Given a field , the -th Weyl algebra is an associative unital algebra over generated by the symbols modulo relations , and (the Kronecker delta).
In characteristic zero, it agrees with the algebra of regular differential operators on the -dimensional affine space.
Sometimes one considers the Weyl algebras over an arbitrary -algebra , including noncommutative , when the definition is simply . Another generalization are the symplectic Weyl algebras.
In quantum physics, one often studies Weyl algebras over the complex numbers; the usual notation there is for (where is the imaginary unit).
Please distinguish from Weil algebra.
Relation to Heisenberg Lie algebra
The Weyl algebra on generators is the quotient of the universal enveloping algebra of the Heisenberg Lie algebra on generators, obtained by identifying the central elements of the Heisenberg Lie algebra with multiples of the identity element.
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