Imaginary Whittaker Modules of the Twisted Affine Nappi-Witten Lie Algebra

China Abstract The Nappi-Witten Lie algebra was first introduced by C. Nappi and E. Witten in the study of Wess-Zumino-Novikov-Witten (WZNW) models. They showed that the WZNW model (NW model) based on a central extension of the two-dimensional Euclidean group describes the homogene-ous four-dimensional space-time corresponding to a gravitational plane wave. The associated Lie algebra is neither abelian nor semisimple. Recently K. Christodoulopoulou studied the irreducible Whittaker modules for finite-and infinite-dimensional Heisenberg algebras and for the Lie algebra ob-tained by adjoining a degree derivation to an infinite-dimensional Heisenberg algebra, and used these modules to construct a new class of modules for non-twisted affine algebras, which are called imaginary Whittaker modules. In this paper, imaginary Whittaker modules of the twisted affine Nap-pi-Witten Lie algebra are constructed based on Whittaker modules of Heisenberg algebras. It is proved that the imaginary Whittaker module with the center acting as a non-zero scalar is


Introduction
The conform field theory (CFT) plays an important role in mathematics and physics. Current algebra [1] [2] has proved to be a valuable tool in understanding CFT and String Theory. All the CFTs we know so far can be constructed one way or another from current algebras. The simplest is the WZW models [3] [4], which realize current algebra as their full symmetry. For obvious reasons, the first type of algebras to be analysed was compact ones, used for compactification purposes in String Theory. Later on, non-compact algebras (of the type SL(N, R), SU(M, N) and SO(M, N)) and their cosets have been considered [5] [6] [7] in order to describe curved Minkowski signature spacetimes. Only recently did current algebras of the non-semisimple type receive some attention [8]. The Nappi-Witten model is a WZW model based on a non-semisimple group. It was discovered by C. Nappi and E. Witten [8]

= = =
Just as the non-twisted affine Kac-Moody Lie algebras given in [13], the non-twisted affine Nappi-Witten Lie algebra is defined as with the bracket defined as follows: for , x y nw ∈ and , m n ∈  .
There exist Lie algebra automorphisms θ of nw and θ  of  nw : , and , λ µ ∈  . The twisted affine Nappi-Witten Lie algebra is defined as follows: The representation theory for the non-twisted affine Nappi-Witten Lie algebra has been well studied in [14]. The irreducible restricted modules for the non-twisted affine Nappi-Witten Lie algebra with some natural conditions have been classified and the extension of the vertex operator algebra ( ) 4 ,0 H V l by the even lattice L has been considered in [15]. Verma modules and vertex operator representations for the twisted affine Nappi-Witten Lie algebra have also been investigated in [16]. Recently K. Christodoulopoulou defined Whittaker mod-ules for Heisenberg algebras and used these modules to construct a new class of modules for non-twisted affine algebras (imaginary Whittaker modules) [17].
[18] studied virtual Whittaker modules of the non-twisted affine Nappi-Witten Lie algebra. Inspired by the works mentioned above, the aim of the present paper is to give a characterization of the imaginary Whittaker modules for the Here is a brief outline of Section 2. First, we obtain a Heisenberg subalgebra Throughout the paper, denote by  , *  ,  ,  and +  the sets of the complex numbers, the non-zero complex numbers, the non-negative integers, the integers and the positive integers, respectively. All linear spaces and algebras in this paper are over  unless indicated otherwise.

The Imaginary Whittaker Modules
In the following, for x nw ∈ and n ∈  , we will denote We first review the Whittaker modules of the Heisenberg algebra H  in [17]. Let Define an action of ( )  [17]. We will assume that ( ) be a Whittaker vector of type ψ . Define a ( ) , 0 for all , any . We assume that ( ) It is easy to see that p as p -modules and we can view

3)
, , W W ϕ χ ψ ϕ ψ ϕ χ α − ∈ = ⊕  and as modules for 2) The map via the adjoint action and It is clear that the isomorphism g of (1) maps holds.
The following proposition is evident for weight modules. , | for all .

Proposition 2.3 Any
can be written in the form We are now in a position to give the main result of this paper as follows.