On the eigenvector algebra of the product of elements with commutator one in the first Weyl algebra
نویسنده
چکیده
Let A1 = K〈X,Y | [Y,X] = 1〉 be the (first) Weyl algebra over a field K of characteristic zero. It is known that the set of eigenvalues of the inner derivation ad(Y X) of A1 is Z. Let A1 → A1, X 7→ x, Y 7→ y, be a K-algebra homomorphism, i.e. [y, x] = 1. It is proved that the set of eigenvalues of the inner derivation ad(yx) of theWeyl algebra A1 is Z and the eigenvector algebra of ad(yx) is K〈x, y〉 (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: is an algebra endomorphism of A1 an automorphism?).
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