An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A h generated by elements x, y, which satisfy yx − xy = h, where h ∈ F[x]. We investigate the family of algebras A h as h ranges over all the polynomials in F [x]. When h = 0, the algebras A h are subalgebras of the Weyl algebra A 1 and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of A h over arbitrary fields F and describe the invariants in A h under the automorphisms. We determine the center, normal elements, and height one prime ideals of A h , localizations and Ore sets for A h , and the Lie ideal [A h , A h ]. We also show that A h cannot be realized as a generalized Weyl algebra over F [x], except when h ∈ F. In two sequels to this work, we completely describe the irreducible modules and derivations of A h over any field.

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A h generated by elements x, y, which satisfy yx − xy = h, where h ∈ F [x]. When h = 0, the algebra A h is subalgebra of the Weyl algebra A1 and can be viewed as differential operators with polynomial coefficients. This paper determines the derivations of A h and the Lie structure of the first Hochschild cohomology group HH 1 (A h ) = Der F (A h )/Inder F (A h ) of outer derivations over an arbitrary field. In characteristic 0, we show that HH 1 (A h ) has a unique maximal nilpotent ideal modulo which it is 0 or a direct sum of simple Lie algebras that are field extensions of the one-variable Witt algebra. In positive characteristic, we obtain decomposition theorems for Der F (A h ) and HH 1 (A h ) and describe the structure of HH 1 (A h ) as a module over the center of A h .

We give algebraic and geometric classifications of 6-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are 14 one-parameter families of 6-dimensional nilpotent anticommutative algebras, complemented by 130 additional isomorphism classes. The corresponding geometric variety is irreducible and determined by the Zariski closure of a one-parameter family of algebras. In particular, there are no rigid 6-dimensional complex nilpotent anticommutative algebras.

A generalization of down-up algebras was introduced by Cassidy and Shelton in [11], the so-called generalized down-up algebras. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f, r, s, γ) such that r is not a root of unity, listing explicitly the elements of the group. In the last section we apply these results to Noetherian down-up algebras, thus obtaining a characterization of the automorphism group of Noetherian down-up algebras A(α, β, γ) for which the roots of the polynomial X 2 − αX − β are not both roots of unity.

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A h generated by elements x, y, which satisfy yxWhen h = 0, the algebras A h are subalgebras of the Weyl algebra A 1 and can be viewed as differential operators with polynomial coefficients. In previous work, we studied the structure of A h and determined its automorphism group Aut F (A h ) and the subalgebra of invariants under Aut F (A h ). Here we determine the irreducible A h -modules. In a sequel to this paper, we completely describe the derivations of A h over any field.