nLab
Weyl 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 relations x 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, it 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; the usual notation there is p jp_j for i j- \mathrm{i} \partial_j (where i\mathrm{i} is the imaginary unit).

Please distinguish from Weil algebra.

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.

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

Revised on March 1, 2012 20:26:42 by Urs Schreiber (82.169.65.155)