Weyl algebra
Encyclopedia
In abstract algebra
, the Weyl algebra is the ring
of differential operator
s with polynomial
coefficients (in one variable),
More precisely, let F be a field
, and let F[X] be the ring of polynomials
in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X is the derivative
with respect to X. The algebra is generated by X and ∂X.
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
.
The Weyl algebra is a quotient
of the free algebra
on two generators, X and Y, by the ideal
generated by the single relation
The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The n-th Weyl algebra, An, is the ring of differential operators with polynomial coefficients in n variables. It is generated by Xi and
.
Weyl algebras are named after Hermann Weyl
, who introduced them to study the Heisenberg
uncertainty principle
in quantum mechanics
. It is a quotient
of the universal enveloping algebra
of the Heisenberg algebra, the Lie algebra
of the Heisenberg group, by setting the element 1 of
the Lie algebra equal to the unit 1 of the universal enveloping algebra.
The Weyl algebra is also referred to as the symplectic Clifford algebra.
Start with an
abstract vector space
V (of dimension 2n) equipped with a symplectic form
.
Define the Weyl algebra W(V) to be
where
is the tensor algebra
on V, and the notation
means "the ideal
generated by". In other words,
is the algebra generated by V subject
only to the relation
. Then, W(V) is isomorphic to 
via the choice of a Darboux basis for 
.
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
, 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 Xi and
(as is frequently done in quantum mechanics
).
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.
In the case of exterior algebra
s, the analogous quantization to the Weyl one is the Clifford algebra
, which is also referred to as the orthogonal Clifford algebra.
Noetherian
domain
. 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
and 
for some finite dimensional representation 
(where [X,Y]=1).
Since the trace of a commutator is zero, and the trace of the identity is the dimension of the matrix, the representation must be zero dimensional.
In fact, there are stronger statements than the absence of finite-dimensional representations. To any f.g. A_n-module M, there is a corresponding subvariety Char(M) of
called the 'characteristic variety' whose size roughly corresponds to the size 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
for the natural symplectic form.
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.
(and an extension using the Fourier transform
to integrable ("most")
functions, not just polynomial functions), see Weyl quantization
.
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
.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, the Weyl algebra is the ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
of differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
s with polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
coefficients (in one variable),
More precisely, let F be a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, and let F[X] be the ring of polynomials
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X is the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
with respect to X. The algebra is generated by X and ∂X.
The Weyl algebra is an example of a simple ring
Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
that is not a matrix ring
Matrix ring
In abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings...
over a division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
. It is also a noncommutative example of a domain
Domain (ring theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
, and an example of an Ore extension
Ore extension
In mathematics, especially in the area of algebra known as ring theory, an Ore extension, named after Oystein Ore, is a special type of a ring extension whose properties are relatively well understood...
.
The Weyl algebra is a quotient
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
of the free algebra
Free algebra
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring .-Definition:...
on two generators, X and Y, by the ideal
Ideal
-In philosophy:* Ideal , values that one actively pursues as goals* Platonic ideal, a philosophical idea of trueness of form, associated with Plato-In mathematics:* Ideal , special subsets of a ring considered in abstract algebra...
generated by the single relation
The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The n-th Weyl algebra, An, is the ring of differential operators with polynomial coefficients in n variables. It is generated by Xi and
.Weyl algebras are named after Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
, who introduced them to study the Heisenberg
Werner Heisenberg
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...
uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
. It is a quotient
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
of the universal enveloping algebra
Universal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
of the Heisenberg algebra, the Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
of the Heisenberg group, by setting the element 1 of
the Lie algebra equal to the unit 1 of the universal enveloping algebra.
The Weyl algebra is also referred to as the symplectic Clifford algebra.
Generators and relations
One may give an abstract construction of the algebras An in terms of generators and relations.Start with an
abstract vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V (of dimension 2n) equipped with a symplectic form
.Define the Weyl algebra W(V) to be
where
is the tensor algebraTensor algebra
In mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
on V, and the notation
means "the idealIdeal
-In philosophy:* Ideal , values that one actively pursues as goals* Platonic ideal, a philosophical idea of trueness of form, associated with Plato-In mathematics:* Ideal , special subsets of a ring considered in abstract algebra...
generated by". In other words,
is the algebra generated by V subjectonly to the relation
. Then, W(V) is isomorphic to via the choice of a Darboux basis for .Quantization
The algebra W(V)is a quantization of the symmetric algebra
Symmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
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
Symmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
Sym(V) equipped with a deformed product — called the Groenewold–Moyal product
Moyal product
In mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℝ2n, equipped with its Poisson bracket .This particular star product is also sometimes called...
(considering the symmetric
algebra to be polynomial functions on
, where the variables span the vector space V, and replacing inthe 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 Xi and (as is frequently done in quantum mechanicsQuantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
).
Thus, the Weyl algebra is a quantization
Quantization (physics)
In physics, quantization is the process of explaining a classical understanding of physical phenomena in terms of a newer understanding known as "quantum mechanics". It is a procedure for constructing a quantum field theory starting from a classical field theory. This is a generalization of the...
of the symmetric algebra, which is essentially the same as the Moyal quantization
Moyal product
In mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ∗, on the functions on ℝ2n, equipped with its Poisson bracket .This particular star product is also sometimes called...
(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.
In the case of exterior algebra
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
s, the analogous quantization to the Weyl one is the Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
, which is also referred to as the orthogonal Clifford algebra.
Properties of the Weyl Algebra
In the case that the ground field F has characteristic zero, the nth Weyl algebra is a simpleSimple ring
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
Noetherian
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
domain
Domain (ring theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
. It has global dimension
Global dimension
In ring theory and homological algebra, the global dimension of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules...
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
and for some finite dimensional representation (where [X,Y]=1).Since the trace of a commutator is zero, and the trace of the identity is the dimension of the matrix, the representation must be zero dimensional.
In fact, there are stronger statements than the absence of finite-dimensional representations. To any f.g. A_n-module M, there is a corresponding subvariety Char(M) of
called the 'characteristic variety' whose size roughly corresponds to the size 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
for the natural symplectic form.Positive Characteristic
The situation is considerably different in the case of a Weyl algebra over a field of characteristicCharacteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
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
Azumaya algebra
In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring...
over its center. As a consequence, there are many finite-dimensional representations which are all built out of simple representations of dimension p.
Generalizations
For more details about this quantization in the case (and an extension using the Fourier transformFourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
to integrable ("most")
functions, not just polynomial functions), see Weyl quantization
Weyl quantization
In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a "quantum mechanical" Hermitian operator with a "classical" kernel function in phase space invertibly...
.
Weyl algebras and Clifford algebras admit a further structure of a *-algebra, and can be unified as even and odd terms of a superalgebra
Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading....
, as discussed in CCR and CAR algebras
CCR and CAR algebras
In Mathematics and Physics the CCR and CAR algebras arise from the study of canonical commutation relations in Bosonic and Fermionic quantum mechanics...
.