Weyl algebra

In mathematics, especially in the field of algebra, a **polynomial ring** or **polynomial algebra** is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.

Polynomial rings occur in many parts of mathematics, and the study of their properties was among the main motivations for the development of commutative algebra and ring theory. Polynomial rings and their ideals are fundamental in algebraic geometry. Many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, are generalizations of polynomial rings.

A closely related notion is that of the ring of polynomial functions on a vector space, and, more generally, ring of regular functions on an algebraic variety.

## Differential and skew-polynomial rings

Main articles: Ore extension
Other generalizations of polynomials are differential and skew-polynomial rings.

A **differential polynomial ring** is a ring of differential operators formed from a ring *R* and a derivation *δ* of *R* into *R*. This derivation operates on *R*, and will be denoted *X*, when viewed as an operator. The elements of *R* also operate on *R* by multiplication. The composition of operators is denoted as the usual multiplication. It follows that the relation *δ*(*ab*) = *aδ*(*b*) + *δ*(*a*)*b* may be rewritten
as

X
⋅
# a
a
⋅
X
δ
(
a
)
.

{\displaystyle X\cdot a = a\cdot X +\delta(a).}

This relation may be extended to define a skew multiplication between two polynomials in *X* with coefficients in *R*, which make them a non-commutative ring.

The standard example, called a Weyl algebra, takes *R* to be a (usual) polynomial ring *k*[*Y*], and *δ* to be the standard polynomial derivative

∂

∂
Y

{\displaystyle \tfrac{\partial}{\partial Y}}

. Taking *a* =*Y* in the above relation, one gets the canonical commutation relation, *X*·*Y* − *Y*·*X* = 1. Extending this relation by associativity and distributivity allows explicitly constructing the Weyl algebra..

The **skew-polynomial ring** is defined similarly for a ring *R* and a ring endomorphism *f* of *R*, by extending the multiplication from the relation *X*·*r* = *f*(*r*)·*X* to produce an associative multiplication that distributes over the standard addition. More generally, given a homomorphism *F* from the monoid **N** of the positive integers into the endomorphism ring of *R*, the formula *X**n*·*r* = *F*(*n*)(*r*)·*X**n* allows constructing a skew-polynomial ring. Skew polynomial rings are closely related to crossed product algebras.

### Ore extension

Ore extensions appear in several natural contexts, including skew and differential polynomial rings, group algebras of polycyclic groups, universal enveloping algebras of solvable Lie algebras, and coordinate rings of quantum groups.
Suppose that *R* is a (not necessarily commutative) ring,

σ
:
R
→
R

{\displaystyle \sigma \colon R \to R}

 is a ring homomorphism, and

δ
:
R
→
R

{\displaystyle \delta\colon R\to R}

 is a  ***σ*-derivation** of *R*, which means that

δ

{\displaystyle \delta}

 is a homomorphism of abelian groups satisfying

δ
(

r
1

r
2

# )
σ
(

r
1

)
δ
(

r
2

)
δ
(

r
1

)

r
2

{\displaystyle \delta(r\_1 r\_2)=\sigma(r\_1)\delta(r\_2)+\delta(r\_1)r\_2}

.
Then the **Ore extension**

R
[
x
;
σ
,
δ
]

{\displaystyle R[x;\sigma,\delta]}

, also called a **skew polynomial ring**, is the noncommutative ring obtained by giving the ring of polynomials

R
[
x
]

{\displaystyle R[x]}

 a new multiplication, subject to the identity

x
# r
σ
(
r
)
x
δ
(
r
)

{\displaystyle x r=\sigma(r)x + \delta(r)}

.
If *δ* = 0 (i.e., is the zero map) then the Ore extension is denoted *R*[*x*; *σ*]. If *σ* = 1 (i.e., the identity map) then the Ore extension is denoted *R*[*x*,*δ*] and is called a **differential polynomial ring**.

## Weyl algebra

In abstract algebra, the **Weyl algebra** is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form

f
m

(
X
)

∂
X
m

f

m
−
1

(
X
)

∂
X

m
−
1

⋯

f
1

(
X
)

∂
X

f
0

(
X
)
.

{\displaystyle f\_m(X) \partial\_X^m + f\_{m-1}(X) \partial\_X^{m-1} + \cdots + f\_1(X) \partial\_X + f\_0(X).}

More precisely, let *F* be the underlying 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 isomorphic to the quotient of the free algebra on two generators, *X* and *Y*, by the ideal generated by the element

Y
X
−
X
Y
−
1

.

{\displaystyle YX - XY - 1~.}

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 *∂Xi*, *i* = 1, ..., *n*.

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 central element of the Heisenberg algebra (namely [*X*,*Y*]) equal to the unit of the universal enveloping algebra (called 1 above).

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

### Symmetric algebra

Clifford algebras are closely related to exterior algebras. Indeed, if *Q* = 0 then the Clifford algebra Cℓ(*V*, *Q*) is just the exterior algebra ⋀(*V*). For nonzero *Q* there exists a canonical *linear* isomorphism between ⋀(*V*) and Cℓ(*V*, *Q*) whenever the ground field *K* does not have characteristic two. That is, they are naturally isomorphic as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally).  Clifford multiplication together with the distinguished subspace is strictly richer than the exterior product since it makes use of the extra information provided by *Q*.

The Clifford algebra is a filtered algebra, the associated graded algebra is the exterior algebra.

 More precisely, Clifford algebras may be thought of as *quantizations* (cf. Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra.

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 algebra, a quantum deformation of the symmetric algebra by a symplectic form
* Clifford algebra, a quantum deformation of the exterior algebra by a quadratic form

 From Wikipedia:q-analog
*q*-analogs also appear in the study of quantum groups and in *q*-deformed superalgebras. The connection here is similar, in that much of string theory is set in the language of Riemann surfaces, resulting in connections to elliptic curves, which in turn relate to *q*-series.
In mathematics, a ***q*-analog** of a theorem, identity or expression is a generalization involving a new parameter *q* that returns the original theorem, identity or expression in the limit as *q* → 1. Typically, mathematicians are interested in *q*-analogs that arise naturally, rather than in arbitrarily contriving *q*-analogs of known results. The earliest *q*-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.

## Generalizations of Pauli matrices

The Pauli matrices

σ
1

{\displaystyle \sigma\_1}

 and

σ
3

{\displaystyle \sigma\_3}

 satisfy the following:

σ
1
2

=

σ
3
2

=
I
,

σ
1

σ
3

=
−

σ
3

σ
1

=

e

π
i

σ
3

σ
1

.

{\displaystyle
\sigma \_1 ^2 = \sigma \_3 ^2 = I, \; \sigma \_1 \sigma \_3 = - \sigma \_3 \sigma \_1 = e^{\pi i} \sigma \_3 \sigma\_1.
}

The so-called Walsh–Hadamard conjugation matrix is

# W

1

2

[

1

1

1

−
1

]

.

{\displaystyle
W = \frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1 \\ 1 & -1
\end{bmatrix}.
}

Like the Pauli matrices, *W* is both Hermitian and unitary.

σ
1

,

σ
3

{\displaystyle \sigma \_1, \; \sigma \_3}

 and *W* satisfy the relation

σ
1

=
W

σ
3

W
∗

.

{\displaystyle \; \sigma \_1 = W \sigma \_3 W^\* .}

The goal now is to extend the above to higher dimensions, *d*, a problem solved by J. J. Sylvester (1882).

### Construction: The clock and shift matrices

Fix the dimension d as before. Let *ω* = exp(2*πi*/*d*), a root of unity. Since *ω**d* = 1 and *ω* ≠ 1, the sum of all roots annuls:

1
ω
⋯

ω

d
−
1

=
0
.

{\displaystyle 1 + \omega + \cdots + \omega ^{d-1} = 0 .}

Integer indices may then be cyclically identified mod d.

Now define, with Sylvester, the
**shift matrix**

Σ
1

=

[

0

0

0

⋯

0

1

1

0

0

⋯

0

0

0

1

0

⋯

0

0

0

0

1

⋯

0

0

⋮

⋮

⋮

⋱

⋮

⋮

0

0

0

⋯

1

0

]

{\displaystyle
\Sigma \_1 =
\begin{bmatrix}
0 & 0 & 0 & \cdots &0 & 1\\
1 & 0 & 0 & \cdots & 0 & 0\\
0 & 1 & 0 & \cdots & 0 & 0\\
0 & 0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots &\vdots &\vdots \\
0 & 0 &0 & \cdots & 1 & 0\\
\end{bmatrix}
}

and the **clock matrix**,

Σ
3

=

[

1

0

0

⋯

0

0

ω

0

⋯

0

0

0

ω
2

⋯

0

⋮

⋮

⋮

⋱

⋮

0

0

0

⋯

ω

d
−
1

]

.

{\displaystyle
\Sigma \_3 =
\begin{bmatrix}
1 & 0 & 0 & \cdots & 0\\
0 & \omega & 0 & \cdots & 0\\
0 & 0 &\omega ^2 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & \omega ^{d-1}
\end{bmatrix}.
}

These matrices generalize *σ*1 and *σ*3, respectively.

Note that the unitarity and tracelessness of the two Pauli matrices is preserved, but not Hermiticity in dimensions higher than two. Since Pauli matrices describe Quaternions, Sylvester dubbed the higher-dimensional analogs "nonions", "sedenions", etc.

These two matrices are also the cornerstone of **quantum mechanical dynamics in finite-dimensional vector spaces** as formulated by Hermann Weyl, and find routine applications in numerous areas of mathematical physics. The clock matrix amounts to the exponential of position in a "clock" of *d* hours, and the shift matrix is just the translation operator in that cyclic vector space, so the exponential of the momentum. They are (finite-dimensional) representations of the corresponding elements of the Weyl-Heisenberg on a *d*-dimensional Hilbert space.

The following relations echo and generalize those of the Pauli matrices:

Σ
1
d

=

Σ
3
d

=
I

{\displaystyle \Sigma \_ 1 ^d = \Sigma \_ 3 ^d = I}

and the braiding relation,

Σ
3

Σ
1

=
ω

Σ
1

Σ
3

=

e

2
π
i

/

d

Σ
1

Σ
3

,

{\displaystyle \; \Sigma\_3 \Sigma \_1 = \omega \Sigma\_1 \Sigma \_3 = e^{2 \pi i / d} \Sigma\_1 \Sigma \_3 ,}

the Weyl formulation of the CCR, and can be rewritten as

Σ
3

Σ
1

Σ
3

d
−
1

Σ
1

d
−
1

=
ω

.

{\displaystyle \; \Sigma\_3 \Sigma \_1 \Sigma \_3^{d-1} \Sigma\_1 ^{d-1} = \omega ~.}

On the other hand, to generalize the Walsh–Hadamard matrix *W*, note

# W

1

2

[

1

1

1

ω

2
−
1

]

=

1

2

[

1

1

1

ω

d
−
1

]

.

{\displaystyle
W = \frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1 \\ 1 & \omega ^{2 -1}
# \end{bmatrix}
\frac{1}{\sqrt{2}}
\begin{bmatrix}
1 & 1 \\ 1 & \omega ^{d -1}
\end{bmatrix}.
}

Define, again with Sylvester, the following analog matrix, still denoted by *W* in a slight abuse of notation,

# W

1

d

[

1

1

1

⋯

1

1

ω

d
−
1

ω

2
(
d
−
1
)

⋯

ω

(
d
−
1

)
2

1

ω

d
−
2

ω

2
(
d
−
2
)

⋯

ω

(
d
−
1
)
(
d
−
2
)

⋮

⋮

⋮

⋱

⋮

1

ω

ω
2

⋯

ω

d
−
1

]

.

{\displaystyle
W =
\frac{1}{\sqrt{d}}
\begin{bmatrix}
1 & 1 & 1 & \cdots & 1\\
1 & \omega^{d-1} & \omega^{2(d-1)} & \cdots & \omega^{(d-1)^2}\\
1 & \omega^{d-2} & \omega^{2(d-2)} & \cdots & \omega^{(d-1)(d-2)}\\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 &\omega &\omega ^2 & \cdots & \omega^{d-1}

\end{bmatrix}~.
}

It is evident that *W* is no longer Hermitian, but is still unitary. Direct calculation yields

Σ
1

=
W

Σ
3

W
∗

,

{\displaystyle \; \Sigma\_1 = W \Sigma\_3 W^\* ~,}

which is the desired analog result. Thus, W , a Vandermonde matrix, arrays the eigenvectors of Σ1, which has the same eigenvalues as Σ3.

When *d* = 2*k*, *W* \* is precisely the matrix of the discrete Fourier transform,
converting position coordinates to momentum coordinates and vice versa.

The complete family of *d*2 unitary (but non-Hermitian) independent matrices

(

Σ
1

)
k

(

Σ
3

)
j

=

∑

# m
0

d
−
1

m
k
⟩

ω

j
m

⟨
m

,

{\displaystyle
(\Sigma\_1)^k (\Sigma\_3)^j =\sum\_{m=0}^{d-1} |m+k\rangle \omega^{jm} \langle m| ,
}

provides Sylvester's well-known trace-orthogonal basis for

g
l

{\displaystyle \mathfrak{gl}}

(*d*,ℂ), known as "nonions"

g
l

{\displaystyle \mathfrak{gl}}

(3,ℂ), "sedenions"

g
l

{\displaystyle \mathfrak{gl}}

(4,ℂ), etc...

This basis can be systematically connected to the above Hermitian basis. (For instance, the powers of Σ3, the Cartan subalgebra,
map to linear combinations of the *h**k**d*s.) It can further be used to identify

g
l

{\displaystyle \mathfrak{gl}}

(*d*,ℂ) , as *d* → ∞, with the algebra of Poisson brackets.

## Ore extension

Suppose that *R* is a (not necessarily commutative) ring,

σ
:
R
→
R

{\displaystyle \sigma \colon R \to R}

 is a ring homomorphism, and

δ
:
R
→
R

{\displaystyle \delta\colon R\to R}

 is a  ***σ*-derivation** of *R*, which means that

δ

{\displaystyle \delta}

 is a homomorphism of abelian groups satisfying

δ
(

r
1

r
2

# )
σ
(

r
1

)
δ
(

r
2

)
δ
(

r
1

)

r
2

{\displaystyle \delta(r\_1 r\_2)=\sigma(r\_1)\delta(r\_2)+\delta(r\_1)r\_2}

.
Then the **Ore extension**

R
[
x
;
σ
,
δ
]

{\displaystyle R[x;\sigma,\delta]}

, also called a **skew polynomial ring**, is the noncommutative ring obtained by giving the ring of polynomials

R
[
x
]

{\displaystyle R[x]}

 a new multiplication, subject to the identity

x
# r
σ
(
r
)
x
δ
(
r
)

{\displaystyle x r=\sigma(r)x + \delta(r)}

.
If *δ* = 0 (i.e., is the zero map) then the Ore extension is denoted *R*[*x*; *σ*]. If *σ* = 1 (i.e., the identity map) then the Ore extension is denoted *R*[*x*,*δ*] and is called a **differential polynomial ring**.

The Weyl algebras are Ore extensions, with *R* any commutative polynomial ring, *σ* the identity ring endomorphism, and *δ* the polynomial derivative. Ore algebras are a class of iterated Ore extensions under suitable constraints that permit to develop a noncommutative extension of the theory of Gröbner bases.

## See also

* Generalized Clifford algebra
* Gamma matrices