Matthew Ondrus

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.

Whittaker modules for generalized Weyl algebras

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, Smiths 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(phi,t) are in bijection with the phi-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 Kostants well-known results on Whittaker modules and their associated annihilators for U(sl_2).

Whittaker Categories for the Virasoro Algebra

This article builds on work from [16], where the authors described Whittaker modules for the Virasoro algebra. Using the framework outlined in [3], the current article investigates a category of Virasoro-algebra modules that includes Whittaker modules. Results in this article include a classification of the simple modules in the category and a description of certain induced modules that are a natural generalization of simple Whittaker modules.

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 Ah generated by elements x, y, which satisfy yx − xy = h, where h ∈ F[x]. We investigate the family of algebras Ah as h ranges over all the polynomials in F[x]. When h 6= 0, these algebras are subalgebras of the Weyl algebra A1 and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of Ah over arbitrary fields F and describe the invariants in Ah under the automorphisms. We determine the center, normal elements, and height one prime ideals of Ah, localizations and Ore sets for Ah, and the Lie ideal [Ah,Ah]. We also show that Ah 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 derivations and irreducible modules of Ah over any field.

Tensor Products and Whittaker Vectors for Quantum Groups

We describe the action of the center of the quantum group Uq (𝔤) on the tensor product V ⊗ L(λ) of an infinite-dimensional representation V having an infinitesimal character χτ and an irreducible finite-dimensional Uq (𝔤) representation L(λ) of highest weight λ. We apply this result in order to describe the tensor product of a Whittaker module and a finite-dimensional simple module for the algebra Uq(𝔰l2).

Automorphisms and Derivations of the Insertion–Elimination Algebra and Related Graded Lie Algebras

This paper addresses several structural aspects of the insertion–elimination algebra

Whittaker modules for Uq(sl2)

{\mathfrak{g}}

Constructing all irreducible Specht modules in a block of the symmetric group

g, a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of