1. Introduction
While vertex algebras are usually assumed to be vector spaces over
, the most important formula Jacobi identity makes sense over any commutative ring, so it is natural to consider vertex algebras over
. An integral basis of a vertex algebra could be considered an analogue of the Chevalley basis in a Lie algebra. Similar to the construction of Lie algebras over
using Chevalley bases, we can create vertex algebras over
. Integral bases for vertex operator algebras associated with lattices have been studied in [1] [2]. In this paper, we are going to investigate integral basis for affine vertex algebras and Virasoro vertex algebra. Let
be a simple Lie algebra over
and
be the corresponding affine Kac-Moody algebra. The vacuum module
at level k has a vertex algebra structure [3] [4] [5], we call it affine vertex algebra. We want to find an integral basis for it when
and
is an integer in Section 3.
Next, we consider the Virasoro vertex algebra. Among the most important vertex algebras are those associated with the Virasoro Lie algebra. It has been studied in [6] [7] [8]. They show that it is generated by the conformal vector
and is minimal in the sense that it does not have any proper vertex operator subalgebra. Besides, any minimal vertex operator algebras of the same central charge are isomorphic. The study of Virasoro vertex algebras is the algebraic foundation of the study of the “minimal modules” in conformal field theory [9]. We use the similar way to analyze an integral basis for Virasoso vertex algebra
when the level is 2k, where
is an integer.
We know that affine Lie algebra and Virasoro Lie algebra have close relationship in physics, so we consider them simultaneously, i.e., as one algebraic structure. Then the definition of affine-Virasoro was introduced [10] [11], which is the semidirect product of the Virasoro algebra and an affine Kac-Moody Lie algebra with a common center. In the last section, we get an integral basis for the universal enveloping algebra of it.
In this paper, we observe that the
-basis of affine vertex algebra
and Virasoro vertex algebra
may be integral basis for them in certain conditions. We create the conditions and confirm that they are exactly the integral bases. Then we utilize the analogue of Chevalley bases for finite dimensional Lie algebras to get an integral basis for the universal enveloping algebra of affine-Virasoro algebra.
2. Preliminaries
We assume that the readers are familiar with the theory of vertex operator algebras [3] [6] [12] [13].
Given an (untwisted) affine Lie algebra
equipped with the bracket relation,
for
and
, together with the condition that
is a nonzero central element of
. Let
,
and
act trivially on
and let
act as the scalar k, making
a
-module, which we denote by
.
By the Poincaré-Birkhoff-Witt theorem, we have that,
(2.1)
as a
-graded vector space. Set
![](//html.scirp.org/file/6-1721880x41.png)
Then
![](//html.scirp.org/file/6-1721880x42.png)
is spanned by the vectors
![](//html.scirp.org/file/6-1721880x44.png)
for
, with
. It can be proved that this is a vertex algebra, more detail can be found in [5]. The vertex algebra structure is determined by
![](//html.scirp.org/file/6-1721880x47.png)
Now we have that,
![](//html.scirp.org/file/6-1721880x48.png)
where
for
.
In the next section, we will consider the integral basis of it when
and
is an integer.
3. Integral Basis of ![](//html.scirp.org/file/6-1721880x53.png)
In this section shall find an integral basis for
. We consider the case when
is an integral number. Firstly, we recall the definition of integral basis of a vertex algebra.
Definition 3.1. Suppose that
is a vertex algebra (over
), an integral basis of it is its
-basis whose
-span can form a vertex algebra over
.
In order to find an integral basis for
, we may firstly find an integral basis for
. That is, we need to find a basis of
whose
-span
is closed under the bracket. When
, it is easy to see that the standard basis elements
satisfy the condition. Now let
be the standard basis of
, we choose an ordered basis of
, that is
![]()
where
. Then by (2.1), we get that
(3.1)
is a
-basis of
.
For convenience, we denote
by V, denote the
-span of a
-basis of
by
.
Theorem 3.2. The formula (3.1) is an integral basis of
.
Proof. It is known that formula (3.1) is a
-basis of
. To check that
is a vertex algebra, we need to prove that the coefficients are still in
of any
(3.2)
for
. Since
is spanned by (3.1) over
, we just need to check formula (3.2) for (3.1). So
![]()
![]()
Since
,
,
,
, the expression
![]()
where
, is a
-linear combination of (3.1), we get that (3.1) is an integral basis of
.
4. Integral Basis of Virasoro Vertex Algebra ![]()
In this section we shall find an integral basis for the Virasoro vertex algebra
when the level is 2k, where
is an integer.
Firstly we recall the definition of Virasoro vertex algebra [6] [14] [15]. As we know, any vertex operator algebra
has the vertex subalgebra
generated by the conformal vector
, and this is in fact the smallest vertex operator subalgebra of
; it is exactly the submodule of
for the Virasoro algebra generated by
:
(4.1)
The Virasoro algebra
is the Lie algebra with basis
equipped with the bracket relations,
(4.2)
together with the condition that
is a central element of
. The Virasoro algebra
equipped with the
-grading
(4.3)
where
(4.4)
is a
-graded Lie algebra, and this grading is given by
-eigenvalus. For the Virasoro algebra
, we have the graded subalgebras
(4.5)
We also have the graded subalgebras
(4.6)
Let
be any complex number. Consider
as an
-module with
acting as the scalar
and with
acting trivially. Denote this
-module by
. Then we form the induced module
(4.7)
From the Poincaré-Birkhoff-Witt theorem, as a vector space,
(4.8)
Set
![]()
Then
(4.9)
where
has a basis consisting of the vectors
(4.10)
for
,
, with
. It can be proved that this is a vertex operator algebra, details can be found in [3]. The vertex algebra structure is determined by
![]()
and we have
![]()
where
.
Next, we will consider the integral basis of it when
acts as 2k and k is integral. We know that
![]()
is a basis of
. Then
![]()
with
,
, with
, is a
-basis of
. We want to check that it is a
-basis of
. When
acts as 2k, we get that,
![]()
for any
.
Just like the affine case, we have that,
![]()
![]()
where
. Since
![]()
and
![]()
the expression
![]()
is a
-linear combination of
, where
,
, with
, so
is an integral basis of
.
5. Integral Basis for the Universal Enveloping Algebra of Affine-Virasoro Algebra
In this section we take the combination of affine algebras and Virasoro Lie algebras into consideration. By analogy with the construction of Lie algebras over
using Chevalley bases, we utilize the
-basis of it whose structure constants are integral to find an integral basis for the universal enveloping algebra of affine-Virasoro algebra
when
.
Firstly we recall the definition of the affine-Virasoro algebra [15].
Definition 5.1. Let L be a finite-dimensional Lie algebra with a non-degenerated invariant normalized symmetric bilinear form
, then the affine-Virasoro Lie algebra is the vector space
![]()
with Lie bracket:
(5.1)
where
.
Now we consider the case of
. Then by Definition (5.1), we get that the corresponding affine-Virasoro algebra
, with Lie bracket:
![]()
![]()
where
.
Lemma 5.2. If
is a
-basis of
, then
![]()
is an integral basis of
.
Proof. Since
is a
-basis of
, we can get that all of the structure constants are integral, then we get the conclusion.
Corollary 5.3. Let
, the integral basis for the universal enveloping algebra of affine-Virasoro algebra is
![]()
where
.
Proof. We only need to check that
is a
-basis of
. By relations (5.1), all coefficients of these brackets are integral except
. Now we use
to replace C, we can get,
![]()
Since
![]()
we conclude that
is a Chevalley basis of
, then we have proved this corollary.
6. Conclusion
In this paper, we get the integral basis for
,
and
. The constructions of affine vertex algebra and Virasoro vertex algebra are key to our proof. Lemma (5.2) is essential in finding integral basis for
. Approaches used here can be easily generalized to tensor product of vertex algebras and universal enveloping algebras. We can also generalize
to general
.