On Free Field Realizations of W(2,2)-Modules
The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra H at level zero as modules for the W(2,2)-algebra by using construction from [J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:1405.1707]. We prove that the irreducible highest weight H-module is irreducible as W(...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2016 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2016
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/148548 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Free Field Realizations of W(2,2)-Modules / D. Adamović, G. Radobolja // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 20 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860167074977939456 |
|---|---|
| author | Adamović, D. Radobolja, G. |
| author_facet | Adamović, D. Radobolja, G. |
| citation_txt | On Free Field Realizations of W(2,2)-Modules / D. Adamović, G. Radobolja // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 20 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra H at level zero as modules for the W(2,2)-algebra by using construction from [J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:1405.1707]. We prove that the irreducible highest weight H-module is irreducible as W(2,2)-module if and only if it has a typical highest weight. Finally, we construct a screening operator acting on the Heisenberg-Virasoro vertex algebra whose kernel is exactly W(2,2) vertex algebra.
|
| first_indexed | 2025-12-07T17:57:06Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 113, 13 pages
On Free Field Realizations of W (2, 2)-Modules
Dražen ADAMOVIĆ † and Gordan RADOBOLJA ‡
† Department of Mathematics, University of Zagreb, Bijenička 30, 10 000 Zagreb, Croatia
E-mail: adamovic@math.hr
URL: https://web.math.pmf.unizg.hr/~adamovic/
‡ Faculty of Science, University of Split, Rudera Boškovića 33, 21 000 Split, Croatia
E-mail: gordan@pmfst.hr
Received June 09, 2016, in final form December 03, 2016; Published online December 06, 2016
http://dx.doi.org/10.3842/SIGMA.2016.113
Abstract. The aim of the paper is to study modules for the twisted Heisenberg–Virasoro
algebraH at level zero as modules for theW (2, 2)-algebra by using construction from [J. Pure
Appl. Algebra 219 (2015), 4322–4342, arXiv:1405.1707]. We prove that the irreducible
highest weight H-module is irreducible as W (2, 2)-module if and only if it has a typical
highest weight. Finally, we construct a screening operator acting on the Heisenberg–Virasoro
vertex algebra whose kernel is exactly W (2, 2) vertex algebra.
Key words: Heisenberg–Virasoro Lie algebra; vertex algebra; W (2, 2) algebra; screening-
operators
2010 Mathematics Subject Classification: 17B69; 17B67; 17B68; 81R10
1 Introduction
Lie algebra W (2, 2) was first introduced by W. Zhang and C. Dong in [20] as part of a clas-
sification of certain simple vertex operator algebras. Its representation theory has been studied
in [14, 15, 18, 19] and several other papers. Although W (2, 2) is an extension of the Vira-
soro algebra, its representation theory is very different. This is most notable with highest
weight representations. It was shown in [19] that some Verma modules contain a cosingular
vector.
Highest weight representation theory of the twisted Heisenberg–Virasoro Lie algebra has also
been studied recently. Representations with nontrivial action of CI have been developed in [6].
Representations at level zero, i.e., with trivial action of CI were studied in [8] due to their
importance in some constructions over the toroidal Lie algebras (see [7, 9]). In this case, a free
field realization of highest weight modules along with the fusion rules for a suitable category of
modules were obtained in [4].
Irreducible highest weight modules of highest weights (0, 0) over these algebras carry the
structure of simple vertex operator algebras. Let us denote these vertex operator algebras
as LW (2,2)(cL, cW ) and LH(cL, cL,I). It was proved in [4] that simple vertex operator algebra
LW (2,2)(cL, cW ) embeds into Heisenberg–Virasoro vertex operator algebra LH(cL, cL,I) so that
cW = −24c2L,I . As a result each highest weight module over H is also a W (2, 2)-module.
In this paper we shall completely describe the structure of the irreducible highest weight H-
modules as W (2, 2)-modules. We show that in generic case the resulting W (2, 2)-module is
irreducible. However, in case of a module of highest weight such that associated Verma module
over W (2, 2) contains cosingular vectors (we shall call this kind of weight atypical), irreducible
H-module is reducible over W (2, 2). We shall denote the irreducible highest weight H-module
mailto:adamovic@math.hr
https://web.math.pmf.unizg.hr/~adamovic/
mailto:gordan@pmfst.hr
http://dx.doi.org/10.3842/SIGMA.2016.113
2 D. Adamović and G. Radobolja
LH(cL, 0, cL,I , h, hI) shortly as LH(h, hI). We also use the following notation1
hp,r =
(
1− p2
)cL − 2
24
+ p(p− 1) + p
1− r
2
for p, r ∈ Z>0. Define
AT H(cL, cL,I) = {(hp,r, (1± p)cL,I) | p, r ∈ Z>0}.
We call a weight (h, hI) atypical for H (resp. typical) if (h, hI) ∈ AT H(cL, cL,I) (resp. (h, hI) /∈
AT H(cL, cL,I)). We shall refer to a highest weight module over H as (a)typical if its highest
weight is (a)typical for H.
The next theorem gives a main result of the paper.
Theorem 1.1. Assume that cL,I 6= 0.
(1) LH(h, hI) is irreducible as a W (2, 2)-module if and only if
(h, hI) /∈ AT H(cL, cL,I).
(2) If (h, hI) ∈ AT H(cL, cL,I) then LH(h, hI) is a non-split extension of two irreducible highest
weight W (2, 2)-modules.
We recall some aspects of representation theories of infinite-dimensional Lie algebras H
and W (2, 2) in Section 2. The main results on the branching rules will be proved in Sec-
tion 3. From the free field realization in [4] follows that irreducible H-modules are pairwise
contragredient. For half of these modules, proofs rely on a W (2, 2)-homomorphism between
Verma modules over W (2, 2) and H which is induced by a homomorphism of vertex operator
algebras. The rest is then proved elegantly by passing to contragredients. We also prove a very
interesting result that the Verma module for H with typical highest weight is an infinite direct
sum of irreducible W (2, 2)-modules (cf. Theorem 3.7). This result presents a W (2, 2)-analogue
of certain Feigin–Fuchs modules for the Virasoro algebra (cf. Remark 3.8).
From the results in the paper, we see that the vertex algebra LW (2,2)(cL, cW ) has many
properties similar to the W-algebras appearing in logarithmic conformal field theory (LCFT):
• LW (2,2)(cL, cW ) admits a free field realization inside of the Heisenberg–Virasoro vertex
algebra LH(cL, cL,I).
• Typical modules are realized as irreducible modules for LH(cL, cL,I).
• In the atypical case, irreducible LH(cL, cL,I)-modules as LW (2,2)(cL, cW )-modules have
semi–simple rank two.
The singlet vertex algebra M(1) has similar properties. M(1) is realized as kernel of a screening
operator inside the Heisenberg vertex algebra M(1) (cf. [1]). In Section 4 we construct the
screening operator
S1 : LH(cL, cL,I)→ LH(1, 0),
which commutes with the action of W (2, 2)-algebra such that
KerLH(cL,cL,I) S1
∼= LW (2,2)(cL, cW ).
Our construction uses an extension Vext of the vertex algebra LH(cL, cL,I) by a non-weight
module for the Heisenberg–Virasoro vertex algebra. In our forthcoming paper [5], we shall
present an explicit realization of Vext and apply this construction to the study of intertwining
operators and logarithmic modules.
1We emphasise a term cL−2
24
for its importance in a free field realization of H (see [4] for details).
On Free Field Realizations of W (2, 2)-Modules 3
2 Lie algebra W (2, 2) and the twisted Heisenberg–Virasoro
Lie algebra at level zero
W (2, 2) is a Lie algebra with basis {L(n),W (n), CL, CW : n ∈ Z} over C, and a Lie bracket
[L(n), L(m)] = (n−m)L(n+m) + δn,−m
n3 − n
12
CL,
[L(n),W (m)] = (n−m)W (n+m) + δn,−m
n3 − n
12
CW ,
[W (n),W (m)] = [·, CL] = [·, CW ] = 0.
Highest weight representation theory over W (2, 2) was studied in [14, 19]. However, represen-
tations treated in these papers have equal central charges CL = CW . These results have recently
been generalised to CL 6= CW in [15]. Here we state the most important results. Verma module
with central charge (cL, cW ) and highest weight (h, hW ) is denoted by V W (2,2)(cL, cW , h, hW ),
its highest weight vector by vh,hW
and irreducible quotient module by LW (2,2)(cL, cW , h, hW ).
Recall the definition of a cosingular vector. Homogeneous vector v ∈ M is called cosingular
(or subsingular) if it is not singular in M and if there is a proper submodule N ⊂M such that
v +N is a singular vector in M/N .
Theorem 2.1 ([15, 19]). Let cW 6= 0.
(i) Verma module V W (2,2)(cL, cW , h, hW ) is reducible if and only if hW = 1−p2
24 cW for some
p ∈ Z>0. In that case, there exists a singular vector u′p ∈ C[W (−1), . . . ,W (−p)]vh,hW
such
that U(W (2, 2))u′p
∼= V W (2,2)(cL, cW , h+ p, hW ).
(ii) A quotient module2
V W (2,2)(cL, cW , h, hW )/U(W (2, 2))u′p =: L̃W (2,2)(cL, cW , hp,r, hW )
is reducible if and only if h = hp,r for some r ∈ Z>0. In that case, there is a cosin-
gular vector urp ∈ V W (2,2)(cL, cW , h, hW )h+rp such that urp := urp + U(W (2, 2))u′p is
a singular vector in L̃W (2,2)(cL, cW , hp,r, hW ) which generates a submodule isomorphic to
LW (2,2)(cL, cW , hp,r + rp, hW ). The short sequence
0→ LW (2,2)(cL, cW , hp,r + rp, hW )→ L̃W (2,2)(cL, cW , hp,r, hW )
→ LW (2,2)(cL, cW , hp,r, hW )→ 0, (2.1)
where the highest weight vector in LW (2,2)(cL, cW , hp,r + rp, hW ) maps to urp is exact.
Define
AT W (2,2)(cL, cW ) =
{(
hp,r,
1− p2
24
cW
)
| p, r ∈ Z>0
}
.
Remark 2.2. We will refer to the (modules of) highest weights (h, hW ) ∈ AT W (2,2)(cL, cW )
as atypical for W (2, 2), and otherwise as typical. Again, we refer to a highest weight W (2, 2)-
module as (a)typical depending on its highest weight. So a Verma module over W (2, 2) contains
a nontrivial cosingular vector if and only if it is atypical.
Proposition 2.3. Let hW = 1−p2
24 cW , p ∈ Z>0.
2This module is denoted by L′ in [15, 19]. We change notation to L̃ due to use of superscript W (2, 2).
4 D. Adamović and G. Radobolja
(i) Let (hp,r, hW ), r ∈ Z>0 be an atypical weight and k ∈ Z. Then (hp,r + kp, hW ) is atypical
if and only if k < r
2 .
(ii) Atypical Verma module V W (2,2)(hp,r, hW ) contains exactly b r+1
2 c cosingular vectors. The
weights of these vectors are hp,r + (r − i)p = hp,−r+2i, i = 0, . . . , b r−12 c.
Proof. (i) Directly from Theorem 2.1 since hp,r + kp = hp,r−2k.
(ii) Follows from (i) since V W (2,2)(hp,r, hW ) contains an infinite chain of submodules isomor-
phic to Verma modules of highest weights hp,r + ip = hp,r−2i, i > 0. Applying Theorem 2.1 to
each of these submodules we obtain cosingular vectors of weights
hp,r−2i + (r − 2i)p = hp,r + (r − i)p = hp,−r+2i
as long as r − 2i > 0. �
Remark 2.4. Standard PBW basis for V W (2,2)(cL, cW , h, hW ) consists of vectors
W (−ms) · · ·W (−m1)L(−nt) · · ·L(−n1)vh,hW
such that ms ≥ · · · ≥ m1 ≥ 1, nt ≥ · · · ≥ n1 ≥ 1. The only nonzero component of urp belonging
to C[L(−1), L(−2), . . .]v is L(−p)rvh,hW
[19].
Define P2(n) =
n∑
i=0
P (n− i)P (i) where P is a partition function with P (0) = 1. We have the
following character formulas [19]
charV W (2,2)(cL, cW , h, hW ) = qh
∑
n≥0
P2(n)qn = qh
∏
k≥1
(
1− qk
)−2
,
for all h, hW ∈ C. If hW = 1−p2
24 cW , then
char L̃W (2,2)(cL, cW , h, hW ) = qh
(
1− qp
)∑
n≥0
P2(n)qn = qh
(
1− qp
)∏
k≥1
(
1− qk
)−2
.
If (h, hW ) is typical for W (2, 2), then this is the character of an irreducible highest weight
module. Finally, the character of atypical irreducible module is
charLW (2,2)(cL, cW , hp,r, hW ) = qhp,r
(
1− qp
)(
1− qrp
)∑
n≥0
P2(n)qn
= qhp,r
(
1− qp
)
(1− qrp)
∏
k≥1
(
1− qk
)−2
.
The twisted Heisenberg–Virasoro algebra H is the universal central extension of the Lie
algebra of differential operators on a circle of order at most one. It is the infinite-dimensional
complex Lie algebra with a basis
{L(n), I(n) : n ∈ Z} ∪ {CL, CLI , CI}
and commutation relations
[L(n), L(m)] = (n−m)L(n+m) + δn,−m
n3 − n
12
CL,
[L(n), I(m)] = −mI(n+m)− δn,−m
(
n2 + n
)
CLI ,
[I(n), I(m)] = nδn,−mCI , [H, CL] = [H, CLI ] = [H, CI ] = 0.
On Free Field Realizations of W (2, 2)-Modules 5
The Lie algebra H admits the following triangular decomposition
H = H− ⊕H0 ⊕H+, (2.2)
H± = spanC{I(±n), L(±n) |n ∈ Z>0}, H0 = spanC{I(0), L(0), CL, CL,I , CI}.
Although they seem to be two similar extensions of the Virasoro algebra, representation
theories of W (2, 2) and H are different. The main reason for that lies in the fact that I(0) is
a central element, while W (0) is not. However, applying free field realization, we shall see that
highest weight modules over the two algebras are related.
Denote by V H(cL, cI , cL,I , h, hI) the Verma module and by vh,hI
its highest weight vector.
CL, CI , CL,I , L(0) and I(0) act on vh,hI
by scalars cL, cI , cL,I , h and hI , respectively. Then
(cL,cI , cL,I) is called a central charge, and (h, hI) a highest weight. In this paper we consider
central charges (cL, 0, cL,I) such that cL,I 6= 0.
Theorem 2.5 ([8]). Let cL,I 6= 0. Verma module V H(cL, 0, cL,I , h, hI) is reducible if and only if
hI = (1± p)cL,I for some p ∈ Z>0. In that case, there is a singular vector v±p of weight p, which
generates a maximal submodule in V H(cL, 0, cL,I , h, hI) isomorphic to V H(cL, 0, cL,I , h+ p, hI).
Remark 2.6. In case hI = (1 + p)cL,I an explicit formula for a singular vector v+p is obtained
using Schur polynomials in I(−1), . . . , I(−p). See [4] for details. Assume that x ∈ U(W (2, 2))−
is such that xvh,hI
∈ V H(cL, 0, cL,I , h, hI) lies in a maximal submodule. Then x does not have
a nontrivial additive component (in PBW basis) that belongs to C[L(−1), L(−2), . . .] [8].
There is an infinite chain of Verma submodules generated by singular vectors v±kp, k ∈ Z>0,
with all the subquotients being irreducible. Note that there is no mention of L̃H since there are
no cosingular vectors in V H.
The following character formulas were obtained in [8]:
charV H(cL, 0, cL, h, hI) = qh
∑
n≥0
P2(n)qn = qh
∏
k≥1
(
1− qk
)−2
,
charLH(cL, 0, cL, h, hI) = qh
(
1− qp
)∑
n≥0
P2(n)qn = qh
(
1− qp
)∏
k≥1
(
1− qk
)−2
.
Remark 2.7. Throughout the rest of the paper we work with highest weight modules over
the Lie algebras W (2, 2) and H so we always denote algebra in superscript. In order to avoid
too cumbersome notation, we omit central charges. Therefore, we write V H(h, hI) for Verma
module over H, V W (2,2)(h, hW ) for Verma module over W (2, 2) and so on. We always assume
that cW and cL,I are nonzero. Moreover, if we work with several modules over both algebras,
cL is equal for all modules.
We shall write 〈x〉W (2,2) for a cyclic submodule U(W (2, 2))x and 〈x〉H for U(H)x. Finally,
∼=W (2,2) denotes an isomorphism of W (2, 2)-modules.
3 Irreducible highest weight modules
In this section we present main results of the paper which completely describe the structure of
(irreducible) highest weight modules for H as W (2, 2)-modules. The main tool is the homomor-
phism between W (2, 2) and the Heisenberg–Virasoro vertex algebras from [4].
LW (2,2)(cL, cW , 0, 0) is a simple universal vertex algebra associated to Lie algebra W (2, 2) (cf.
[19, 20]) which we denote by LW (2,2)(cL, cW ). It is generated by fields
L(z) = Y (ω, z) =
∑
n∈Z
L(n)z−n−2, W (z) = Y (W, z) =
∑
n∈Z
W (n)z−n−2,
6 D. Adamović and G. Radobolja
where ω = L(−2)1 and W = W (−2)1. Each highest weight W (2, 2)-module is also a module
over a vertex operator algebra LW (2,2)(cL, cW ).
Likewise (see [7]) LH(cL, 0, cL,I , 0, 0) is a simple Heisenberg–Virasoro vertex operator algebra,
which we denote by LH(cL, cL,I). This algebra is generated by the fields
L(z) = Y (ω, z) =
∑
n∈Z
L(n)z−n−2, I(z) = Y (I, z) =
∑
n∈Z
I(n)z−n−1,
where ω = L(−2)1 and I = I(−1)1. Moreover, highest weight H-modules are modules over
a vertex operator algebra LH(cL,cL,I).
It was shown in [4] that there is a monomorphism of vertex operator algebras
Ψ: LW (2,2)(cL, cW )→ LH(cL, cL,I), (3.1)
ω 7→ L(−2)1,
W 7→ (I(−1)2 + 2cL,II(−2))1,
where cW = −24c2L,I . By means of Ψ, each highest weight module over H becomes an
LW (2,2)(cL, cW )-module and therefore a module over W (2, 2). In particular, Ψ induces a non-
trivial W (2, 2)-homomorphism (which we shall denote by the same letter)
Ψ: V W (2,2)(cL, cW , h, hW )→ V H(cL, 0, cL,I , h, hI),
where cW = −24c2L,I and hW = hI(hI − 2cL,I). Ψ maps the highest weight vector vh,hW
to the
highest weight vector vh,hI
and the action of W (−n) on V H(cL, 0, cL,I , h, hI) is given by
W (−n) ≡ 2cL,I(n− 1)I(−n) +
∑
i∈Z
I(−i)I(−n+ i), (3.2)
W (−n) ≡ 2cL,I
(
n− 1 +
hI
cL,I
)
I(−n) +
∑
i 6=0,n
I(−i)I(−n+ i).
Note that hW = 1−p2
24 cW if and only if hI = (1 ± p)cL,I , so either both of these Verma
modules are irreducible, or they are reducible with singular vectors at equal levels. Moreover,
(h, hW ) ∈ AT W (2,2)(cL, cW ) if and only if (h, hI) ∈ AT H(cL, cL,I).
Throughout the rest of this section we assume that cW = −24c2L,I .
Lemma 3.1 ([4, Lemma 7.2]). Suppose that hI 6= (1 − p)cL,I for all p ∈ Z>0. Then Ψ is an
isomorphism of W (2, 2)-modules. In particular, if hI 6= (1± p)cL,I for p ∈ Z>0, then
LH(h, hI) ∼=W (2,2) L
W (2,2)(h, hW ),
where hW = hI(hI − 2cL,I).
Lemma 3.2. Suppose that x ∈ V H(h, hI) is H-singular. Then x is W (2, 2)-singular as well.
In particular, if y is a homogeneous vector such that x = Ψ(y), then y is either singular or
cosingular vector in V W (2,2)(h, hW ).
Proof. Let x ∈ V H(h, hI) be a H-singular vector, i.e., L(k)x = I(k)x = 0 for all k ∈ Z>0.
From (3.2) we have
W (n)x = −2cL,I(n+ 1)I(n)x+
∑
i∈Z
I(−i)I(n+ i)x,
so W (n)x = 0 for all n ∈ Z>0. Therefore, x is W (2, 2)-singular. If x = Ψ(y), then L(k)y,W (k)y
∈ Ker Ψ for k > 0. Therefore y + Ker Ψ is a singular vector in V W (2,2)(h, hW )/Ker Ψ. �
On Free Field Realizations of W (2, 2)-Modules 7
Theorem 3.3. Let p ∈ Z>0.
(i) If (h, (1 + p)cL,I) is typical for H (equivalently if
(
h, 1−p
2
24 cW
)
is typical for W (2, 2)) then
LH(h, (1 + p)cL,I) ∼=W (2,2) L
W (2,2)
(
h,
1− p2
24
cW
)
. (3.3)
(ii) If (hp,r, (1 + p)cL,I) ∈ AT H(cL, cL,I) (equivalently if (hp,r,
1−p2
24 cW ) ∈ AT W (2,2)(cL, cW ))
then
LH(hp,r, (1 + p)cL,I) ∼=W (2,2) L̃
W (2,2)
(
hp,r,
1− p2
24
cW
)
and the short sequence of W (2, 2)-modules
0→ LW (2,2)
(
hp,r + rp,
1− p2
24
cW
)
→ LH(hp,r, (1 + p)cL,I) (3.4)
→ LW (2,2)
(
hp,r,
1− p2
24
cW
)
→ 0
is exact.
Proof. By Lemma 3.1, Ψ is an isomorphism of Verma modules and thus by Lemma 3.2 it maps
a W (2, 2)-singular vector u′p to an H-singular vector v+p . If h 6= hp,r, both of these vectors
generate maximal submodules in respective Verma modules so (3.3) follows.
Now suppose that h = hp,r. We need to show that a cosingular vector urp is not mapped
into a maximal submodule of V H(hp,r, hI). But urp has L(−p)rv as an additive component
(see Remark 2.4), and by construction (3.1), Ψ(urp) also must have this additive component.
However, Ψ(urp) can not lie in a maximal H-submodule of V H(h, hI) (see Remark 2.6). This
means that isomorphism Ψ of Verma modules induces a W (2, 2)-isomorphism of L̃W (2,2)(h, hW )
and LH(h, hI) for all h ∈ C. Exactness of (3.4) is just an application of (2.1). �
Remark 3.4. Note that the image Ψ(urp) of a W (2, 2)-cosingular vector is neither H-singular,
nor H-cosingular in V H(hp,r, (1 + p)cL,I). For example, L(−1)v0,0 in V H(0, 2cL,I) is W (2, 2)-
cosingular, but not H-singular since I(1)L(−1)v0,0 = 2cL,Iv0,0.
If hI = (1 − p)cL,I , then Ψ is not an isomorphism. We shall present a W (2, 2)-structure of
Verma module later. In order to examine irreducible W (2, 2)-modules we apply the properties
of contragredient modules.
Let us recall the definition of contragredient module (see [12]). Assume that (M,YM ) is
a graded module over a vertex operator algebra V such that M = ⊕∞n=0M(n), dimM(n) < ∞
and suppose that there is γ ∈ C such that L(0)|M(n) ≡ (γ + n) Id. The contragredient module
(M∗, YM∗) is defined as follows. For every n ∈ Z>0 let M(n)∗ be the dual vector space and let
M∗ = ⊕∞n=0M(n)∗ be a restricted dual of M . Consider the natural pairing 〈·, ·〉 : M∗⊗M → C.
Define the linear map YM∗ : V → EndM∗[[z, z−1]] such that
〈YM∗(v, z)m′,m〉 =
〈
m′, YM
(
ezL(1)
(
−z−2
)L(0)
v, z−1
)
m
〉
(3.5)
for each v ∈ V , m ∈M , m′ ∈M∗. Then (M∗, YM∗) is a V -module.
In particular, choosing v = ω = L−21 in (3.5) one gets
〈L(n)m′,m〉 = 〈m′, L(−n)m〉.
Simple calculation with I ∈ LH(cL, cL,I) and W ∈ LW (2,2)(cL, cW ) shows that
〈I(n)m′,m〉 = 〈m′,−I(−n)m+ δn,02cL,I〉, 〈W (n)m′,m〉 = 〈m′,W (−n)m〉.
Therefore we get the following result (the first and third relations were given in [4]):
8 D. Adamović and G. Radobolja
Lemma 3.5.
LH(h, hI)∗ ∼= LH(h,−hI + 2cL,I), LW (2,2)(h, hW )∗ ∼= LW (2,2)(h, hW ).
In particular,
LH(h, (1± p)cL,I)∗ ∼= LH(h, (1∓ p)cL,I).
Directly from Theorem 3.3 and Lemma 3.5 follows
Corollary 3.6. Let p ∈ Z>0.
(i) If (h, (1− p)cL,I) is typical for H (equivalently if
(
h, 1−p
2
24 cW
)
is typical for W (2, 2)) then
LH(h, (1− p)cL,I) ∼=W (2,2) L
W (2,2)
(
h,
1− p2
24
cW
)
.
(ii) If (hp,r, (1 − p)cL,I) ∈ AT H(cL, cL,I) (equivalently if
(
hp,r,
1−p2
24 cW
)
∈ AT W (2,2)(cL, cW ))
then
LH(hp,r, (1− p)cL,I) ∼=W (2,2) L̃
W (2,2)
(
hp,r,
1− p2
24
cW
)∗
and the short sequence of W (2, 2)-modules
0→ LW (2,2)
(
hp,r,
1− p2
24
cW
)
→ LH (hp,r, (1− p)cL,I)
→ LW (2,2)
(
hp,r + rp,
1− p2
24
cW
)
→ 0
is exact.
From Lemma 3.1, Theorem 3.3 and Corollary 3.6 follow assertions of Theorem 1.1.
Finally, we show that Verma module over H is an infinite direct sum of irreducible W (2, 2)-
modules. Recall that V H(h, (1 − p)cL,I) has a series of singular vectors v−ip, i ∈ Z≥0 (for i = 0,
we set v−0 = vh,hI
) which generate a descending chain of Verma submodules over H:
〈vh,hI
〉H =V H(h, hI)
⊆
〈v−p 〉H ∼=V H(h+ p, hI)
⊆
...
⊆
〈v−ip〉H ∼=V
H(h+ ip, hI)
⊆
〈v−(i+1)p〉H ∼=V
H(h+ (i+ 1)p, hI)
⊆
...
Therefore one may identify V H(h+ ip, hI) with a submodule of V H(h, hI) and a singular vector
v−ip ∈ V H(h, hI) with the highest weight vector vh+ip,hI
∈ V H(h+ ip, hI). We will prove that in
a typical case each of those vectors generates an irreducible W (2, 2)-submodule.
On Free Field Realizations of W (2, 2)-Modules 9
Theorem 3.7. Let p ∈ Z>0. Suppose that (h, (1 − p)cL,I) /∈ AT H(cL, cL,I). Then we have the
following isomorphism of W (2, 2)-modules
V H(h, (1− p)cL,I) ∼=W (2,2)
⊕
i≥0
LW (2,2)
(
h+ ip,
1− p2
24
cW
)
.
Proof. First we notice that the vertex algebra homorphism Ψ: LW (2,2)(cW , cL)→ LH(cW , cL),
for every i ∈ Z≥0 induces the following non-trivial homomorphism of W (2, 2)-modules:
Ψ(i) : V W (2,2)
(
h+ ip,
1− p2
24
cW
)
→ 〈v−ip〉W (2,2) ⊂ V H(h+ ip, (1− p)cL,I),
which maps the highest weight vector of V W (2,2)
(
h + ip, 1−p
2
24 cW
)
to v−ip. Since (h, 1−p
2
24 cW ) is
typical it follows from Proposition 2.3(i) that (h+ ip, 1−p
2
24 cW ) are typical for all i ∈ Z>0 as well.
Let hW = 1−p2
24 cW . Consider the homomorphism Ψ(i) : V W (2,2)(h+ ip, hW )→ V H(h+ ip, hI)
above. Applying (3.2), we get
Ψ(i)(W (−p)vh+ip,hW
) =
p−1∑
i=1
I(−i)I(i− p)vh+ip,hI
,
so I(−p)vh+ip,hI
/∈ Im Ψ(i). Since the Verma modules V W (2,2)(h + ip, hW ) and V H(h + ip, hI)
have equal characters, it follows that Ker Ψ(i) contains a singular vector in V W (2,2)(h+ ip, hW )
of conformal weight h+(i+1)p. Since the weight (h+ip, hW ) is typical, the maximal submodule
in V W (2,2)(h + ip, hW ) is generated by this singular vector so we conclude that Ker Ψ(i) is the
maximal submodule in V W (2,2)(h+ ip, hW ). Therefore
Im Ψ(i) = 〈vh+ip,hI
〉W (2,2)
∼= LW (2,2)(h+ ip, hW ).
In this way we get a series of W (2, 2)-monomorphisms
LW (2,2)(h+ ip, hW ) ↪→ V H(h, (1− p)cL,I), i ∈ Z≥0 (3.6)
mapping vh+ip,hW
to a singular vector v−ip. Let v−jp be an H-singular vector in V H(h + ip, (1 −
p)cL,I) of weight h + jp, for j > i. By Lemma 3.2, v−jp is singular for W (2, 2) and therefore
v−jp /∈ 〈vh+ip,hI
〉W (2,2) for j > i. We conclude that the images of morphisms (3.6) have trivial
pairwise intersections (since these images are non-isomorphic irreducible modules), so their sum
is direct. The assertion follows from the observation that the character of this sum is
∞∑
i=0
qh+ip
(
1− qp
)∏
k≥1
(
1− qk
)−2
= qh
∏
k≥1
(
1− qk
)−2
= charV H(h, (1− p)cL.I). �
Remark 3.8. It is interesting to notice that our Theorem 3.7 shows that V H(h, hI) can be
considered as a W (2, 2)-analogue of certain Feigin–Fuchs modules for the Virasoro algebra which
are also direct sums of infinitely many irreducible modules (cf. [11], [2, Theorem 5.1]).
From the previous theorem follows
V H(h, hI)
=
〈vh,hI
〉W (2,2) =LW (2,2)(h, hW )
⊕
10 D. Adamović and G. Radobolja
〈v−p 〉W (2,2)
∼=LW (2,2)(h+ p, hW )
⊕
...
⊕
〈v−ip〉W (2,2)
∼=LW (2,2)(h+ ip, hW )
⊕
...
In atypical case however, the W (2, 2)-submodules generated by H–singular vectors are nested
as follows. Consider V H(hp,r, hI) where (hp,r, hI) ∈ AT H(cL, cL,I). Then Ψ0 maps a cosingular
vector urp ∈ V W (2,2)(hp,r, hW ) to a singular vector v−rp. In other words we have
〈v−rp〉W (2,2) ⊆ 〈vhp,r,hI
〉 ∼=W (2,2) L̃
W (2,2)(hp,r, hW ).
Using the same argument in view of Proposition 2.3 we see that
〈v−(r−i)p〉W (2,2) ⊆ 〈v−ip〉W (2,2)
∼= L̃W (2,2)(hp,r + ip, hW ), i = 1, . . . ,
⌊r − 1
2
⌋
.
Therefore,
〈vhp,r,hI
〉W (2,2)/〈v−rp〉W (2,2)
∼= LW (2,2)(hp,r, hW ),
〈v−ip〉W (2,2)/〈v−(r−i)p〉W (2,2)
∼= LW (2,2)(hp,r + ip, hW ), i <
r − 1
2
,
〈v−ip〉W (2,2)
∼= LW (2,2)(hp,r + ip, hW ), i ≥ r − 1
2
.
In this case, I(−p)r−ivhp,r,hI
are W (2, 2)-cosingular vectors in V H(hp,r, hI).
Example 3.9. Consider p = 1 case. Singular vector in V H(h, 0) is u′1 =
(
L(−1)+ h
cL,I
I(−1)
)
vh,0,
and u′1 generates a copy of V H(h+ 1, 0).
r = 1: Ψ: V W (2,2)(0, 0) → V H(0, 0) maps a singular vector u′1 = W (−1)v0,0 to 0 and
a cosingular vector u1 = L(−1)v0,0 to H-singular vector v−1 = L(−1)v0,0. We get the short exact
sequence of W (2, 2)-modules
0→ LW (2,2)(0, 0)→ LH(0, 0)→ LW (2,2)(1, 0)→ 0,
which is an expansion of (3.1) considered as a homomorphism of W (2, 2)-modules. The rightmost
module is generated by a projective image of I(−1)v0,0. Therefore, LH(cL, cL,I) is generated
over W (2, 2) by v0,0 and I(−1)v0,0.
r ∈ Z>0: In general, a cosingular vector urp ∈ V W (2,2)
(
1−r
2 , 0
)
maps to a singular vector
v−r ∈ V H
(
1−r
2 , 0
)
of weight 1+r
2 .
v−r =
r−1∏
i=0
(
L(−1) +
1− r + 2i
2cL,I
I(−1)
)
v 1−r
2
,0.
4 Screening operators and W (2, 2)-algebra
We think that the vertex algebra LW (2,2)(cL, cW ) is a very interesting example of non-rational
vertex algebra, which admits similar fusion ring of representations as someW-algebras appearing
in LCFT (cf. [1, 2, 10, 13]). Since W-algebras appearing in LCFT are realized as kernels of
On Free Field Realizations of W (2, 2)-Modules 11
screening operators acting on certain modules for Heisenberg vertex algebras, it is natural to
ask if LW (2,2)(cL, cW ) admits similar realization. In [4] we embedded the W (2, 2)-algebra as
a subalgebra of the Heisenberg–Virasoro vertex algebra. In this section we shall construct
a screening operator S1 such that the kernel of this operator is exactly LW (2,2)(cL, cW ).
Let us first construct a non-semisimple extension of the vertex algebra LH(cL, cL,I). Recall
that the Lie algebra H admits the triangular decomposition (2.2). Let E = spanC{v0, v1} be
2-dimensional H≥0 = H0 ⊕H+-module such that H+ acts trivially on E and
L(0)vi = vi, i = 0, 1, I(0)v1 = v0, I(0)v0 = 0,
CLv
i = cLv
i, CL,Iv
i = cL,Iv
i, CIv
i = 0, i = 1, 2.
Consider now induced H-module
Ẽ = U(H)⊗U(H≥0) E.
By construction, Ẽ is a non-split self-extension of the Verma module V H(1, 0):
0→ V H(1, 0)→ Ẽ → V H(1, 0)→ 0.
Moreover, Ẽ is a restricted module for H and therefore it is a module over vertex operator
algebra LH(cL, cL,I). Since
Ẽ ∼= E ⊗ U(H−)
as a vector space, the operator L(0) defines a Z≥0-gradation on Ẽ.
Note that (L(−1) + I(−1)/cL,I)v0 is a singular vector in Ẽ and it generates the proper
submodule. Finally we define the quotient module
U =
Ẽ
U(H).(L(−1) + I(−1)/cL,I)v0
.
Proposition 4.1. U is a Z≥0-graded module for the vertex operator algebra LH(cL, cL,I):
U =
⊕
m∈Z≥0
U(m), L(0)|U(m) ≡ (m+ 1) Id .
The lowest component U(0) ∼= E. Moreover, U is a non-split extension of the Verma module
V H(1, 0) by the simple highest weight module LH(1, 0):
0→ LH(1, 0)→ U → V H(1, 0)→ 0.
Proof. By construction U is a graded quotient of a Z≥0-graded LH(cL, cL,I)-module Ẽ. The
lowest component is U(0) ∼= E. Submodule U(H).v0 is isomorphic to LH(1, 0), and the projective
image of v1 generates the Verma module V H(1, 0) since I(0)v1 = v0. For the same reason, this
exact sequence does not split. �
Now we consider LH(cL, cL,I)-module
Vext := LH(cL, cL,I)⊕ U .
By using [16, Theorem 4.8.1] (see also [3, 17]) we have that Vext has the structure of a vertex
operator algebra with vertex operator map Yext defined as follows:
Yext(a1 + w1, z)(a2 + w2) = Y (a1, z)(a2 + w2) + ezL(−1)Y (a2,−z)w1,
where a1, a2 ∈ LH(cL, cL,I), w1, w2 ∈ U .
12 D. Adamović and G. Radobolja
Take now vi ∈ E ⊂ U , i = 0, 1 as above and define
Si(z) = Yext
(
vi, z
)
=
∑
n∈Z
Si(n)z−n−1.
By construction
S1(z) ∈ End
(
LH(cL, cL,I), LH(1, 0)
)
((z)).
Proposition 4.2. For all n,m ∈ Z we have:
[L(n), Si(m)] = −mSi(n+m), i = 0, 1,
[W (n), S0(m)] = 0, [W (n), S1(m)] = 2mcL,IS0(n+m).
In particular, S0(0) and S1(0) are screening operators. Moreover,
S1 = S1(0) : LH(cL, cL,I)→ LH(1, 0)
is nontrivial and S1(0)I(−1)1 = −v0.
Proof. Since L(k)vi = δk,0v
i for k ≥ 0, commutator formula gives that
[L(n), Si(m)] = −mSi(n+m).
Next we calculate [W (n), S1(m)]. We have
W (−1)v1 = 2I(−1)v0 = −2cL,IL(−1)v0,
W (0)v1 = −2cL,Iv
0, W (n)v1 = 0, n ≥ 0.
This implies that
[W (n), S1(m)] = 2cL,ImS0(n+m).
Since W (n)v0 = 0 for n ≥ −1 we get
[W (n), S0(m)] = 0.
Therefore we have proved that Si(0), i = 0, 1 are screening operators. Next we have
S1(0)I(−1)1 = Resz Yext
(
v1, z
)
I(−1) = Resz e
zL(−1)Y (I(−1)1,−z)v1 = −v0.
The proof follows. �
Theorem 4.3. S1 is a derivation of the vertex algebra Vext and we have
KerLH(cL,cL,I) S1
∼= LW (2,2)(cL, cW ).
Proof. By construction S1 = Resz Yext(v
1, z), so S1 is a derivation so W = KerLH(cL,cL,I) S1 is
a vertex subalgebra of LH(cL, cL,I). Since
S1L(−2)1 = S1W (−2)1 = 0
we have that LW (2,2)(cL, cW ) ⊂ W . Since S1I(−1)1 6= 0, we have that I(−1)1 does not belong
to W . By using the fact that LH(cL, cL,I) is as W (2, 2)-module generated by singular vector 1
and cosingular vector I(−1)1 (see Example 3.9) we get that W = LW (2,2)(cL, cW ). The proof
follows. �
Remark 4.4. Of course, every Vext-module becomes a W (2, 2)-module with screening opera-
tor S1. Similar statement holds for intertwining operators. Constructions of such modules and
intertwining operators require different techniques which we will present in our forthcoming
paper [5].
On Free Field Realizations of W (2, 2)-Modules 13
Acknowledgements
The authors are partially supported by the Croatian Science Foundation under the project 2634
and by the Croatian Scientific Centre of Excellence QuantixLie.
References
[1] Adamović D., Classification of irreducible modules of certain subalgebras of free boson vertex algebra,
J. Algebra 270 (2003), 115–132, math.QA/0207155.
[2] Adamović D., Milas A., Logarithmic intertwining operators and W (2, 2p − 1) algebras, J. Math. Phys. 48
(2007), 073503, 20 pages, math.QA/0702081.
[3] Adamović D., Milas A., Lattice construction of logarithmic modules for certain vertex algebras, Selecta
Math. (N.S.) 15 (2009), 535–561, arXiv:0902.3417.
[4] Adamović D., Radobolja G., Free field realization of the twisted Heisenberg–Virasoro algebra at level zero
and its applications, J. Pure Appl. Algebra 219 (2015), 4322–4342, arXiv:1405.1707.
[5] Adamović D., Radobolja G., Self-dual and logarithmic representations of the twisted Heisenberg–Virasoro
algebra at level zero, in preparation.
[6] Arbarello E., De Concini C., Kac V.G., Procesi C., Moduli spaces of curves and representation theory,
Comm. Math. Phys. 117 (1988), 1–36.
[7] Billig Y., Energy-momentum tensor for the toroidal Lie algebra, math.RT/0201313.
[8] Billig Y., Representations of the twisted Heisenberg–Virasoro algebra at level zero, Canad. Math. Bull. 46
(2003), 529–537, math.RT/0201314.
[9] Billig Y., A category of modules for the full toroidal Lie algebra, Int. Math. Res. Not. 2006 (2006), 68395,
46 pages, math.RT/0509368.
[10] Creutzig T., Milas A., False theta functions and the Verlinde formula, Adv. Math. 262 (2014), 520–545,
arXiv:1309.6037.
[11] Feigin B.L., Fuchs D.B., Representations of the Virasoro algebra, in Representation of Lie Groups and
Related Topics, Adv. Stud. Contemp. Math., Vol. 7, Gordon and Breach, New York, 1990, 465–554.
[12] Frenkel I.B., Huang Y.-Z., Lepowsky J., On axiomatic approaches to vertex operator algebras and modules,
Mem. Amer. Math. Soc. 104 (1993), viii+64 pages.
[13] Gainutdinov A.M., Semikhatov A.M., Tipunin I.Yu., Feigin B.L., The Kazhdan–Lusztig correspondence for
the representation category of the triplet W -algebra in logorithmic conformal field theories, Theoret. Math.
Phys. 148 (2006), 1210–1235, math.QA/0512621.
[14] Jiang W., Pei Y., On the structure of Verma modules over the W -algebra W (2, 2), J. Math. Phys. 51 (2010),
022303, 8 pages.
[15] Jiang W., Zhang W., Verma modules over the W (2, 2) algebras, J. Geom. Phys. 98 (2015), 118–127.
[16] Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progress in Mathe-
matics, Vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004.
[17] Li H.S., Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 (1994),
279–297.
[18] Liu D., Zhu L., Classification of Harish-Chandra modules over the W -algebra W (2, 2), arXiv:0801.2601.
[19] Radobolja G., Subsingular vectors in Verma modules, and tensor product of weight modules over the
twisted Heisenberg–Virasoro algebra and W (2, 2) algebra, J. Math. Phys. 54 (2013), 071701, 24 pages,
arXiv:1302.0801.
[20] Zhang W., Dong C., W -algebra W (2, 2) and the vertex operator algebra L( 1
2
, 0) ⊗ L( 1
2
, 0), Comm. Math.
Phys. 285 (2009), 991–1004, arXiv:0711.4624.
http://dx.doi.org/10.1016/j.jalgebra.2003.07.011
http://arxiv.org/abs/math.QA/0207155
http://dx.doi.org/10.1063/1.2747725
http://arxiv.org/abs/math.QA/0702081
http://dx.doi.org/10.1007/s00029-009-0009-z
http://dx.doi.org/10.1007/s00029-009-0009-z
http://arxiv.org/abs/0902.3417
http://dx.doi.org/10.1016/j.jpaa.2015.02.019
http://arxiv.org/abs/1405.1707
http://dx.doi.org/10.1007/BF01228409
http://arxiv.org/abs/math.RT/0201313
http://dx.doi.org/10.4153/CMB-2003-050-8
http://arxiv.org/abs/math.RT/0201314
http://dx.doi.org/10.1155/IMRN/2006/68395
http://arxiv.org/abs/math.RT/0509368
http://dx.doi.org/10.1016/j.aim.2014.05.018
http://arxiv.org/abs/1309.6037
http://dx.doi.org/10.1090/memo/0494
http://dx.doi.org/10.1007/s11232-006-0113-6
http://dx.doi.org/10.1007/s11232-006-0113-6
http://arxiv.org/abs/math.QA/0512621
http://dx.doi.org/10.1063/1.3290646
http://dx.doi.org/10.1016/j.geomphys.2015.07.029
http://dx.doi.org/10.1007/978-0-8176-8186-9
http://dx.doi.org/10.1007/978-0-8176-8186-9
http://dx.doi.org/10.1016/0022-4049(94)90104-X
http://arxiv.org/abs/0801.2601
http://dx.doi.org/10.1063/1.4813439
http://arxiv.org/abs/1302.0801
http://dx.doi.org/10.1007/s00220-008-0562-x
http://dx.doi.org/10.1007/s00220-008-0562-x
http://arxiv.org/abs/0711.4624
1 Introduction
2 Lie algebra W(2,2) and the twisted Heisenberg–Virasoro Lie algebra at level zero
3 Irreducible highest weight modules
4 Screening operators and W(2,2)-algebra
References
|
| id | nasplib_isofts_kiev_ua-123456789-148548 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T17:57:06Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Adamović, D. Radobolja, G. 2019-02-18T15:11:09Z 2019-02-18T15:11:09Z 2016 On Free Field Realizations of W(2,2)-Modules / D. Adamović, G. Radobolja // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 20 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B69; 17B67; 17B68; 81R10 DOI:10.3842/SIGMA.2016.113 https://nasplib.isofts.kiev.ua/handle/123456789/148548 The aim of the paper is to study modules for the twisted Heisenberg-Virasoro algebra H at level zero as modules for the W(2,2)-algebra by using construction from [J. Pure Appl. Algebra 219 (2015), 4322-4342, arXiv:1405.1707]. We prove that the irreducible highest weight H-module is irreducible as W(2,2)-module if and only if it has a typical highest weight. Finally, we construct a screening operator acting on the Heisenberg-Virasoro vertex algebra whose kernel is exactly W(2,2) vertex algebra. The authors are partially supported by the Croatian Science Foundation under the project 2634
 and by the Croatian Scientific Centre of Excellence QuantixLie. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Free Field Realizations of W(2,2)-Modules Article published earlier |
| spellingShingle | On Free Field Realizations of W(2,2)-Modules Adamović, D. Radobolja, G. |
| title | On Free Field Realizations of W(2,2)-Modules |
| title_full | On Free Field Realizations of W(2,2)-Modules |
| title_fullStr | On Free Field Realizations of W(2,2)-Modules |
| title_full_unstemmed | On Free Field Realizations of W(2,2)-Modules |
| title_short | On Free Field Realizations of W(2,2)-Modules |
| title_sort | on free field realizations of w(2,2)-modules |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148548 |
| work_keys_str_mv | AT adamovicd onfreefieldrealizationsofw22modules AT radoboljag onfreefieldrealizationsofw22modules |