Abstract. Whittaker modules have been well studied in the setting of complex semisimple Lie algebras. Their definition can easily be generalized to certain other Lie algebras with triangular decomposition, including the Virasoro algebra. We define Whittaker modules for the Virasoro algebra and obtain analogues to several results from the classical setting, including a classification of simple Whittaker modules by central characters and composition series for general Whittaker modules.

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.

Abstract. We investigate Whittaker modules for generalized Weyl algebras, a class of associative algebras which includes the quantum plane, Weyl algebras, the universal enveloping algebra of sl 2 and of Heisenberg Lie algebras, Smith's generalizations of U (sl 2 ), various quantum analogues of these algebras, and many others. We show that the Whittaker modules V = Aw of the generalized Weyl algebra A = R(φ, t) are in bijection with the φ-stable left ideals of R. We determine the annihilator Ann A (w) of the cyclic generator w of V . We also describe the annihilator ideal Ann A (V ) under certain assumptions that hold for most of the examples mentioned above. As one special case, we recover Kostant's well-known results on Whittaker modules and their associated annihilators for U (sl 2 ).

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 .

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.