nLab
Weyl algebra

Definition

Given a field $k$, the $n$-th Weyl algebra $A_{n,k}$ is an associative unital algebra over $k$ generated by the symbols $x^1,\dots ,x^n,{\partial }_1,\dots ,{\partial }_n$ modulo relations $x^i{x}^j=x^j{x}^i$, ${\partial }_i{\partial }_j={\partial }_j{\partial }_i$ and ${\partial }_i{x}^j-x^j{\partial }_i={\delta }_i^j$ (the Kronecker delta?).

In characteristic zero, it agrees with the algebra of regular differential operators on the $n$-dimensional affine space.

Sometimes one considers the Weyl algebras over an arbitrary $k$-algebra $R$, including noncommutative $R$, when the definition is simply $A_{n,k}{\otimes }_{k}R$. Another generalization are the symplectic Weyl algebras.

In quantum physics, one often studies Weyl algebras over the complex numbers; the usual notation there is $p_j$ for $-\mathrm{i}{\partial }_j$ (where $\mathrm{i}$ is the imaginary unit).

Please distinguish from Weil algebra.

Properties

Relation to Heisenberg Lie algebra

The Weyl algebra on $2n$ generators is the quotient of the universal enveloping algebra of the Heisenberg Lie algebra on $2n$ generators, obtained by identifying the central elements of the Heisenberg Lie algebra with multiples of the identity element.

References

• S. C. Coutinho, A primer of algebraic D-modules, London Math. Soc. Stud. Texts 33, Cambridge University Press 1995. xii+207 pp.

• eom: J.-E. Björk, Weyl algebra