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 be a field, and let be the ring of polynomials in one variable with coefficients in . Then the corresponding Weyl algebra consists of differential operators of form

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

Alternatively, can be constructed as the quotient of the free algebra on two generators, q and p, by the ideal generated by . Similarly, is obtained by quotienting the free algebra on 2n generators by the ideal generated bywhere is the Kronecker delta.

More generally, let be a partial differential ring with commuting derivatives . The Weyl algebra associated to is the noncommutative ring satisfying the relations for all . The previous case is the special case where and where is a field.

This article discusses only the case of with underlying field 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

edit

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  . These coordinates satisfy the Poisson bracket relations: 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: where   denotes the commutator. Here,   and   are the operators corresponding to   and   respectively. Erwin Schrödinger proposed in 1926 the following:[1]

  •   with multiplication by  .
  •   with  .

With this identification, the canonical commutation relation holds.

Constructions

edit

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

Representation

edit

The Weyl algebra   can be concretely constructed as a representation.

In the differential operator representation, similar to Schrödinger's canonical quantization, let   be represented by multiplication on the left by  , and let   be represented by differentiation on the left by  .

In the matrix representation, similar to the matrix mechanics,   is represented by[2] 

Generator

edit

  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

 

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 vuuv = ω(v, u). Then, W(V) is isomorphic to An via the choice of a Darboux basis for ω.

  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

edit

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 in the Moyal product formula with 1).

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

 

If one prefers to have the and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by qi 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  , then the Weyl algebra is isomorphic to  .[3]

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.[4][5]

The Weyl algebra is also referred to as the symplectic Clifford algebra.[4][5][6] Weyl algebras represent for symplectic bilinear forms the same structure that Clifford algebras represent for non-degenerate symmetric bilinear forms.[6]

D-module

edit

The Weyl algebra can be constructed as a D-module.[7] Specifically, the Weyl algebra corresponding to the polynomial ring   with its usual partial differential structure is precisely equal to Grothendieck's ring of differential operations  .[7]

More generally, let   be a smooth scheme over a ring  . Locally,   factors as an étale cover over some   equipped with the standard projection.[8] Because "étale" means "(flat and) possessing null cotangent sheaf",[9] this means that every D-module over such a scheme can be thought of locally as a module over the   Weyl algebra.

Let   be a commutative algebra over a subring  . The ring of differential operators   (notated   when   is clear from context) is inductively defined as a graded subalgebra of  :

  •  
  •  

Let   be the union of all   for  . This is a subalgebra of  .

In the case  , the ring of differential operators of order   presents similarly as in the special case   but for the added consideration of "divided power operators"; these are operators corresponding to those in the complex case which stabilize  , but which cannot be written as integral combinations of higher-order operators, i.e. do not inhabit  . One such example is the operator  .

Explicitly, a presentation is given by

 

with the relations

 
 
 

where   by convention. The Weyl algebra then consists of the limit of these algebras as  .[10]: Ch. IV.16.II 

When   is a field of characteristic 0, then   is generated, as an  -module, by 1 and the  -derivations of  . Moreover,   is generated as a ring by the  -subalgebra  . In particular, if   and  , then  . As mentioned,  .[11]

Properties of An

edit

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

General Leibniz rule

edit

Theorem (general Leibniz rule) —  

Proof

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

In particular,   and  .

Corollary — The center of Weyl algebra   is the underlying field of constants  .

Proof

If the commutator of   with either of   is zero, then by the previous statement,   has no monomial   with   or  .

Degree

edit

Theorem —   has a basis  .[12]

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   with nonzero coefficients, group it in descending order:  , where   is a nonzero polynomial. This operator applied to   results in  .

This allows   to be a graded algebra, where the degree of   is   among its nonzero monomials. The degree is similarly defined for  .

Theorem — For  :[13]

  •  
  •  
  •  
Proof

We prove it for  , as the   case is similar.

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

Theorem —   is a simple domain.[14]

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

Proof

Because  , it has no zero divisors.

Suppose for contradiction that   is a nonzero two-sided ideal of  , with  . Pick a nonzero element   with the lowest degree.

If   contains some nonzero monomial of form  , then   contains a nonzero monomial of form   Thus   is nonzero, and has degree  . As   is a two-sided ideal, we have  , which contradicts the minimality of  .

Similarly, if   contains some nonzero monomial of form  , then   is nonzero with lower degree.

Derivation

edit

Theorem — The derivations of   are in bijection with the elements of   up to an additive scalar.[15]

That is, any derivation   is equal to   for some  ; any   yields a derivation  ; if   satisfies  , then  .

The proof is similar to computing the potential function for a conservative polynomial vector field on the plane.[16]

Proof

Since the commutator is a derivation in both of its entries,   is a derivation for any  . Uniqueness up to additive scalar is because the center of   is the ring of scalars.

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

Base case: Let   be a linear map that is a derivation. We construct an element   such that  . Since both   and   are derivations, these two relations generate   for all  .

Since  , there exists an element   such that  

 

Thus,   for some polynomial  . Now, since  , there exists some polynomial   such that  . Since  ,   is the desired element.

For the induction step, similarly to the above calculation, there exists some element   such that  .

Similar to the above calculation,   for all  . Since   is a derivation in both   and  ,   for all   and all  . Here,   means the subalgebra generated by the elements.

Thus,  ,  

Since   is also a derivation, by induction, there exists   such that   for all  .

Since   commutes with  , we have   for all  , and so for all of  .

Representation theory

edit

Zero characteristic

edit

In the case that the ground field F has characteristic zero, the nth Weyl algebra is a simple Noetherian domain.[17] 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 [q,p] = 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 An-module M, there is a corresponding subvariety Char(M) of V × V called the 'characteristic variety'[clarification needed] whose size roughly corresponds to the size[clarification needed] of M (a finite-dimensional module would have zero-dimensional characteristic variety). Then Bernstein's inequality states that for M non-zero,

 

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

Positive characteristic

edit

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

In this case, for any element D of the Weyl algebra, the element Dp 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

edit

The ideals and automorphisms of   have been well-studied.[18][19] The moduli space for its right ideal is known.[20] However, the case for   is considerably harder and is related to the Jacobian conjecture.[21]

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

edit

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

 

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

 

See also

edit

Notes

edit
  1. ^ Landsman 2007, p. 428.
  2. ^ Coutinho 1997, pp. 598–599.
  3. ^ Coutinho 1997, pp. 602–603.
  4. ^ a b Lounesto & Ablamowicz 2004, p. xvi.
  5. ^ a b Micali, Boudet & Helmstetter 1992, pp. 83–96.
  6. ^ a b Helmstetter & Micali 2008, p. xii.
  7. ^ a b Coutinho 1997, pp. 600–601.
  8. ^ "Section 41.13 (039P): Étale and smooth morphisms—The Stacks project". stacks.math.columbia.edu. Retrieved 2024-09-29.
  9. ^ "etale morphism of schemes in nLab". ncatlab.org. Retrieved 2024-09-29.
  10. ^ Grothendieck, Alexander (1964). "Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20: 5–259. ISSN 1618-1913.
  11. ^ Coutinho 1995, pp. 20–24.
  12. ^ Coutinho 1995, p. 9, Proposition 2.1.
  13. ^ Coutinho 1995, pp. 14–15.
  14. ^ Coutinho 1995, p. 16.
  15. ^ Dirac 1926, pp. 415–417.
  16. ^ Coutinho 1997, p. 597.
  17. ^ Coutinho 1995, p. 70.
  18. ^ Berest & Wilson 2000, pp. 127–147.
  19. ^ Cannings & Holland 1994, pp. 116–141.
  20. ^ Lebruyn 1995, pp. 32–48.
  21. ^ Coutinho 1995, section 4.4.

References

edit