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Weyl Algebra

The Weyl algebra A_n(F) over a field F of characteristic 0 is the associative algebra generated by x_1, x_2, ..., x_n and partial_1, partial_2, ..., partial_n, with commuting x generators, commuting partial generators, and relations

partial_ix_j-x_jpartial_i=delta_(ij).

The Weyl algebra consists of the polynomial-coefficient differential operators on F[x_1,...,x_n]. For example, in A_1(F), the relation partialx=xpartial+1 is the product rule. Every element has a unique finite expansion in the normally ordered monomials x^alphapartial^beta.

The Dixmier conjecture concerns endomorphisms of Weyl algebras. Their commutator relations are the noncommutative counterpart of the canonical relations in the Poisson conjecture.

See also

Differential Operator, Dixmier Conjecture, Poisson Conjecture

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References

Dixmier, J. "Sur les algèbres de Weyl." Bull. Soc. Math. France 96, 209-242, 1968. https://doi.org/10.24033/bsmf.1667.Long, C. D. "An Explicit Counterexample to the Rank-Two Poisson Conjecture." 22 Jul 2026. https://arxiv.org/abs/2608.23777.

Cite this as:

Weisstein, Eric W. "Weyl Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WeylAlgebra.html

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