Generators and Relations
One may give an abstract construction of the algebras An in terms of generators and relations. Start with an abstract vector space V (of dimension 2n) equipped with a symplectic form ω. Define the Weyl algebra W(V) to be
where T(V) is the tensor algebra on V, and the notation means "the ideal generated by". In other words, W(V) is the algebra generated by V subject only to the relation vu − uv = ω(v, u). Then, W(V) is isomorphic to An via the choice of a Darboux basis for ω.
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