr}([\sigma(q), \sigma(p)]) = \text{tr}(1)$$

Since the trace of a commutator is always zero, and the trace of the identity matrix is the dimension of the representation, this equation implies that the dimension of the representation must be zero.

In fact, the absence of finite-dimensional representations is just the tip of the iceberg. For any finitely generated $A_n$-module $M$, there exists a corresponding subvariety $\text{Char}(M)$ of $V \times V^*$—termed the ‘characteristic variety’—whose “size” is roughly correlated with the “size” of $M$. A finite-dimensional module would, by definition, have a characteristic variety of zero dimension. Bernstein’s inequality states that for a non-zero module $M$,

$$\dim(\text{char}(M)) \geq n$$

An even more profound result is Gabber’s theorem, which asserts that $\text{Char}(M)$ is a co-isotropic subvariety of $V \times V^*$ with respect to the natural symplectic form.

Positive characteristic

The landscape shifts dramatically when the Weyl algebra is defined over a field of characteristic $p > 0$.

In this scenario, for any element $D$ in the Weyl algebra, the element $D^p$ becomes central. Consequently, the Weyl algebra exhibits a remarkably large center. It functions as a finitely generated module over its center and, even more strikingly, as an Azumaya algebra over its center. This leads to a plethora of finite-dimensional representations, all constructed from simple representations of dimension $p$.

Generalizations

The ideals and automorphisms of $A_1$ have been meticulously studied, and the moduli space for its right ideals is well-understood. However, the case for $A_n$ presents a far greater challenge and is intimately connected to the notoriously difficult Jacobian conjecture .

For a more in-depth exploration of this quantization, particularly for $n=1$ and its extension to a class of integrable functions beyond polynomials using the Fourier transform , consult the Wigner–Weyl transform .

Weyl algebras and Clifford algebras can be further endowed with the structure of a *-algebra . They can be unified within the framework of a superalgebra , as discussed in CCR and CAR algebras .

Affine varieties

Weyl algebras also find a natural generalization in the context of algebraic varieties. Consider a polynomial ring:

$$R = \frac{\mathbb{C}[x_1, \dots, x_n]}{I}$$

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

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

This construction allows us to study differential operators acting on functions defined on more complex geometric objects than simple affine spaces.