The conform field theory (CFT) plays an important role in mathematics and physics. Current algebra   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  , 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    in order to describe curved Minkowski signature spacetimes. Only recently did current algebras of the non-semisimple type receive some attention . The Nappi-Witten model is a WZW model based on a non-semisimple group. It was discovered by C. Nappi and E. Witten  that the WZW model based on the Heisenberg group coincides with the σ-model of the maximally symmetric gravitational wave in four dimensions. The corresponding Lie algebra is called the Nappi-Witten Lie algebra nw, which is neither abelian nor semisimple. More results on NW model were presented in    .
The Lie algebra nw is a four-dimensional vector space over with generators and the following Lie bracket:
There is a non-degenerate invariant symmetric bilinear form on nw defined by
Just as the non-twisted affine Kac-Moody Lie algebras given in , the non-twisted affine Nappi-Witten Lie algebra is defined as
with the bracket defined as follows:
for and .
There exist Lie algebra automorphisms of nw and of :
for , and . 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 . 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 by the even lattice L has been considered in . Verma modules and vertex operator representations for the twisted affine Nappi-Witten Lie algebra have also been investigated in . Recently K. Christodoulopoulou defined Whittaker modules for Heisenberg algebras and used these modules to construct a new class of modules for non-twisted affine algebras (imaginary Whittaker modules) .  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 twisted affine Nappi-Witten Lie algebra .
Here is a brief outline of Section 2. First, we obtain a Heisenberg subalgebra by the decomposition of the Lie algebra . Second, we construct the imaginary Whittaker module of by the Whittaker module of . Finally, we give the properties of the module (see Propositions 2.2 and 2.3) and prove that with K acting as a non-zero scalar is irreducible (see Theorem 2.4).
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.
2. The Imaginary Whittaker Modules
In the following, for and , we will denote by . It is clear that has the following decomposition
We first review the Whittaker modules of the Heisenberg algebra in .
Let , where
Thus is an infinite-dimensional Heisenberg subalgebra of .
Assume that and is an algebra homomorphism such that . Set . Let be a one-dimensional vector space viewed as a -module by
Define an action of on by left multiplication. Then and is a Whittaker module of type for .
Lemma 2.1 ( ) Let . If for infinitely many , then is irreducible as a -module.
In the following we define the imaginary Whittaker module of according to . We will assume that is such that and . Let is an algebra homomorphism such that for infinitely many .
Set . is a parabolic subalgebra of . It is obvious that and is an ideal of . Let be a Whittaker vector of type . Define a -module structure on by letting
Define an action of on by left multiplication. Then is called an imaginary Whittaker module of type for .
We assume that is such that , , . Let . Set
It is easy to see that . , . Set
for any .
1) is a free -module, and
2) as -modules and we can view as the -submodule of under this isomorphism.
3) and as modules for ,
In particular, .
Proof. 1) Since , the PBW Theorem implies that and thus as vector space over . So the map defined by is an isomorphism of left -modules.
2) The map defines a -isomorphism of onto the -submodule of .
3) acts semisimply on via the adjoint action and . It is clear that the isomorphism g of (1) maps isomorphically to for every . In particular, if , then because . Thus (3) holds.
The following proposition is evident for weight modules.
Proposition 2.3 Any -submodule V of has a weight space decomposition
relative to .
Proof. Set . Then by Proposition 2.2 (3), we have
Any can be written in the form , where , and there exists such that are distinct. We have for ,
This is a system of linear equations with a nondegenerate matrix. Hence all lie in V.
We are now in a position to give the main result of this paper as follows.
Theorem 2.4 Let and be an algebra homomorphism such that and for infinitely many . Then is irreducible as a -module.
Proof. Let be a -submodule of . We next show that . By Proposition 2.2 (2), we can identify with . Since is irreducible as a -module and , it suffices to show that .
By Proposition 2.3, for some , we have . Let and . We assume
where , , , , I is a finite index set, , , .
Claim There exists such that .
It is clear that
for all . It is easy to check that, for each such that , the coefficient of the basis element
in is . Thus .
Since , we can use induction on r and conclude that there exists such that . The theorem is proved.
We construct the imaginary Whittaker module of the twisted affine Nappi-Witten Lie algebra by its Heisenberg subalgebra . We study the structure of the module and prove that with the center acting as a non-zero scalar is irreducible. Our future work is to determine the maximal submodule of when it is reducible.
The author is supported by the Natural Science Foundation of Fujian Province (2017J05016) and is very thankful for everything.
 Balog, J., O’Raifeartaigh, L., Forgacs, P. and Wipf, A. (1989) Consistency of String Propagation on Curved Space-Times: An SU (1,1) Based Counterexample. Nuclear Physics B, 325, 225-241.
 Christodoulopoulou, K. (2008) Whittaker Modules for Heisenberg Algebras and Imaginary Whittaker Modules for Affine Lie Algebras. Journal of Algebra, 339, 2871-2890.