nLab
Weyl algebra

To be distinguished from Weil algebra.

Contents

Definition

Given a field kk, the nn-th Weyl algebra A n,kA_{n,k} is an associative unital algebra over kk generated by the symbols x 1,,x n, 1,, nx^1,\ldots,x^n,\partial_1,\ldots,\partial_n modulo the “canonical commutation relationsx ix j=x jx ix^i x^j = x^j x^i, i j= j i\partial_i\partial_j = \partial_j\partial_i and ix jx j i=δ i j\partial_i x^j - x^j \partial_i = \delta_i^j (the Kronecker delta).

In characteristic zero, this agrees with the algebra of regular differential operators on the nn-dimensional affine space.

Sometimes one considers the Weyl algebras over an arbitrary kk-algebra RR, including noncommutative RR, when the definition is simply A n,k kRA_{n,k}\otimes_k R. Another generalization are the symplectic Weyl algebras.

In quantum physics, one often studies Weyl algebras over the complex numbers (see below); the usual notation there is p jp_j for i j- \mathrm{i} \partial_j (where i\mathrm{i} is the imaginary unit).

Properties

Relation to Heisenberg Lie algebra

The Weyl algebra on 2n2n generators is the quotient of the universal enveloping algebra of the Heisenberg Lie algebra on 2n2n generators, obtained by identifying the central elements of the Heisenberg Lie algebra with multiples of the identity element.

Accordingly, given a Heisenberg Lie n-algebra it makes sense to call its universal enveloping E-n algebra a Weyl n-algebra.

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

A categorification of the Weyl algebra is introduced in operadic language in

Revised on August 4, 2017 06:43:02 by Urs Schreiber (94.220.75.2)