First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators
UDC 515.12 We investigate the first cohomology space associated with the embedding of the Lie orthosymplectic superalgebra $\mathfrak{osp}(n|2)$ on the $(1,n)$-dimensional superspace $\mathbb{R}^{1|n}$ in the Lie superalgebra $ \mathcal{S}\Psi\mathcal{DO}(n)$ (for $n \geq 4$ ) of superpseudodifferen...
Gespeichert in:
| Datum: | 2022 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2022
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6052 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512258442919936 |
|---|---|
| author | Boujelben, M. Boujelben, M. |
| author_facet | Boujelben, M. Boujelben, M. |
| author_sort | Boujelben, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-10-24T09:23:09Z |
| description | UDC 515.12 We investigate the first cohomology space associated with the embedding of the Lie orthosymplectic superalgebra $\mathfrak{osp}(n|2)$ on the $(1,n)$-dimensional superspace $\mathbb{R}^{1|n}$ in the Lie superalgebra $ \mathcal{S}\Psi\mathcal{DO}(n)$ (for $n \geq 4$ ) of superpseudodifferential operators with smooth coefficients. Following Ovsienko and Roger, we give explicit expressions of the basis cocycles.This work is the simplest generalization of a result by Basdouri [First space cohomology of the orthosymplectic Lie superalgebra in the Lie superalgebra of superpseudodifferential operators, Algebras and Representation Theory, 16, 35-50 (2013)]. |
| doi_str_mv | 10.37863/umzh.v74i6.6052 |
| first_indexed | 2026-03-24T03:25:56Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i6.6052
UDC 515.12
M. Boujelben (Univ. de Sfax, Tunisia)
FIRST COHOMOLOGY SPACE OF THE ORTHOSYMPLECTIC
LIE SUPERALGEBRA osp(\bfitn | \bftwo ) IN THE LIE SUPERALGEBRA
OF SUPERPSEUDODIFFERENTIAL OPERATORS
ПРОСТIР ПЕРШОЇ КОГОМОЛОГIЇ ОРТОСИМПЛЕКТИЧНОЇ
СУПЕРАЛГЕБРИ ЛI osp(\bfitn | \bftwo ) У СУПЕРАЛГЕБРI ЛI
СУПЕРПСЕВДОДИФЕРЕНЦIАЛЬНИХ ОПЕРАТОРIВ
We investigate the first cohomology space associated with the embedding of the Lie orthosymplectic superalgebra osp(n| 2)
on the (1, n)-dimensional superspace \BbbR 1| n in the Lie superalgebra \scrS \Psi \scrD \scrO (n) (for n \geq 4 ) of superpseudodifferential
operators with smooth coefficients. Following Ovsienko and Roger, we give explicit expressions of the basis cocycles. This
work is the simplest generalization of a result by Basdouri [First space cohomology of the orthosymplectic Lie superalgebra
in the Lie superalgebra of superpseudodifferential operators, Algebras and Representation Theory, 16, 35 – 50 (2013)].
Вивчається простiр першої когомологiї, пов’язаний з вкладенням ортосимплектичної супералгебри Лi osp(n| 2) на
(1, n)-вимiрному суперпросторi \BbbR 1| n у супералгебрi Лi \scrS \Psi \scrD \scrO (n) (для n \geq 4 ) суперпсевдодиференцiальних
операторiв з гладкими коефiцiєнтами. Наслiдуючи Овсiєнка та Роджера, ми наводимо точнi вирази для базису
коциклiв. Ця робота є найпростiшим узагальненням результату Basdouri [First space cohomology of the orthosymplectic
Lie superalgebra in the Lie superalgebra of superpseudodifferential operators, Algebras and Representation Theory, 16,
35 – 50 (2013)].
1. Introduction. The procedure of contraction is opposite to deformation. This procedure is
important in physics because it explains, in terms of Lie algebras, why some theories arise as a limit
regime of more “exact” theories. Motivated by the need to relate the symmetries underlying Einstein’s
mechanics and Newtonian mechanics, Inönü and Wigner introduced the concept of contraction, which
consists in multiplying the generators of the symmetry by “contraction parameters”, such that when
these parameters reach some singularity point, one obtains a “different” Lie algebra with the same
dimension [13]. A similar procedure had been mentioned previously by Segal [16]. The method
has been generalized a few years later by Saletan [17]. Another physical example is the contraction
of the de Sitter algebras to the Poincaré algebra, in the limit of large (universe) radius. These
examples suggest that deformations are likely to be more useful than contractions in the investigation
of fundamental theories [10].
In the 1960, deformation theory of Lie algebras began with the works of Gerstenhaber and,
Nijenhuis and Richardson. Recently, multiparameter deformations of Lie (super)algebras and their
modules were intensively studied.
To study the formal and polynomial deformations of the natural embedding of the Lie algebra
vect(S1) of smooth vector fields on the circle S1 into the Lie algebra \Psi \scrD \scrO (S1) of pseudodifferential
operators, Ovsienko and Roger [15] calculate the first cohomology space \mathrm{H}1(vect(S1), \Psi \scrD \scrO (S1)),
where the action is given by the standard embedding. The graded space \mathrm{G}\mathrm{r}(\Psi \scrD \scrO (S1)) associated
with the natural filtration given by order of pseudodifferential operators coincides with \scrP the Lie –
Poisson algebra of symbols of pseudodifferential operators (formal Laurent series in the symbol \xi
c\bigcirc M. BOUJELBEN, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6 761
762 M. BOUJELBEN
of
\partial
\partial x
with coefficients in the space C\infty (S1) of smooth functions on S1). Since the \mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}(S1)-
module \scrP is isomorphic to a direct sum of \mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}(S1)-modules of tensor densities on S1, the space
\mathrm{H}1(\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}(S1), \scrP ) can be deduced from the cohomology of \mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}(S1) with coefficients in tensor
densities computed by Feigin and Fuchs [11, 12]. The method of Ovsienko and Roger goes over
from the graded space \scrP to the filtered space \Psi \scrD \scrO (S1) using the spectral sequence.
In paper [1, 2], using the same methods as in the paper [15] the authors computed
\mathrm{H}1
diff(\scrK (1),\scrS \Psi \scrD \scrO (1)) and \mathrm{H}1
diff(\scrK (2),\scrS \Psi \scrD \scrO (2)). The spaces \mathrm{H}1
diff(osp(n| 2);\scrS \scrP (n)) and
\mathrm{H}1
diff(osp(n| 2);\scrS \Psi \scrD \scrO (n)) was calculated in [4] for 0 \leq n \leq 2 and for n = 3 in [3].
In this paper, we restrict ourselves to the cases n \geq 4 and we restrict the action to the orthosymp-
lectic Lie (super)algebra osp(n| 2) and we consider the spaces \scrS \scrP (n) as osp(n| 2)-modules. We
compute the cohomology spaces \mathrm{H}1(osp(n| 2),\scrS \scrP (n)) and \mathrm{H}1 (osp(n| 2),\scrS \Psi \scrD \scrO (n)) . We show
that these cohomology spaces are nontrivial. These cohomology spaces are closely related to the
deformation theory (see, e.g., [6, 7, 9, 10, 14, 15]. These spaces arise in the classification of
infinitesimal deformations of the osp(n| 2)-modules. We hope to be able to describe in the future all
the deformations of these modules \scrS \Psi \scrD \scrO (n) .
2. Definitions and notations. 2.1. The Lie superalgebra of contact vector fields on \BbbR \bfone | \bfitn .
Let \BbbR 1| n be the superspace with coordinates (x, \theta 1, . . . , \theta n), where x is an even indeterminate and
\theta 1, . . . , \theta n are odd indeterminates: \theta i\theta j = - \theta j\theta i. This superspace is equipped with the standard
contact structure given by the distribution D = \langle \eta 1, . . . , \eta n\rangle generated by the vector fields \eta i =
= \partial \theta i - \theta i\partial x. That is, the distribution D is the kernel of the following 1-form:
\alpha n = dx+
n\sum
i=1
\theta id\theta i.
Consider the superspace C\infty (\BbbR 1| n) which is the space of functions F of the form
F =
\sum
1\leq i1<...<ik\leq n
fi1,...,ik(x)\theta i1 . . . \theta ik , where fi1,...,ik \in C\infty (\BbbR ). (2.1)
Of course, even (resp., odd) elements in C\infty (\BbbR 1| n) are the functions (2.1) for which the summation
is only over even (resp., odd) integer k. Denote by p(F ) the parity of a homogeneous function F.
On C\infty (\BbbR 1| n), we consider the contact bracket
\{ F,G\} = FG\prime - F \prime G - 1
2
( - 1)p(F )
n\sum
i=1
\eta i(F )\eta i(G), (2.2)
where the superscript \prime stands for
\partial
\partial x
. Consider the superspace \scrK (n) of contact vector fields on \BbbR 1| n.
That is, \scrK (n) is the superspace of vector fields on \BbbR 1| n preserving the distribution \langle \eta 1, . . . , \eta n\rangle :
\scrK (n) =
\bigl\{
X \in \mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}(\BbbR 1| n) | [X, \eta i] = FX\eta i for some FX \in C\infty (\BbbR 1| n)
\bigr\}
.
The Lie superalgebra \scrK (n) is spanned by the vector fields of the form
XF = F\partial x -
1
2
( - 1)p(F )
n\sum
i=1
\eta i(F )\eta i, where F \in C\infty (\BbbR 1| n).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
FIRST COHOMOLOGY SPACE OF THE ORTHOSYMPLECTIC LIE SUPERALGEBRA osp(n| 2) . . . 763
The vector field XF has the same parity as F. The bracket in \scrK (n) can be written as
[XF , XG] = X\{ F,G\} .
For every contact vector fields XF , one define a one-parameter family of first-order differential
operators on C\infty (\BbbR 1| n):
\frakL \lambda
XF
= XF + \lambda F \prime , \lambda \in \BbbR .
We easily check that
[\frakL \lambda
XF
,\frakL \lambda
XG
] = \frakL \lambda
X\{ F,G\}
.
We thus obtain a one-parameter family of \scrK (n)-modules on C\infty (\BbbR 1| n) that we denote \frakF n
\lambda , the space
of all weighted densities on C\infty (\BbbR 1| n) of weight \lambda with respect to \alpha n :
\frakF n
\lambda =
\Bigl\{
F\alpha \lambda
n | F \in C\infty (\BbbR 1| n)
\Bigr\}
.
In particular, we have \frakF 0
\lambda = \scrF \lambda . Obviously the adjoint \scrK (n)-module is isomorphic to the space of
weighted densities on C\infty (\BbbR 1| n) of weight - 1.
The orthosymplectic Lie superalgebra osp(n| 2) can be realized as a subalgebra of \scrK (n):
osp(n| 2) = Span(X1, Xx, Xx2 , Xx\theta i , X\theta i , X\theta i\theta j ), 1 \leq i, j \leq n.
We easily see that osp(n - 1| 2) is a subalgebra of osp(n| 2):
osp(n - 1| 2) =
\bigl\{
XF \in osp(n| 2) | \partial \theta nF = 0
\bigr\}
.
Note also that, for any i \in \{ 1, 2, . . . , n - 1\} , osp(n - 1| 2) is isomorphic to
osp(n - 1| 2)i =
\bigl\{
XF \in osp(n| 2) | \partial \theta iF = 0
\bigr\}
.
Therefore, the spaces of weighted densities \frakF n
\lambda are also osp(n - 1| 2)-modules. In [5], it was proved
that, as osp(n - 1| 2)-modules, we have
\frakF n
\lambda \simeq \frakF n - 1
\lambda \oplus \Pi
\biggl(
\frakF n - 1
\lambda + 1
2
\biggr)
, (2.3)
where \Pi is the change of parity operator.
As osp(n - 1| 2)i-isomorphism
osp(n| 2) \simeq osp(n - 1| 2)i \oplus \Pi (\scrH i),
where \scrH i is the subspace of \frakF i
- 1
2
spanned by \{ \theta i\alpha
- 1
2
1 , x\alpha
- 1
2
1 , \alpha
- 1
2
1 \} , where i = 1, 2, . . . , n - 1. To
be more precise, any element XF is decomposed into XF = XFi + XFn - i\theta n - i
where \partial \theta n - i
Fi =
= \partial \theta n - i
Fn - i = 0, and then XFi \in osp(n - 1| 2)i and XFn - i\theta n - i
can be identified to \Pi (Fn - i\alpha
- 1
2
1 ) \in
\in \Pi (\scrH i). Moreover, we can see easily that
[osp(n - 1| 2)i,\Pi (\scrH i)] \subset \Pi (\scrH i) and [\Pi (\scrH i),\Pi (\scrH i)] \subset osp(n - 1| 2)i.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
764 M. BOUJELBEN
2.2. Superpseudodifferential operators on \BbbR \bfone | \bfitn . The superspace of the supercommutative al-
gebra of superpseudodifferential symbols on \BbbR 1| n with its natural multiplication is spanned by the
series
\scrS \scrP (n) =
=
\left\{ F =
\infty \sum
k= - M
\sum
\varepsilon =(\varepsilon 1,...,\varepsilon n)
ak, \varepsilon (x, \theta )\xi
- k \=\theta 1
\varepsilon 1 . . . \=\theta n
\varepsilon n : ak, \varepsilon \in C\infty (\BbbR 1| n); \varepsilon i = 0, 1; M \in \BbbN
\right\} ,
where \xi corresponds to \partial x and \=\theta i corresponds to \partial \theta i (p( \=\theta i) = 1).
The space \scrS \scrP (n) has a structure of the Poisson – Lie superalgebra given by the following bracket:
\{ F, G\} =
\partial (F )
\partial \xi
\partial (G)
\partial x
- \partial (F )
\partial x
\partial (G)
\partial \xi
- ( - 1)p(F )
n\sum
i=1
\Bigl( \partial (F )
\partial \theta i
\partial (G)
\partial \=\theta i
+
\partial (F )
\partial \=\theta i
\partial (G)
\partial \theta i
\Bigr)
.
It endows \scrS \scrP (n) with a Lie superalgebra structure (still denoted \scrS \scrP (n)).
The space \scrS \scrP (n) is \BbbZ -graded where the degrees of x and \theta are equal to 0 and the degrees of \xi
and \theta are equal to 1. A homogeneous element of degree m has the following form:
Am = F0\xi
m +
n\sum
k=1
\sum
1\leq i1<...<ik\leq n
Fi1...ik\xi
m - k\theta i1 . . . \theta ik , where F0, Fi1...ik \in C\infty (\BbbR 1| n).
We will denote \scrS \scrP m(n) the space of homogeneous elements of degree - m.
This definition endows the space \scrS \scrP (n) with a \BbbZ -grading:
\scrS \scrP (n) =
\widetilde \bigoplus
m\in \BbbZ
\scrS \scrP m(n),
where
\widetilde \bigoplus
m\in \BbbZ
= (
\bigoplus
m<0
)
\bigoplus \prod
m\geq 0
and
\scrS \scrP m(n) =
=
\Bigl\{
F\xi - m +G1\xi
- m - 1\=\theta 1 +G2\xi
- m - 1\=\theta 2 + . . .+H1,2\xi
- m - 2\=\theta 1\=\theta 2 + . . . | F,Gi, Hi,j \in C\infty (\BbbR 1| n)
\Bigr\}
is the homogeneous subspace of degree - m.
The associative superalgebra of superpseudodifferential operators \scrS \Psi \scrD \scrO (n) on \BbbR 1| n has the
same underlying vector space as \scrS \scrP (n) by the multiplication is now defined by the following rule:
F \circ G =
\sum
k\geq 0, \nu i=0, 1
( - 1)\nu i(p(F )+1)
k!
(\partial k\xi \partial
\nu i
\=\theta i
F )(\partial kx\partial
\nu i
\theta i
G).
This composition rule induces the supercommutator defined by
[F, G] = F \circ G - ( - 1)p(F )p(G)G \circ F.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
FIRST COHOMOLOGY SPACE OF THE ORTHOSYMPLECTIC LIE SUPERALGEBRA osp(n| 2) . . . 765
Of course, the case n = 0 corresponds to the classical setting: \scrK (0) = \mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}(\BbbR ) and the corre-
sponding orthosymplectic Lie algebra osp(0| 2) is nothing but the classical Lie algebra sl(2), which
is isomorphic to the Lie subalgebra of \mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}(\BbbR ) generated by
sl(2) = Span
\biggl(
d
dx
, x
d
dx
, x2
d
dx
\biggr)
and \frakF 0
\lambda are the classical \lambda -densities, usually denoted
\scrF \lambda =
\Bigl\{
f(dx)\lambda | f \in C\infty (\BbbR )
\Bigr\}
.
\scrS \scrP (0) is the classical spaces of symbols, usually denoted
\scrP =
\Biggl\{
F (x, \xi ), F (x, \xi ) =
\sum
k\in \BbbZ
fk(x)\xi
k
\Biggr\}
,
and \scrS \Psi \scrD \scrO (0) is the classical associative algebra of pseudodifferential operators, usually denoted
\Psi \scrD \scrO .
2.3. The structure of \bfscrS \bfscrP (\bfitn ) as a osp(\bfitn | \bftwo )-module. The natural embedding of osp(n| 2) into
\scrS \scrP (n) defined by
\pi (XF ) = F\xi +
( - 1)p(F )+1
2
n\sum
i=1
\=\eta i(F )\=\zeta i, where \=\zeta i = \=\theta i - \theta i\xi ,
and \pi (XF ) = F\xi for n = 0, induces an osp(n| 2)-module structure on \scrS \scrP (n). Setting \mathrm{d}\mathrm{e}\mathrm{g} x =
= \mathrm{d}\mathrm{e}\mathrm{g} \theta i = 0, \mathrm{d}\mathrm{e}\mathrm{g} \xi = \mathrm{d}\mathrm{e}\mathrm{g} \=\theta i = 1 for all i.
Each element of \scrS \Psi \scrD \scrO (m) can be expressed as
A =
\sum
k\in \BbbZ
(Fk +G1
k\xi
- 1\=\theta 1 + . . .+H1,2
k \xi - 2\=\theta 1\=\theta 2 + . . .)\xi - k,
where Fk, G
i
k, H
i,j
k \in C\infty (\BbbR 1| n). We define the order of A to be
\mathrm{o}\mathrm{r}\mathrm{d}(A) = \mathrm{s}\mathrm{u}\mathrm{p}\{ k | Fk \not = 0 or Gi
k \not = 0 or H i,j
k \not = 0\} .
This definition of order equips \scrS \Psi \scrD \scrO (n) with a decreasing filtration as follows: set
\bfF m = \{ A \in \scrS \Psi \scrD \scrO (n), \mathrm{o}\mathrm{r}\mathrm{d}(A) \leq - m\} ,
where m \in \BbbZ . So, one has
. . . \subset \bfF m+1 \subset \bfF m \subset . . . .
This filtration is compatible with the multiplication and the Poisson bracket, that is, for A \in \bfF p
and B \in \bfF q, one has A \circ B \in \bfF p+q and \{ A,B\} \in \bfF p+q - 1. This filtration makes \scrS \Psi \scrD \scrO (n) an
associative filtered superalgebra. Moreover, this filtration is compatible with the natural osp(n| 2)-
action on \scrS \Psi \scrD \scrO (n). Indeed, if XF \in osp(n| 2) and A \in \bfF m, then
XF .A = [XF , A] \in \bfF m.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
766 M. BOUJELBEN
The induced osp(n| 2)-module structure on the quotient \bfF m/\bfF m+1 is isomorphic to that the osp(n| 2)-
module \scrS \scrP m(n). Therefore,
\scrS \scrP (n) \simeq \widetilde \bigoplus
m\in \BbbZ
\bfF m/\bfF m+1.
Therefore, \scrS \scrP (n) is osp(n - 1| 2)i-module.
For i \in \{ 1, 2, . . . , n - 1\} , let \frakF n - 1,i
\lambda be the osp(n - 1| 2)i-module of weighted densities of weight
\lambda on \BbbR 1| n - 1.
3. Theory of cohomology. Let us first recall some fundamental concepts from cohomology
theory (see, e.g., [12]).
Let g = g\=0 \oplus g\=1 be a Lie superalgebra acting on a superspace V = V\=0 \oplus V\=1 . The space of
k-cochains of g with values in V is the g-module
Ck(g, V ) := \mathrm{H}\mathrm{o}\mathrm{m}(\Lambda kg;V ).
The coboundary operator \delta k : Ck(g, V ) - \rightarrow Ck+1(g, V ) is a g-map satisfying \delta n \circ \delta k - 1 = 0. The
kernel of \delta k, denoted Zk(g, V ), is the space of k-cocycles, among them, the elements in the range
of \delta k - 1 are called k-coboundaries. We denote Bk(g, V ) the space of k-coboundaries. By definition,
the kth cohomology space is the quotient space
\mathrm{H}k(g, V ) = Zk(g, V )/Bk(g, V ).
We will only need the formula of \delta n (which will be simply denoted \delta ) in degrees 0 and 1: for
v \in C0(g, V ) = V, \delta v(x) := ( - 1)p(x)p(v)x \cdot v, for \Upsilon \in C1(g, V ),
\delta (\Upsilon )(x, y) := ( - 1)p(x)p(\Upsilon )x \cdot \Upsilon (y) - ( - 1)p(y)(p(x)+p(\Upsilon ))y \cdot \Upsilon (x) - \Upsilon ([x, y]).
The spaces \mathrm{H}1
diff(osp(n| 2);\scrS \scrP (n)) and \mathrm{H}1
diff(osp(n| 2);\scrS \Psi \scrD \scrO (n)) for 0 \leq n \leq 3 was calculated
in [4] for 0 \leq n \leq 2 and for n = 3 in [3].
In this paper, we study the differential cohomology spaces \mathrm{H}1(osp(n| 2),\scrS \scrP (n)) and
\mathrm{H}1(osp(n| 2),\scrS \Psi \scrD \scrO (n)) for n \geq 4.
We recall that the space \mathrm{H}1(osp(n| 2),\scrS \Psi \scrD \scrO (n)) is equal to the space \mathrm{H}1(osp(n| 2),\scrP ), and
this two spaces are spanned by the same generators.
Proposition 3.1 [2]. 1. As a osp(n - 1| 2)i-module, i \in \{ 1, 2, . . . , n - 1\} , we have
\scrS \scrP m(n) \simeq \frakF n
m \oplus \Pi (\frakF n
m+ 1
2
\oplus \frakF n
m+ 1
2
)\oplus \frakF n
m+1 for m = 0, - 1.
2. For m \not = 0, - 1:
a) the following subspace of \scrS \scrP m(n):
\scrS \scrP m, i(n) =
=
\biggl\{
B
(m,i)
F = F\theta n - i
\=\theta n - i\xi
- m - 1 + \theta n - i
\biggl(
\=\eta n - i -
1
2
\=\eta i
\biggr)
(F )\=\zeta i\=\zeta n - i\xi
- m - 2 | F \in C\infty (\BbbR 1| n - 1)
\biggr\}
is a osp(n - 1| 2)i- module, i = 1, 2, . . . , n - 1, isomorphic to \frakF n
m+1;
b) as a osp(n - 1| 2)i-module we get
\scrS \scrP m(n)/\scrS \scrP m, i(n) \simeq \frakF n
m \oplus \Pi (\frakF n
m+ 1
2
\oplus \frakF n
m+ 1
2
), i = 1, 2, . . . , n - 1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
FIRST COHOMOLOGY SPACE OF THE ORTHOSYMPLECTIC LIE SUPERALGEBRA osp(n| 2) . . . 767
To prove Proposition 3.1, we need the following result (see [2]).
The space \mathrm{H}1(osp(n| 2), \scrS \scrP (n)) inherits the grading (2.2) of \scrS \scrP (n), so it suffices to compute
it in each degree.
4. The space \bfH \bfone
\bfd \bfi ff(osp(\bfitn | \bftwo );\bfscrS \bfscrP (\bfitn )) for \bfitn \geq \bffour . In this section, we will compute the first
differential cohomology spaces \mathrm{H}1
diff(osp(n| 2);\scrS \scrP (n)) for n \geq 4. Our main result is the following
theorem.
Theorem 4.1.
\mathrm{H}1(osp(n| 2),\scrS \scrP (n)) \simeq \BbbR 4.
The nontrivial spaces \mathrm{H}1(osp(n| 2),\scrS \scrP (n)) are spanned by the cohomology classes of the 1-
cocycles \Lambda 1, \Lambda 2, \Lambda 3 and \Lambda 4 :
\Lambda 1(XF ) = F \prime ,
\Lambda 2(XF ) = F \prime \xi - 1\=\zeta 1 . . . \=\zeta n,
\Lambda 3(XF ) =
\Bigl(
\=\eta 1(F
\prime )\zeta 1 + . . .+ \=\eta n(F
\prime )\zeta n
\Bigr)
\xi - 1,
\Lambda 4(XF ) = F \prime \prime \xi - 2\=\zeta 1 . . . \=\zeta n.
We know that any element \Upsilon \in Z1(osp(n| 2),\scrS \scrP m(n)) is decomposed into \Upsilon = \Upsilon \prime +\Upsilon \prime \prime where
\Upsilon \prime \in Z1(osp(n - 1| 2)i,\scrS \scrP m(n)) and \Upsilon \prime \prime \in \mathrm{H}\mathrm{o}\mathrm{m}(\Pi (\scrH i),\scrS \scrP m(n)).
To prove the Theorem 4.1 we need first to proof the following lemma and propositions.
The first cohomology space \mathrm{H}1(osp(n| 2),\frakF n
\lambda ) was computed in [8]. The result is the following.
Theorem 4.2. The space \mathrm{H}1
diff(osp(n| 2);\frakF n
\lambda ) has the following structure:
\mathrm{H}1
diff(osp(n| 2);\frakF n
\lambda ) \simeq
\left\{
\BbbR 2, if n = 2 and \lambda = 0,
\BbbR , if
\left\{
n = 0 and \lambda = 0, 1,
n = 1 and \lambda = 0,
1
2
,
n \geq 3 and \lambda = 0,
0 otherwise.
Moreover, basis for nontrivial cohomology spaces are given in the following table:
(n, \lambda ) 1-cocycles
(n, 0) \Upsilon n
\lambda (XF ) = F \prime
(0, 1) \Upsilon 0
1(XF ) = F \prime \prime dx1\biggl(
1,
1
2
\biggr)
\Upsilon 1
1
2
(XF ) = \=\eta 1(F
\prime )\alpha
1
2
1
(2, 0) \Lambda 2
0(XF ) = \=\eta 1\=\eta 2(F )
Proposition 4.1. The space \mathrm{H}1
diff(osp(n - 1| 2)i;\frakF n
\lambda ) has the following structure:
\mathrm{H}1
diff(osp(n - 1| 2)i;\frakF n
\lambda ) \simeq
\left\{
\BbbR , if \lambda = 0,
\BbbR , if \lambda = - 1
2
,
0 otherwise.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
768 M. BOUJELBEN
Moreover, basis for nontrivial cohomology spaces are given in the following:
C1(XF ) = F \prime , if \lambda = 0,
C2(XF ) = F \prime \theta n - i, if \lambda = - 1
2
.
(4.1)
Proof. From the isomorphism (2.3), we have
\mathrm{H}1
diff(osp(n - 1| 2)i;\frakF n
\lambda ) \simeq \mathrm{H}1
diff(osp(n - 1| 2)i;\frakF n - 1
\lambda )\oplus \mathrm{H}1
diff
\biggl(
osp(n - 1| 2)i; \Pi
\biggl(
\frakF n - 1
\lambda + 1
2
\biggr) \biggr)
.
By using the Theorem 4.2, we get
\mathrm{H}1
diff(osp(n - 1| 2)i;\frakF n - 1
\lambda ) \simeq \BbbR , if \lambda = 0,
and
\mathrm{H}1
diff
\biggl(
osp(n - 1| 2)i; \Pi
\biggl(
\frakF n - 1
\lambda + 1
2
\biggr) \biggr)
\simeq \BbbR , if \lambda = - 1
2
.
Proposition 4.1 is proved.
The proof of Theorem 4.1 need the following lemma.
Lemma 4.1. The 1-cocycle \Upsilon \in \mathrm{Z}1(osp(n| 2),\scrS \scrP m(n)), m \in \BbbZ , is a coboundary if and only if
\Upsilon | osp(n - 1| 2)i
, 1 \leq i \leq n, is a coboundary.
Proof. It is easy to see that if \Upsilon is a couboundary for osp(n| 2) then \Upsilon | osp(n - 1| 2)i
is a coboundary
over \Upsilon | osp(n - 1| 2)i
, 1 \leq i \leq n. Now assume that \Upsilon | osp(n - 1| 2)i
, 1 \leq i \leq n, is a coboundary over
\Upsilon | osp(n - 1| 2)i
, 1 \leq i \leq n, that is, there exists A \in \scrS \scrP m(n) such that, for all XFi \in osp(n - 1| 2)i,
\Upsilon (XFi) = \{ XFi , A\} .
Using the condition of a 1-cocycle, we have
\Upsilon (X\theta i\theta j ) = \{ X\theta i\theta j , A\} .
We prove that \Upsilon (XF ) = \{ XF , A\} for any XF \in osp(n| 2), and, therefore, \Upsilon is a coboundary of
osp(n| 2).
4.1. Proof of Theorem 4.1. According to Lemma 4.1, the restriction of any nontrivial 1-cocycle
of osp(n| 2) with coefficients in \scrS \scrP m(n) to osp(n - 1| 2)i is a nontrivial 1-cocycle.
We see that if m \not = 0, - 1, and by Lemma 4.1, the corresponding cohomology \mathrm{H}1(osp(n| 2),
\scrS \scrP m(n)) vanishes.
If m \in \{ 0, - 1\} , from the Proposition 3.1, we have
\mathrm{H}1
diff(osp(n - 1| 2)i;\scrS \scrP m(n)) \simeq \mathrm{H}1
diff(osp(n - 1| 2)i;\frakF n
m)\oplus \mathrm{H}1
diff(osp(n - 1| 2)i; \Pi (\frakF n
m+ 1
2
) \oplus
\oplus \mathrm{H}1
diff(osp(n - 1| 2)i; \Pi (\frakF n
m+ 1
2
)\oplus \mathrm{H}1
diff(osp(n - 1| 2)i;\frakF n
m+1).
By using the Proposition 4.1, we obtain
\mathrm{H}1
diff(osp(n - 1| 2)i;\scrS \scrP m(n)) \simeq
\left\{
\BbbR 3, if m = - 1,
\BbbR , if m = 0,
0 otherwise.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
FIRST COHOMOLOGY SPACE OF THE ORTHOSYMPLECTIC LIE SUPERALGEBRA osp(n| 2) . . . 769
Case 1: m = - 1, the space \mathrm{H}1(osp(n - 1| 2)i,\scrS \scrP - 1(n)) is spanned by the following 1-cocycles:
\Phi i
1(XF ) = \psi i
- 1,1(C1(XF )),
\Phi i
2(XF ) = \psi i
- 1, 1
2
(\Pi (C2(XF ))),
\Phi i
3(XF ) = \widetilde \psi i
- 1, 1
2
(\Pi (C2(XF ))).
Case 2: m = 0, the space \mathrm{H}1(osp(n - 1| 2)i,\scrS \scrP 0(n)) is spanned by the following 1-cocycle:
\Phi i
4(XF ) = \psi i
0,0(C1(XF )),
where the cocycles C1 and C2 are defined by the formulae (4.1) and \psi i
m, j ,
\widetilde \psi i
m, j are as in [2].
Now, any a nontrivial 1-cocycle of osp(n| 2) with coefficients in \scrS \scrP m(n) can be decomposed as
\Upsilon = (\Upsilon \prime , \Upsilon \prime \prime ) and
\Upsilon \prime : osp(n - 1| 2)i - \rightarrow \scrS \scrP m(n),
\Upsilon \prime \prime : \Pi (\scrH i) - \rightarrow \scrS \scrP m(n),
where \Upsilon \prime , \Upsilon \prime \prime are linear maps.
The space \mathrm{H}1(osp(n - 1| 2)i,\scrS \scrP n(2)), i = 1, 2, . . . , n, determines the linear maps \Upsilon \prime . Then
\Upsilon \prime = \Phi i. More precisely, we get:
case 1: m = 0, \Upsilon \prime = \alpha 1\Phi
i
4,
case 2: m = - 1, \Upsilon \prime = \alpha 2\Phi
i
1 + \alpha 3\Phi
i
2 + \alpha 4\Phi
i
3, where the coefficients \alpha k are constants.
In each case, the 1-cocycle conditions determines \Upsilon \prime \prime . We obtain, for m = 0, \Upsilon 0 = \alpha 1\Lambda 1 and
m = - 1, \Upsilon - 1 = \alpha 2\Lambda 2 + \alpha 3\Lambda 3 + \alpha 4\Lambda 4.
Thus, the space \mathrm{H}1(osp(n| 2),\scrS \scrP 0(n)) is spanned by the nontrivial cocycle \Lambda 1 and the space
\mathrm{H}1(osp(n| 2),\scrS \scrP - 1(n)) is generated by the nontrivial cocycles: \Lambda 2, \Lambda 3 and \Lambda 4.
Theorem 4.1 is proved.
5. Cohomology of osp(\bfitn | \bftwo ) in \bfscrS \bfPsi \bfscrD \bfscrO (\bfitn ). 5.1. The spectral sequence for a filtered module
over a Lie superalgebra [15]. The reader should refer to [15], for details on homological algebra
used to construct spectral sequences. We will merely quote the results for a filtered module M with
decreasing filtration \{ Mn\} n\in \BbbZ over a Lie (super)algebra g so that Mn+1 \subset Mn, \cup n\in \BbbZ Mn =M and
gMn \subset Mn.
Consider the natural filtration induced on the space of cochains by setting:
Fn(C\ast (g, M)) = C\ast (g, Mn),
then we have
dFn(C\ast (g, M)) \subset Fn(C\ast (g, M)) (i.e., the filtration is preserved by d),
Fn+1(C\ast (g, M)) \subset Fn(C\ast (g, M)) (i.e., the filtration is decreasing).
Then there is a spectral sequence (E\ast ,\ast
r , dr) for r \in \BbbN with dr of degree (r, 1 - r) and
Ep,q
0 = F p(Cp+q(g, M))/F p+1(Cp+q(g, M)), Ep,q
1 = Hp+q(g, \mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}p(M)).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
770 M. BOUJELBEN
To simplify the notations, we have to replace Fn(C\ast (g, M)) by FnC\ast . We define
Zp,q
r = F pCp+q
\bigcap
d - 1(F p+rCp+q+1),
Bp,q
r = F pCp+q
\bigcap
d(F p - rCp+q - 1),
Ep,q
r = Zp,q
r /(Zp+1,q - 1
r - 1 +Bp,q
r - 1).
The differential d maps Zp,q
r into Zp+r,q - r+1
r , and hence includes a homomorphism
dr : E
p,q
r - \rightarrow Ep+r,q - r+1
r .
The spectral sequence converges to H\ast (C, d), that is,
Ep,q
\infty \simeq F pHp+q(C, d)/F p+1Hp+q(C, d),
where F pH\ast (C, d) is the image of the map H\ast (F pC, d) \rightarrow H\ast (C, d) induced by the inclusion
F pC \rightarrow C.
5.2. Computing \bfH \bfone (osp(\bfitn | \bftwo ),\bfscrS \bfPsi \bfscrD \bfscrO (\bfitn )). Now we can check the behavior of the cocycles
\Lambda 1, . . . ,\Lambda 4 under the successive differentials of the spectral sequence. The cocycle \Lambda 1 belongs to
E0.1
1 and this cocycles \Lambda 2, \Lambda 3, \Lambda 4 belong to E - 1.2
1 . Consider a cocycle in \scrS \scrP (n), by compute its
differential as if it were with values in \scrS \Psi \scrD \scrO (n) and keep the symbolic part of the result. This gives
a new cocycle of degree equal to the degree of the previous one plus one, and its class will represent
its image under d1. The higher order differentials dr can be calculated by iteration of this procedure,
the space Ep+r,q - r+1
r contains the subspace coming from \mathrm{H}p+q+1(osp(n| 2);\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}p+1(\scrS \Psi \scrD \scrO (n))).
It is now easy to see that the cocycles \Lambda 1, . . . ,\Lambda 4 will survive in the same form, we obtain the
following corollary.
Corollary 5.1.
\mathrm{H}1(osp(n| 2),\scrS \Psi \scrD \scrO (n)) \simeq \BbbR 4.
The nontrivial spaces \mathrm{H}1(osp(n| 2),\scrS \Psi \scrD \scrO (n)) are spanned by the cohomology classes of the 1-
cocycles \Delta 1, \Delta 2, \Delta 3 and \Delta 4 :
\Delta 1(XF ) = F \prime ,
\Delta 2(XF ) = F \prime \xi - 1\=\zeta 1 . . . \=\zeta n,
\Delta 3(XF ) =
\Bigl(
\=\eta 1(F
\prime )\zeta 1 + . . .+ \=\eta n(F
\prime )\zeta n
\Bigr)
\xi - 1,
\Delta 4(XF ) = F \prime \prime \xi - 2\=\zeta 1 . . . \=\zeta n.
References
1. B. Agrebaoui, N. Ben Fraj, On the cohomology of Lie superalgebra of contact vector fields on S1| 1, Bull. Soc. Roy.
Sci. Liége, 73 (2004).
2. B. Agrebaoui, N. Ben Fraj, S. Omri, On the cohomology of Lie superalgebra of contact vector fields on S1| 2, J.
Nonlinear Math. Phys., 13, № 4, 523 – 534 (2006).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
FIRST COHOMOLOGY SPACE OF THE ORTHOSYMPLECTIC LIE SUPERALGEBRA osp(n| 2) . . . 771
3. B. Agrebaoui, I. Basdouri, N. Elghomdi, S. Hammami, First space cohomology of the orthosymplectic Lie
superalgebra osp(3| 2) in the Lie superalgebra of superpseudodifferential operators, J. Pseudo-Different. Oper. and
Appl., 7, 141 – 155 (2016).
4. I. Basdouri, First space cohomology of the orthosymplectic Lie superalgebra in the Lie superalgebra of superpseudodi-
fferential operators, Algebras and Representation Theory, 16, 35 – 50 (2013); DOI: 10.1007/s10468-011-9292-4.
5. M. Ben Ammar, N. Ben Fraj, S. Omri, The binary invariant differential operators on weighted densities on the
superspace \BbbR 1| n and cohomology, J. Math. Phys., 51, № 4 (2009); DOI.10.1063/1.3355127.
6. N. Ben Fraj, S. Omri, Deforming the Lie superalgebra of contact vector fields on S1| 1, J. Nonlinear Math. Phys., 13,
№ 1, 19 – 33 (2006).
7. N. Ben Fraj, S. Omri, Deformating the Lie superalgebra of contact vector fields on S1| 2 inside the Lie superalgebra
of pseudodifferential operators on S1| 2, Theore. and Math. Phys., 163, № 2, 618 – 633 (2010).
8. N. El Gomdi and R. Messaoud, Cohomology of orthosymplectic Lie superalgebra acting on \lambda -densities on \BbbR 1| n, Int.
J. Geom. Methods Mod. Phys., 14, Issue 01 (2017).
9. A. Fialowski, An example of formal deformations of Lie algebras, Proc. NATO, Conf. Deformations Theory of
Algebras, Kluwer (1988), p. 3.
10. A. Fialowski, M. de Montigny, On deformations and contractions of Lie algebras, SIGMA, 2, Article 048 (2006).
11. B. L. Feigin, D. B. Fuks, Homology of the Lie algebra of vector fields on the line, Funct. Anal. and Appl., 14,
201 – 212 (1980).
12. D. B. Fuchs, Cohomology of infinite-dimensional Lie algebras, Plenum Publ., New York (1986).
13. E. Inonu, E. P. Wigner, On the contraction of groups and their representations, Proc. Nat. Acad. Sci. USA, 39, № 6,
510 – 524 (1953).
14. V. Ovsienko, C. Roger, Deforming the Lie algebra of vector fields on S1 inside the Lie algebra of pseudodifferential
symbols on S1, Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Amer. Math. Soc. Transl.
Ser. 2, 211 – 226 (1999).
15. V. Ovsienko, C. Roger, Deforming the Lie algebra of vector fields on S1 inside the Poisson algebra on \.T \ast S1 ,
Comm. Math. Phys., 198, 97 – 110 (1998).
16. I. E. Segal, A class of operator algebras which are determined by groups, Duke Math. J., 18, № 1, 221 – 265 (1951).
17. E. J. Saletan, Contraction of Lie groups, J. Math. Phys., 2, 1 – 21 (1961).
Received 29.03.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 6
|
| id | umjimathkievua-article-6052 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:56Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4d/42f735b79f760ffad186e6008667c24d.pdf |
| spelling | umjimathkievua-article-60522022-10-24T09:23:09Z First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators Boujelben, M. Boujelben, M. Cohomology, Orthosymplectic superalgebra, superpseudodifferential operators, Poisson superalgebra Cohomology, Orthosymplectic superalgebra, superpseudodifferential operators, Poisson superalgebra UDC 515.12 We investigate the first cohomology space associated with the embedding of the Lie orthosymplectic superalgebra $\mathfrak{osp}(n|2)$ on the $(1,n)$-dimensional superspace $\mathbb{R}^{1|n}$ in the Lie superalgebra $ \mathcal{S}\Psi\mathcal{DO}(n)$ (for $n \geq 4$ ) of superpseudodifferential operators with smooth coefficients. Following Ovsienko and Roger, we give explicit expressions of the basis cocycles.This work is the simplest generalization of a result by Basdouri [First space cohomology of the orthosymplectic Lie superalgebra in the Lie superalgebra of superpseudodifferential operators, Algebras and Representation Theory, 16, 35-50 (2013)]. УДК 515.12 Простiр першої когомологiї ортосимплектичної супералгебри Лi $\mathfrak{osp}(n|2)$ у супералгебрi Лi суперпсевдодиференцiальних операторiв Вивчається простiр першої когомологiї, пов’язаний з вкладенням ортосимплектичної супералгебри Лi $\mathfrak{osp}(n|2)$ на $(1, n)$-вимiрному суперпросторi $\mathbb{R}^{1|n}$ у супералгебрi Лi $ \mathcal{S}\Psi\mathcal{DO}(n)$ (для $n \geq 4$) суперпсевдодиференцiальних операторiв з гладкими коефiцiєнтами. Наслiдуючи Овсiєнка та Роджера, ми наводимо точнi вирази для базису коциклiв. Ця робота є найпростiшим узагальненням результату Basdouri[First space cohomology of the orthosymplectic Lie superalgebra in the Lie superalgebra of superpseudodifferential operators, Algebras and Representation Theory, 16, 35-50 (2013)]. Institute of Mathematics, NAS of Ukraine 2022-07-07 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6052 10.37863/umzh.v74i6.6052 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 6 (2022); 761 - 771 Український математичний журнал; Том 74 № 6 (2022); 761 - 771 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6052/9249 Copyright (c) 2022 Maha |
| spellingShingle | Boujelben, M. Boujelben, M. First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators |
| title | First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators |
| title_alt | First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators |
| title_full | First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators |
| title_fullStr | First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators |
| title_full_unstemmed | First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators |
| title_short | First cohomology space of the orthosymplectic Lie superalgebra $\mathfrak{osp}(n|2)$ in the Lie superalgebra of superpseudodifferential operators |
| title_sort | first cohomology space of the orthosymplectic lie superalgebra $\mathfrak{osp}(n|2)$ in the lie superalgebra of superpseudodifferential operators |
| topic_facet | Cohomology Orthosymplectic superalgebra superpseudodifferential operators Poisson superalgebra Cohomology Orthosymplectic superalgebra superpseudodifferential operators Poisson superalgebra |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6052 |
| work_keys_str_mv | AT boujelbenm firstcohomologyspaceoftheorthosymplecticliesuperalgebramathfrakospn2intheliesuperalgebraofsuperpseudodifferentialoperators AT boujelbenm firstcohomologyspaceoftheorthosymplecticliesuperalgebramathfrakospn2intheliesuperalgebraofsuperpseudodifferentialoperators |