Weyl algebra

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

:$f\_n(X)\; partial\_X^n\; +\; cdots\; +\; f\_1(X)\; partial\_X\; +\; f\_0(X).$

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 "f_i" 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

:"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, "A_n", is the ring of differential operators with polynomial coefficients in "n" variables. It is generated by "X_i" and $part\_\{X\_i\}$.

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 Lie algebra of the Heisenberg group, by setting the element "1" ofthe Lie algebra equal to the unit "1" of the universal enveloping algebra.

One may give an abstract construction of the algebras "A_n" in terms of generators and relations.We do so in a more sophisticated way: Start with anabstract vector space "V" (of dimension "2n") equipped with a symplectic form $omega$.Define the Weyl algebra "W(V)" to be

:$W(V)\; :=\; T(V)\; /\; (!(\; v\; otimes\; w\; -\; w\; otimes\; v\; -\; omega(v,w),\; ext\{\; for\; \}\; v,w\; in\; V\; )!),$

where the notation $(!(\; )!)$ means "the ideal generated by". In other words, $W(V)$ is the algebra generated by "V" subjectonly to the relation $vw\; -\; wv\; =\; omega(v,w)$. Then, "W(V)" is isomorphic to $A\_\{n\},$ (it does not depend on the choiceof $omega$). In this form, one sees that "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 symmetric algebra "Sym(V)" equipped with the deformed Moyal product (considering the symmetricalgebra to be polynomial functions on $V^*$, where the variables span the vector space "V", and replacing $i\; hbar$ inthe Moyal product formula with "1"). The isomorphism is given bythe 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$i\; hbar$ and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by "X_i" and$i\; hbar\; part\_\{X\_i\}$ (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), butthe 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 algebras, the analogous quantization to the Weyl one is the Clifford algebra.

For more details about this quantization in the case $n=1$ (and an extension using the Fourier transform to integrable ("most") functions, not just polynomial functions), see Weyl quantization.

References

* M. Rausch de Traubenberg, M. J. Slupinski, A. Tanasa, " [http://arxiv.org/abs/math/0504224 Finite-dimensional Lie subalgebras of the Weyl algebra] ", (2005) "(Classifies subalgebras of the one dimensional Weyl algebra over the complex numbers; shows relationship to SL(2,C))"