1. 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] 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 [9] [10] [11] [12].
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 [13], 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 [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
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 modules 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 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
where
We first review the Whittaker modules of the Heisenberg algebra
in [17].
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
Set
Define an action of
on
by left multiplication. Then
and
is a Whittaker module of type
for
.
Lemma 2.1 ( [17]) Let
. If
for infinitely many
, then
is irreducible as a
-module.
In the following we define the imaginary Whittaker module of
according to [17]. 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
Set
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
.
Proposition 2.2
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
.
Set
It is clear that
Moreover,
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.
3. Conclusion
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.
Acknowledgements
The author is supported by the Natural Science Foundation of Fujian Province (2017J05016) and is very thankful for everything.