20 research outputs found
A Parametric Family of Subalgebras of the Weyl Algebra II. Irreducible 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 is in F[x]. When h
is nonzero, these algebras 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 and
the subalgebra of invariants under the automorphism group. 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.Comment: 30 pages, a few of the sections have been placed in a different order
at the suggestion of the refere
Whittaker Modules for the Virasoro Algebra
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.Comment: 14 pages; revised descriptions of references [4] and [5
A PARAMETRIC FAMILY OF SUBALGEBRAS OF THE WEYL ALGEBRA I. STRUCTURE AND AUTOMORPHISMS
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 is an element of F[x]. We investigate the family of algebras A(h) as h ranges over all the polynomials in F[x]. When h not equal 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 is an element of F. In two sequels to this work, we completely describe the irreducible modules and derivations of A(h) over any field
Derivations of a parametric family of subalgebras of the Weyl algebra
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 is an element of F[x]. When h not equal 0, the algebra A(h) is subalgebra of the Weyl algebra A(1) 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 HH1(A(h)) = Der(F)(A(h))/Inder(F)(A(h)) of outer derivations over an arbitrary field. In characteristic 0, we show that HH1(A(h)) has a unique maximal nilpotent ideal modulo which HH1(Ah) 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 HH1(A(h)) and describe the structure of HH1(A(h)) as a module over the center of A(h)