The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities
UDC 515.1 Over the $(1,2)$-dimensional supercircle, we investigate the second cohomology space associated the lie superalgebra $\mathcal{K}(2)$ of vector fields on the supercircle $S^{1|2}$ with coefficients in the space of weighted densities. We explicitly give 2-cocycle spanning these cohomology s...
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Institute of Mathematics, NAS of Ukraine
2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512232231665664 |
|---|---|
| author | Basdouri, O. Braghtha , A. Hammami , S. O. A. S. Basdouri, O. Braghtha , A. Hammami , S. |
| author_facet | Basdouri, O. Braghtha , A. Hammami , S. O. A. S. Basdouri, O. Braghtha , A. Hammami , S. |
| author_sort | Basdouri, O. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:49:43Z |
| description | UDC 515.1
Over the $(1,2)$-dimensional supercircle, we investigate the second cohomology space associated the lie superalgebra $\mathcal{K}(2)$ of vector fields on the supercircle $S^{1|2}$ with coefficients in the space of weighted densities. We explicitly give 2-cocycle spanning these cohomology spaces.
  |
| doi_str_mv | 10.37863/umzh.v72i10.6030 |
| first_indexed | 2026-03-24T03:25:31Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i10.6030
UDC 515.1
O. Basdouri (Dep. Math., Faculté Sci. Gafsa, Tunisia),
A. Braghtha (Univ. Bourgogne, Inst. Math. Bourgogne, France),
S. Hammami (Univ. Sfax, Tunisia)
THE SECOND COHOMOLOGY SPACES \bfscrK (\bftwo ) WITH COEFFICIENTS
IN THE SUPERSPACE OF WEIGHTED DENSITIES
ПРОСТОРИ ДРУГОЇ КОГОМОЛОГIЇ \bfscrK (\bftwo ) З КОЕФIЦIЄНТАМИ,
ЩО НАЛЕЖАТЬ ДО СУПЕРПРОСТОРУ ЗВАЖЕНИХ ЩIЛЬНОСТЕЙ
Over the (1, 2)-dimensional supercircle, we investigate the second cohomology space associated the lie superalgebra \scrK (2)
of vector fields on the supercircle S1| 2 with coefficients in the space of weighted densities. We explicitly give 2-cocycle
spanning these cohomology spaces.
Над (1, 2)-вимiрним суперколом вивчаються простори другої когомологiї, якi пов’язанi з супералгеброю Лi \scrK (2)
векторних полiв на суперколi S1| 2 з коефiцiєнтами у просторi зважених щiльностей. Ми явно отримали 2-коцикл,
що охоплює цi простори когомологiї.
1. Introduction. Let g be a Lie algebra and M a g-module. We shall associate a cochain complex
known as the Chevalley – Eilenberg differential. The nth space of this complex will be denoted by
Cn(g, M).
It is trivial if n < 0, and if n > 0, it is the space of n-linear antisymmetric mappings of g into
M : they will be called n-cochains of g with coefficients in M. The space of 0-cochains C0(g,M)
reduces to M. The differential \delta n will be defined by the following formula: for c \in Cn(g, ), the
(n+ 1)-cochain \delta n(c) evaluated on g1, g2, . . . , gn+1 \in g gives
\delta nc(g1, . . . , gn+1) =
\sum
1\leq s<t\leq n+1
( - 1)s+t - 1c ([gs, gt], g1, . . . , \^gs, . . . , \^gt, . . . , gq+1)+
+
\sum
1\leq s\leq n+1
( - 1)sgsc (g1, . . . , \^gs, . . . , gn+1) ,
the notation \^gi indicates that the ith term is omitted.
We check that \delta n+1 \circ \delta n = 0, so we have a complex
0 \rightarrow C0(g,M) \rightarrow . . . \rightarrow Cn - 1(g,M)
dn - 1
\rightarrow Cn(g,M) \rightarrow . . . .
We note by Hn(g,M) = \mathrm{k}\mathrm{e}\mathrm{r} dn/\mathrm{I}\mathrm{m}dn - 1 the quotient space. This space is called the space of
n-cohomology from g with coefficients in M. We denoted by:
Zn(g,M) = \mathrm{k}\mathrm{e}\mathrm{r} \delta n : the space of n-cocycles,
Bn(g,M) = \Im \delta n - 1 : the space of n-coboundaries.
For M = \BbbR (or \BbbC ) considered as a trivial module, we denote the cohomology in this case,
Hn(g).
We shall now recall classical interpretations of cohomology spaces of low degrees:
c\bigcirc O. BASDOURI, A. BRAGHTHA, S. HAMMAMI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10 1323
1324 O. BASDOURI, A. BRAGHTHA, S. HAMMAMI
The space H0(g,M) \simeq \mathrm{I}\mathrm{n}\mathrm{v}g(M) := \{ m \in M ; X.m = 0 \forall X \in g, \} .
The space H1(g,M) classifies derivations of g with values in M modulo inner ones. This result
is particularly useful when M = g with the adjoint representation. In this case, a derivation is a map
\varrho : g - \rightarrow g such that
\varrho ([X,Y ]) - [\varrho (X), Y ] - [X, \varrho (Y )] = 0,
while an inner derivation is given by the adjoint action of some element Z \in g.
The space H2(g,M) classifies extensions of Lie algebra g by M, i.e., short exact sequences of
Lie algebras
0 \rightarrow M \rightarrow \^g \rightarrow g \rightarrow 0,
in which M is considered as an Abelian Lie algebra. We shall mainly consider two particular cases
of this situation which will be extensively studied in the sequel:
If M is a trivial g-module (typically M = \BbbR or \BbbC ), H2(g,M) classifies central extensions
modulo trivial ones. Recall that a central extension of g by \BbbR produces a new Lie bracket on
\^g = g\oplus M by setting that
[(X,\lambda ), (Y, \mu )] = ([X,Y ], c(X,Y )).
It is trivial if the cocycle c = dl is a coboundary of a 1-cochain l, in which case the map
(X,\lambda ) \rightarrow (X,\lambda - l(X)) yields a Lie isomorphism between \^g and g\oplus M considered as a direct sum
of Lie algebras.
If M = g with the adjoint representation, then H2(g, g) classifies infinitesimal deformations
modulo trivial ones. By definition, a (formal) series
(X,Y ) \rightarrow \Phi \lambda (X,Y ) := [X,Y ] + \lambda f1(X,Y ) + \lambda 2f2(X,Y ) + . . .
is a deformation of Lie bracket [, ] if \Phi \lambda is a Lie bracket for every \lambda , i.e., is an antisymmetric bilinear
form in X, Y and satisfies the Jacobi identity. If one sets simply that
[X,Y ]\lambda = [X,Y ] + \lambda c(X,Y ),
c being a 2-cochain with values in g and \lambda being a scalar, then this bracket satisfies Jacobi identity
modulo terms of order O
\bigl(
\lambda 2
\bigr)
if and only if c is a 2-cocycle. Thus, one gets what is called
an infinitesimal deformation of the bracket of g, which is trivial if c is a coboundary, by which
we mean ( as in the case of central extensions) that an adequate linear isomorphism from g to
g transforms the initial bracket [, ] into the deformed bracket [, ]\lambda . The infinitesimal deformation
associated to a cocycle c does not always give rise to an actual deformation coinciding with the
infinitesimal deformation to order 1, i.e., such that f1 = c, as one may check by looking inductively
for functions f2, f3, . . . which satisfy Jacobi’s identity to order 2, 3, . . . Cohomological obstructons
to prolongations of deformations are contained in H3(g, g).
The natural generalization of the Virasoro algebra is given by extensions of the Lie algebra
vect(S1) of the vector fields on the circle by modules \scrF \lambda of \lambda -densities on the circle. The
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
THE SECOND COHOMOLOGY SPACES \scrK (2) WITH COEFFICIENTS . . . 1325
problem of classifying such extensions is equivalent to that of the calculation of the cohomo-
logy H2
\bigl(
\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}
\bigl(
S1
\bigr)
;\scrF \lambda
\bigr)
. In [4, 5], V. Ovsienko, C. Roger and P. Marcel, calculated the space
\mathrm{H}2
\bigl(
\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}
\bigl(
S1
\bigr)
;\scrF \lambda
\bigr)
and where \mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}
\bigl(
S1
\bigr)
is the algebra of smooth vector field on the circle S1 and
\scrF \lambda is the space of \lambda densities. Following V. Ovsienko and C. Roger, B. Agrebaoui, I. Basdouri and
M. Boujelben [1] computed H2
diff
\bigl(
\scrK (1);\frakF 1
\lambda
\bigr)
, where \scrK (1) is the lie superalgebra of contact vector
fields on the supercircle S1| 1 with coefficients in the space of weighted densities.
In this paper, we explicitly compute H2
diff
\bigl(
\scrK (2);\frakF 2
\lambda
\bigr)
, where \scrK (2) is the lie superalgebra of
contact vector fields in S1| 2 with coefficients in the spaces of weighted densities \frakF 2
\lambda .
The present paper is organized as follows. After some preliminary definitions and explanation
of notation in Section 2. In Section 3, we compute the 2-cohomology space H2
diff
\bigl(
\scrK (2);\frakF 2
\lambda
\bigr)
, we
classify the extensions of a Lie superalgebra \scrK (2) by \frakF 2
\lambda .
2. Preliminaries. In this section, we recall some tools pertaining to the problem of cohomology
such as weighted densities, superfunctions, contact projective vector fields on S1| n.
2.1. Standard contact structure on \bfitS \bfone | \bfitn . Let S1| n be the supercircle with coordinates (x, \theta 1, . . .
. . . , \theta n), where x is an even indeterminate and \theta 1, . . . , \theta n are odd indeterminate: \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.
2.2. Superfunctions on \bfitS \bfone | \bfitn . We define the geometry of the superspace S1| n, where n \in \BbbN , by
describing its associative supercommutative superalgebra of superfunctions on S1| n which we denote
by C\infty (S1| 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 \bigl( S1
\bigr)
. (2.1)
Of course, even (respectively, odd) elements in C\infty \bigl( S1| n\bigr) are the functions (2.1) for which the
summation is only over even (respectively, odd) integer k. Note p(F ) the parity of a homogeneous
function F. On C\infty \bigl( S1| n\bigr) , 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),
where the superscript \prime stands for
\partial
\partial x
.
2.3. Vector fields on \bfitS \bfone | \bfitn . A vector field on S1| n is a superderivation of the associative
supercommutative superalgebra C\infty \bigl( S1| n\bigr) . In coordinates, it can be expressed as
X = f\partial x +
n\sum
i=1
gi\partial \theta i ,
where f and gi are the elements of C\infty \bigl( S1| n\bigr) .
The superspace of all vector fields on C\infty \bigl( S1| n\bigr) is a Lie superalgebra which we shall denote by
\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}
\bigl(
C\infty \bigl( S1| n\bigr) \bigr) .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1326 O. BASDOURI, A. BRAGHTHA, S. HAMMAMI
2.4. Lie superalgebra of contact vector fields on \bfitS \bfone | \bfitn . Consider the superspace \scrK (n) of contact
vector fields on S1| n. That is, \scrK (n) is the superspace of vector fields on S1| n with respect to the
1-form \alpha n. The Lie superalgebra of contact vector fields is by definition
\scrK (n) =
\Bigl\{
X \in \mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}
\Bigl(
S1| n
\Bigr) \bigm| \bigm| \bigm| there exists FX \in C\infty
\Bigl(
S1| n
\Bigr)
such that \frakL XF
(\alpha n) = F\alpha n
\Bigr\}
.
Let us define the vector fields \eta i and \eta i by \eta i = \partial \theta i + \theta i\partial x \eta i = \partial \theta i - \theta i\partial x. Then any contact vector
field on S1| n can be written in the following explicit form:
XF = F\partial x -
1
2
( - 1)p(F )
n\sum
i=1
\eta i(F )\eta i, where F \in C\infty
\Bigl(
S1| n
\Bigr)
.
The \scrK (n) acts on S1| n through
\frakL XF
(XG) = F\partial xXG + ( - 1)p(F )+1 1
2
n\sum
i=1
\eta i(F )\eta i(G).
The vector field XF has the same parity as F. The bracket in \scrK (n) can be written as
[XF , XG] = X\{ F,G\} .
The Lie superalgebra osp(2| n) is called the Lie superalgebra of the contact projective vector fields.
Thus osp(2| n) is a (n + 2| 2n)-dimensional Lie superalgebra spanned by the following contact
projective vector fields: \bigl\{
Xx, Xx2 , X1, 2X\theta i\theta j , X\theta i , Xx\theta i , i, j = 1, . . . , n
\bigr\}
.
2.5. Modules of weighted densities. Now, consider the 1-parameter action of \scrK (n) on
C\infty \bigl( S1| n\bigr) given by the rule
\frakL \lambda
XF
= XF + \lambda F \prime .
We denote this \scrK (n)-module by \frakF n
\lambda , the space of all weighted densities on S1| n of weight \lambda :
\frakF n
\lambda =
\Bigl\{
F\alpha \lambda
n | F \in C\infty (S1| n)
\Bigr\}
.
The superspace \frakF n
\lambda has \scrK (n)-module structure defined by the Lie derivative:
\frakL \lambda
XG
\Bigl(
F\alpha \lambda
n
\Bigr)
=
\bigl(
XG + \lambda G\prime \bigr) (F )\alpha n
\lambda ,
where G\prime :=
\partial G
\partial x
. Obviously, \scrK (n) is isomorphic to \frakF n
- 1 as \scrK (n)-module and
\frakF n
\lambda \simeq \frakF n - 1
\lambda \oplus \Pi
\biggl(
\frakF n - 1
\lambda + 1
2
\biggr)
,
where \Pi is the change of parity function.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
THE SECOND COHOMOLOGY SPACES \scrK (2) WITH COEFFICIENTS . . . 1327
3. Space \bfH \bftwo
\bigl(
\bfscrK (\bftwo );\bffrakF \bftwo
\bfitlambda
\bigr)
. In this paper, we study the differential cohomology spaces
\mathrm{H}2
diff
\bigl(
\scrK (2);\frakF 2
\lambda
\bigr)
. That is, we consider only cochains (XF , XG) \rightarrow \Omega (F,G)\alpha 2
\lambda where \Omega is a diffe-
rential operator.
3.1. Main theorem. The main result of this paper is the following theorem.
Theorem 3.1.
\mathrm{H}2
diff
\bigl(
\scrK (2);\frakF 2
\lambda
\bigr)
\simeq
\left\{ \BbbK , if \lambda = 0,
1
2
, 1,
3
2
, 2, 3,
0 otherwise.
The nontrivial spaces \mathrm{H}2
\bigl(
\scrK (2);\frakF 2
\lambda
\bigr)
are spanned by the following 2-cocycles:
\Omega 0(XF , XG) = (\=\eta 1(F )\=\eta 2(G) - \=\eta 2(F )\=\eta 1(G)) \theta 1\theta 2,
\Omega 1
2
(XF , XG) =
1
2
(\=\eta 1\=\eta 2(F )\=\eta 1(G) - \=\eta 1(F )\=\eta 1\=\eta 1(G)) \theta 1\theta 2,
\Omega 1(XF , XG) = (F \=\eta 1\=\eta 2(G) - \=\eta 1\=\eta 2(F )G+ \=\eta 1(F )\=\eta 2(G) + \=\eta 2(F )\=\eta 1(G)) \theta 1\theta 2,
\Omega 3
2
(XF , XG) = (\=\eta 1\=\eta 2(F )\=\eta 1(G) + \=\eta 1\=\eta 2(F )\=\eta 2(G) - \=\eta 1(F )\=\eta 1\=\eta 2(G) - \=\eta 2(F )\=\eta 1\=\eta 2(G)) \theta 1\theta 2,
\Omega 2(XF , XG) = \=\eta 1\=\eta 2
\bigl(
F \prime \bigr) \=\eta 1\=\eta 2 \bigl( G\prime \bigr) ,
\Omega 3(XF , XG) =
\Bigl(
( - 1)| F | \bigl( \=\eta 1 \bigl( F \prime \prime \bigr) \=\eta 1 \bigl( G\prime \prime \bigr) + \=\eta 2
\bigl(
F \prime \prime \bigr) \=\eta 2(G\prime \prime )
\bigr)
+
+2
\bigl(
\=\eta 1\=\eta 2
\bigl(
F \prime \bigr) \=\eta 1\=\eta 2 \bigl( G\prime \prime \bigr) - \=\eta 1\=\eta 2
\bigl(
F \prime \prime \bigr) \=\eta 1\=\eta 2 \bigl( G\prime \bigr) \bigr) \Bigr) .
Corollary 3.1.
\mathrm{H}2
diff(\scrK (2), \scrK (2)) \simeq 0. (3.1)
3.2. Relationship between \bfH \bftwo
\bfd \bfi ff
\bigl(
\bfscrK (\bftwo ),\bffrakF \bftwo
\bfitlambda
\bigr)
and \bfH \bftwo
\bfd \bfi ff
\bigl(
\bfscrK (\bfone ),\bffrakF \bfone
\bfitlambda
\bigr)
. Before proving the Theo-
rem 3.1 we present some results illustrating the relation between the cohomology space in supercircle
S1| 1 and S1| 2.
Proposition 3.1 [1].
H2
diff
\bigl(
\scrK (1);\frakF 1
\lambda
\bigr)
\simeq
\left\{
\BbbK , if \lambda = 0, 3, 5,
\BbbK 2, if \lambda =
1
2
,
3
2
,
0 otherwise.
The nontrivial space H2
\bigl(
\scrK (1);\frakF 1
\lambda
\bigr)
are spanned by the 2-cocycles:
\omega 0(XF , XG) = FG\prime - F \prime G -
\biggl(
1
4
+
3
4
( - 1)p(F )p(G)
\biggr)
\=\eta 1(F )\eta 1(G),
\omega 1
2
(XF , XG) = ( - 1)p(F )+p(G)
\bigl(
F \prime \eta 1(G
\prime ) - \eta 1(F
\prime )G\prime \bigr) \alpha 1
2
1 ,
\widetilde \omega 1
2
(XF , XG) =
\biggl(
1
2
+
1
4
\Bigl(
1 + ( - 1)p(F )p(G)
\Bigr) \biggr)
( - 1)p(F )+p(G)
\bigl(
F\eta 1(G
\prime ) - \eta 1(F
\prime )G
\bigr)
\alpha
1
2
1 ,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1328 O. BASDOURI, A. BRAGHTHA, S. HAMMAMI
\omega 3
2
(XF , XG) =
\Bigl(
\=\eta 1
\bigl(
F \prime \prime \bigr) G - ( - 1)p(F )F \prime \=\eta 1
\bigl(
G\prime \prime \bigr) \Bigr) - 1
2
\theta 1
\bigl(
\eta 1(F )\eta 1
\bigl(
G\prime \prime \bigr) + \eta 1
\bigl(
F \prime \prime \bigr) \eta 1(G)
\bigr)
\alpha
3
2
1 ,
\widetilde \omega 3
2
(XF , XG) =
\bigl(
F \prime \=\eta 1
\bigl(
G\prime \prime \bigr) - \=\eta 1
\bigl(
F \prime \prime \bigr) G\prime \bigr) \alpha 3
2
1 ,
\omega 3(XF , XG) =
\bigl(
\eta 1
\bigl(
F \prime \prime \bigr) \=\eta 1 \bigl( G\prime \prime \bigr) G\prime \bigr) \alpha 3
1,
\omega 5(XF , XG) =
\Bigl( \Bigl(
F (3)G(4)F (4)G(3)
\Bigr)
+
3
2
\Bigl(
\eta 1(F
(4))\eta 1
\Bigl(
G(2)
\Bigr)
- \eta 1
\Bigl(
F (2)
\Bigr)
\eta 1
\Bigl(
G(4)
\Bigr) \Bigr)
-
- 4\eta 1
\Bigl(
F (3)
\Bigr)
\eta 1
\Bigl(
G(3)
\Bigr) \Bigr)
\alpha 5
1.
The following lemma gives the general form of each \Omega .
Lemma 3.1. The 2-cocycle \Omega belongs to \mathrm{Z}2
\bigl(
\scrK (2),\frakF 2
\lambda
\bigr)
. Up to a coboundary, the map \Omega is
given by
\Omega (XF , XG) =
\sum
i,j,k,l
ai,j,k,l\eta
i
1\eta
j
2(F )\eta k1\eta
l
2(G)\alpha \lambda
2 ,
where ai,j,k,l depends only on \theta 1, \theta 2 and the parity of F and G.
Proof. Every differential operator \Omega can be expressed in the form
\Omega (XF , XG) =
\sum
ai,j,k,l\eta
i
1\eta
j
2(F )\eta k1\eta
l
2(G)\alpha \lambda
2 ,
where the coefficients ai,j,k,l are arbitrary function. By using the 2-cocycle equation, we can show
that
\partial
\partial x
ai,j,k,l = 0. The dependence on the parity of F and G comes from the fact that \Omega is
skew-symmetric:
ai,j,k,l(F,G) = ( - 1)\varepsilon ij(F,G)ai,j,k,l(F,G),
where
\varepsilon ij(F,G) = ij(p(F ) + 1)(p(G) + 1) + p(F )p(G) + 1.
Lemma 3.1 is proved.
Now, to prove Theorem 3.1, we also need to compute the cohomology space vanishing on \scrK (1).
We will be interested in cohomology space vanishing on \scrK (1), that is, we assume
\Omega (X,Y ) = 0, if X,Y \in \scrK (1).
Therefore, the relevant cohomology space is
\mathrm{H}2
diff
\bigl(
\scrK (2),\scrK (1),\frakF 2
\lambda
\bigr)
.
Theorem 3.2. The space
\mathrm{H}2
diff
\bigl(
\scrK (2),\scrK (1),\frakF 2
\lambda
\bigr)
\simeq
\Biggl\{
\BbbK , if \lambda = 2,
0 otherwise.
(3.2)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
THE SECOND COHOMOLOGY SPACES \scrK (2) WITH COEFFICIENTS . . . 1329
Proof. Let \Omega a 2-cocycle of \scrK (2) vanishing on \scrK (1). The expressions of \Omega are given by
Lemma 3.1. We check with “MATHEMATICA” that the 2-cocycle condition has the solution
\Omega (XF , XG) =
\Biggl\{
0 if \lambda \not = 2,
\nu \=\eta 1\=\eta 2 (F
\prime ) \=\eta 1\=\eta 2 (G
\prime )\alpha 2
2, if \lambda = 2,
where \nu is constant. Assume that the map \Omega is trivial 2-cocycle vanishing on \scrK (1). Thus, there
exists an even operator b : \scrK (2) \rightarrow \frakF 2
2, given by
b(XF ) =
\Biggl( \sum
k
\kappa k(x, \theta 1, \theta 2)\eta 1\eta 2
\Bigl(
F (k)
\Bigr)
+
\sum
l
\mu l(x, \theta 1, \theta 2)F
(l)
\Biggr)
\alpha \lambda
2 ,
where the coefficients \kappa k(x, \theta 1, \theta 2) and \mu l(x, \theta 1, \theta 2) are arbitrary such that \Omega is equal to \delta (b), that is
\Omega (XF , XG) := ( - 1)p(XF )p(b)\frakL 2
XF
(b(XG)) -
- ( - 1)p(XG)(p(XF ))\frakL 2
XG
(b(XF )) - b([XF , XG]). (3.3)
The condition (3.3) implies that its coefficients are constant.
We check with “MATHEMATICA” that the condition (3.3) has no solution. We can see that
the expression (3.2) never appears on the right-hand side of (3.3). This is a contradiction with our
assumption.
Theorem 3.2 is proved.
Proof of Theorem 3.1. Consider a 2-cocycles \Omega \in Z2
diff
\bigl(
\scrK (2);\frakF 2
\lambda
\bigr)
. If \Omega | \scrK (1)\otimes \scrK (1) is trivial then
the 2-cocycle \Omega is completely described by Theorem 3.2. Thus, assume that \Omega | \scrK (1)\otimes \scrK (1) is nontrivial.
Of course, by considering Proposition 3.1, we deduce that nontrivial space \mathrm{H}2
diff
\bigl(
\scrK (2);\frakF 2
\lambda
\bigr)
only can
appear if \lambda \in
\biggl\{
- 1
2
, 0,
1
2
, 1,
3
2
,
5
2
,
9
2
, 3, 5
\biggr\}
. The \scrK (1)-isomorphism:
\mathrm{H}2
diff
\bigl(
\scrK (1);\frakF 2
\lambda
\bigr)
\simeq \mathrm{H}2
diff
\bigl(
\scrK (1);\frakF 1
\lambda
\bigr)
\oplus \mathrm{H}2
diff
\Bigl(
\scrK (1);
\prod \Bigl(
\frakF 1
\lambda + 1
2
\Bigr) \Bigr)
.
Together with Proposition 3.1 that describes up to a coboundary and up to a scalar factor the restric-
tion of any 2-cocycle \Omega to \scrK (1). In inception, we consider separately the even and odd cases. Even
cohomology spaces only can appear if \lambda \in \{ 0, 1, 3, 5\} and odd cohomology spaces only can appear
if \lambda \in
\biggl\{
- 1
2
,
1
2
,
3
2
,
5
2
,
9
2
\biggr\}
.
In each case, the restriction of \Omega to \scrK (1) is a linear combination of corresponding 2-cocycles
given in Proposition 3.1. First, the operators \Omega labeled by semi-integer are odd and given by
\Omega (XF , XG) =
\sum
i,j,k,l
ai,j,k,l\eta
i
1\eta
j
2(F )\eta k1\eta
l
2(G)\alpha \lambda
2 ,
where i + j + k + l \in \{ 1, 3\} and the coefficient aijkl are arbitrary functions independent on the
variable x, but they are depending on \theta and parity of F and G. By using “MATHEMATICA”, we
will investigate the dimension of the space of operators that satisfy the 2-cocycle condition:
\delta (\Omega )(XF , XG, XH) := ( - 1)p(F )XF .\Omega (XG, XH) - ( - 1)p(G)(1+p(F ))XG.\Omega (XF , XH)+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1330 O. BASDOURI, A. BRAGHTHA, S. HAMMAMI
+( - 1)p(H)(1+p(G)+p(F ))XH .\Omega (XF , XG) - \Omega ([XF , XG], XH)+
+( - 1)p(G)p(H)\Omega ([XF , XH ], XG) - ( - 1)p(F )(p(G)+p(H))\Omega ([XG, XH ], XF ) = 0, (3.4)
where XF .\Omega (XF , XH) = \frakL \lambda
XF
(\Omega (XG, XH)) and F,G,H \in \scrC \infty \bigl( S1| 2\bigr) .
The number of variables generating any 2-cocycle is much smaller than the number of equations
coming out from the 2-cocycle condition for particular values of aijkl. We have:
For \lambda =
1
2
:
Therefore, by a direct computation, we can see that the 2-cocycle condition is always satisfied
with particular values:
a1000 = 0, a0100 = 0, a0010 = 0, a0001 = 0,
a1110 =
1
2
\theta 1\theta 2, a1101 = 0, a0111 = 0, a1011 = - 1
2
\theta 1\theta 2.
We will study all trivial 2-cocycles, namely, operators of the form \delta b, where b is a linear operator
given by
b(XF ) = (\kappa \eta 1\eta 2(F ) + \mu F )\alpha \lambda
2 .
A direct computation proves that
\delta b(XF , XG) =
1
2
\Bigl(
\kappa
\bigl(
3f1(x)g1(x) + 3f2(x)g2(x) + g0(x)f
\prime
0(x) - f0(x)g
\prime
0(x)
\bigr)
-
- \kappa \theta 1
\bigl(
3f12(x)g2(x) - 3f2(x)g12(x) - 3g1(x)f
\prime
0(x) + 3g0(x)f
\prime
1(x) + f1(x)g
\prime
0(x) - f0(x)g
\prime
1(x)
\bigr)
-
- \kappa \theta 2
\bigl(
3f12(x)g1(x) - 3f1(x)g12(x) - 6g2(x)f
\prime
0(x) - g0(x)f
\prime
2(x) + 6f2(x)g
\prime
0(x) - f0(x)g
\prime
2(x)
\bigr)
+
+\kappa \theta 1\theta 2
\bigl(
g12(x)f
\prime
0(x) + 2g2(x)f
\prime
1(x) - 2g1(x)f
\prime
2(x)+
+g0(x)f
\prime
12(x) - f12(x)g
\prime
0(x) + 2f2(x)g
\prime
1(x) - 2f1(x)g
\prime
2(x) - f0(x)g
\prime
12(x)
\bigr)
+
+\mu
\bigl(
3g12f
\prime
0(x) + 2g2(x)f
\prime
1(x) - 2g1(x)f
\prime
2(x) - 3f12(x)g
\prime
0 + 4f2(x)g
\prime
1(x) - 4f1(x)g
\prime
2(x)
\bigr)
+
+\mu \theta 1
\bigl(
- g12(x)f
\prime
1(x) + 2g1(x)f
\prime
12(x) + f \prime
2(x)g
\prime
0(x) -
- f12(x)g
\prime
1(x) - f \prime
0(x)g
\prime
2(x) - 4f1(x)g
\prime
12(x) - 4g2(x)f
\prime \prime
0 (x) + 4f2(x)g
\prime \prime
0(x)
\bigr)
+
+\mu \theta 2
\bigl(
- g12(x)f
\prime
2(x) + 4g2(x)f
\prime
12(x) - f \prime
1(x)g
\prime
0(x) + f \prime
0(x)g
\prime
1(x)+
+3f12(x)g
\prime
2(x) - 4f1(x)g
\prime
12(x) + 2g1(x)f
\prime \prime
0 (x) - 2f2(x)g
\prime \prime
12(x)
\bigr)
+
+\mu \theta 1\theta 2
\bigl(
- g12(x)f
\prime
12(x) - 2f \prime
1(x)g
\prime
1(x) - 2f \prime
2(x)g
\prime
2(x) + f12(x)g
\prime
12(x) + g\prime 0(x)f
\prime \prime
0 (x)+
+2g1(x)f
\prime \prime
1 (x) - g2(x)f
\prime \prime
2 (x) - f \prime
0(x)g
\prime \prime
0(x) - 2f1(x)g
\prime \prime
1(x) - 2f2(x)g
\prime \prime
2(x)
\bigr) \Bigr)
.
It is now easy to check that the equation \Omega - \delta b = 0 has no solutions. So the 2-cocycle is nontrivial
and \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{H}2
diff(\scrK (2);\frakF 2
\lambda ) = \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{Z}2
diff
\bigl(
\scrK (2);\frakF 2
\lambda
\bigr)
. Hence, the cohomology space is one-dimensional.
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THE SECOND COHOMOLOGY SPACES \scrK (2) WITH COEFFICIENTS . . . 1331
For \lambda =
3
2
:
Therefore, by a direct computation, we can see that the 2-cocycle condition is always satisfied
with particular values:
a1000 = 0, a0100 = 0, a0010 = 0, a0001 = 0,
a1110 = \theta 1\theta 2, a1101 = \theta 1\theta 2, a0111 = - \theta 1\theta 2, a1011 = - \theta 1\theta 2.
Let us study the triviality of this 2-cocycle. We can see that any coboundary \delta b(XF , XG) can be
expressed as follows:
\delta b(XF , XG) = \kappa
\biggl(
3
2
f1(x)g1(x) +
3
2
f2(x)g2(x) +
3
2
g0(x)f
\prime
0(x) -
3
2
f0(x)g
\prime
0(x)
\biggr)
+
+\kappa \theta 1
\biggl(
- 1
2
f12(x)g2(x) +
3
2
f2(x)g12(x) + 4g1(x)f0I(x) -
3
2
g0(x)f1I(x)+
+4f1(x)g0I(x) +
3
2
f0(x)g1I(x)
\biggr)
\kappa \theta 2
\biggl(
- 1
2
f12(x)g1(x) +
3
2
f1(x)g12(x) + 4g2(x)f
\prime
0(x)+
+
3
2
g0(x)f
\prime
2(x) - 4f2(x)g
\prime
0(x) +
3
2
f0(x)g
\prime
2(x)
\biggr)
+ \kappa \theta 1\theta 2
\biggl(
3
2
g12(x)f
\prime
0(x) +
3
2
g0(x)f
\prime
12(x)+
+
3
2
f12(x)g
\prime
0(x) + f2(x)g
\prime
1(x) - f1(x)g
\prime
2(x) -
3
2
f0(x)g
\prime
12(x)
\biggr)
+
+\mu
\biggl(
5
2
g12(x)f
\prime
0(x) + g2(x)f
\prime
1(x) - g1(x)f
\prime
2(x) -
5
2
f12(x)g
\prime
0(x) + 2f2(x)g
\prime
1(x) - 2f1(x)g
\prime
2(x)
\biggr)
+
+\mu \theta 2
\Biggl(
- 3
2
g12(x)f
\prime
2(x) + 2g2(x)f
\prime
12(x) -
3
2
f \prime
1(x)g
\prime
0(x) +
3
2
f \prime
0(x)g
\prime
1(x) +
5
2
f12(x)g
\prime
2(x) -
- f2(x)g
\prime
12(x) + g1(x)f
\prime \prime
0 (x) - f1(x)g
\prime \prime
0(x)
\Biggr)
+ \mu \theta 1
\biggl(
- 3
2
g12(x)f
\prime
1(x) + g1(x)f
\prime
12(x)+
+
3
2
f \prime
2(x)g
\prime
0(x) +
3
2
f12(x)g
\prime
1(x) -
3
2
f \prime
0(x)g
\prime
2(x) - 2f1(x)g
\prime
12(x) - 2g2(x)f
\prime \prime
0 (x) + 2f2(x)g
\prime \prime
0(x)
\biggr)
+
+\mu \theta 1\theta 2
\biggl(
1
2
g12(x)f
\prime
12(x) - 3f \prime
1(x)g
\prime
1(x) - 3f \prime
2(x)g
\prime
2(x) -
1
2
f12(x)g
\prime
12(x) -
1
2
g\prime 0(x)f
\prime \prime
0 (x) -
- g1(x)f
\prime \prime
1 (x) - g2(x)f
\prime \prime
2 (x) +
1
2
f \prime
0(x)g
\prime \prime
0(x) - f1(x)g
\prime \prime
1(x) - f2(x)g
\prime \prime
2(x)
\biggr)
.
So, in the same way as before, the equation \Omega - \delta b = 0 has no solutions. So the 2-cocycle is
nontrivial and \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{H}2
diff
\bigl(
\scrK (2);\frakF 2
\lambda
\bigr)
= \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{Z}2
diff
\bigl(
\scrK (2);\frakF 2
\lambda
\bigr)
. We deduce that the cohomology space
is one-dimensional.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1332 O. BASDOURI, A. BRAGHTHA, S. HAMMAMI
For \lambda \in
\biggl\{
- 1
2
,
5
2
,
9
2
\biggr\}
, the equation (3.4) has no solutions. Then
\mathrm{H}2
diff
\bigl(
\scrK (2),\frakF 2
\lambda
\bigr)
\simeq 0.
By applying the 2-cocycles equation to \Omega , using “MATHEMATICA”, we deduce the expressions
of \Omega . To be more precise, we get
\Omega =
\left\{
1
2
(\=\eta 1\=\eta 2(F )\=\eta 1(G) - \=\eta 1(F )\=\eta 1\=\eta 1(G)) \theta 1\theta 2, if \lambda =
1
2
,
(\=\eta 1\=\eta 2(F )\=\eta 1(G) + \=\eta 1\=\eta 2(F )\=\eta 2(G) - \=\eta 1(F )\=\eta 1\=\eta 2(G) - \=\eta 2(F )\=\eta 1\=\eta 2(G)) \theta 1\theta 2, if \lambda =
3
2
.
Next, the proof here is the same as in odd 2-cocycle. The operators \Omega labeled by integer are even
and given by
\Omega (XF , XG) =
\sum
i,j,k,l
ai,j,k,l\eta
i
1\eta
j
2(F )\eta k1\eta
l
2(G)\alpha \lambda
2 ,
where i + j + k + l \in \{ 0, 2, 4\} and the coefficient aijkl are arbitrary functions independent on the
variable x, but they are depending on \theta and the parity of F and G.
Using “MATHEMATICA”, this map satisfies the 2-cocycles equation
\delta (\Omega )(XF , XG, XH) := XF .\Omega (XG, XH) - ( - 1)p(G)p(F )XG.\Omega (XF , XH)+
+( - 1)p(H)(p(G)+p(F ))XH .\Omega (XF , XG) - \Omega ([XF , XG], XH)+
+( - 1)p(G)p(H)\Omega ([XF , XH ], XG) - ( - 1)p(F )(p(G)+p(H))\Omega ([XG, XH ], XF ) = 0, (3.5)
where F,G,H \in \scrC \infty \bigl( S1| 2\bigr) .
For \lambda = 0:
Therefore, by a direct computation, we can see that the 2-cocycle condition is always satisfied
with particular values:
a0000 = 0, a1100 = 0, a0011 = 0, a1001 = \theta 1\theta 2,
a0110 = - \theta 1\theta 2, a1010 = 0, a0101 = 0, a0111 = 0, a1111 = 0.
On the other hand, we can see that the coboundary \delta b(XF , XG) can be expressed as follows:
\delta b(XF , XG) = \kappa
\biggl(
- 1
2
f1(x)g1(x) -
1
2
f2(x)g2(x)
\biggr)
+
+\kappa \theta 1
\biggl(
1
2
f12(x)g2(x) +
1
2
f2(x)g12(x) -
1
2
g1(x)f
\prime
0(x) +
1
2
f1(x)g
\prime
0(x)
\biggr)
+
+\kappa \theta 2
\biggl(
1
2
f12(x)g1(x) +
1
2
f1(x)g12(x) -
1
2
g2(x)f
\prime
0(x) +
1
2
f2(x)g
\prime
0(x)
\biggr)
+
+\kappa \theta 1\theta 2
\biggl(
- 1
2
g2(x)f
\prime
1(x) +
1
2
g1(x)f
\prime
2(x) +
1
2
f2(x)g
\prime
1(x) -
1
2
f1(x)g
\prime
2(x)
\biggr)
+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
THE SECOND COHOMOLOGY SPACES \scrK (2) WITH COEFFICIENTS . . . 1333
+\mu
\bigl(
g12(x)f
\prime
0(x) - f12(x)g
\prime
0(x) + f2(x)g
\prime
1(x) - f1(x)g
\prime
2(x)
\bigr)
+
+\mu \theta 2
\bigl(
- g2(x)f
\prime
12(x) + f12(x)g
\prime
2(x) + 2f2(x)g
\prime
12(x) + 2g1(x)f
\prime \prime
0 (x) - 2f1(x)g
\prime \prime
0(x)
\bigr)
+
+\mu \theta 1
\bigl(
- g1(x)f
\prime
12(x) - f12(x)g
\prime
1(x) + f1(x)g
\prime
12(x) - f \prime
1(x)g12(x) - g2(x)f
\prime \prime
0 (x) + f2(x)g
\prime \prime
0(x)
\bigr)
+
+\mu \theta 1\theta 2
\bigl(
- g12(x)f
\prime
12(x) + f \prime
1(x)g
\prime
1(x) + 2f \prime
2(x)g
\prime
2(x) + f12(x)g
\prime
12(x) + g\prime 0(x)f
\prime \prime
0 (x)+
+2g1(x)f
\prime \prime
1 (x) + 2g2(x)f
\prime \prime
2 (x) - f \prime
0(x)g
\prime \prime
0(x) + 2f1(x)g
\prime \prime
1(x) + 2f2(x)g
\prime \prime
2(x)
\bigr)
.
So, the cohomology space is one-dimensional since the equation \Omega - \delta b = 0 has no solutions.
Hence, the 2-cocycle is nontrivial and \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{H}2
diff
\bigl(
\scrK (2);\frakF 2
0
\bigr)
= \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{Z}2
diff
\bigl(
\scrK (2);\frakF 2
0
\bigr)
= 1.
For \lambda = 1:
Therefore, by a direct computation, we can see that the 2-cocycle condition is always satisfied
with particular values:
a0000 = 0, a1100 = - \theta 1\theta 2, a0011 = \theta 1\theta 2, a1001 = \theta 1\theta 2,
a0110 = \theta 1\theta 2, a1010 = 0, a0101 = 0, a1111 = 0.
By a direct computation, we get
\delta b(XF , XG) = \kappa
\biggl(
- 1
2
f1(x)g1(x) -
1
2
f2(x)g2(x) + g0(x)f
\prime
0(x) - f0(x)g
\prime
0(x)
\biggr)
+
+\kappa \theta 1
\biggl(
1
2
f12(x)g2(x) +
1
2
f2(x)g12(x) +
1
2
g1(x)f
\prime
0(x)+
+g0(x)f
\prime
1(x) -
1
2
f1(x)g
\prime
0(x) - f0(x)g
\prime
1(x)
\biggr)
+
+\kappa \theta 2
\biggl(
1
2
f12(x)g1(x) +
1
2
f1(x)g12(x) +
1
2
g2(x)f
\prime
0(x)+
+g0(x)f
\prime
2(x) -
1
2
f2(x)g
\prime
0(x) - f0(x)g
\prime
2(x)
\biggr)
+
+\kappa \theta 1\theta 2
\biggl(
g12(x)f
\prime
0(x) +
1
2
g2(x)f
\prime
1(x) -
1
2
g1(x)f
\prime
2(x) + g0(x)f
\prime
12(x) -
- f12(x)g
\prime
0(x) +
3
2
f2(x) g
\prime
1(x) -
3
2
f1(x)g
\prime
2(x) - f0(x)g12
\prime (x)
\biggr)
+
+\mu
\bigl(
2g12(x)f
\prime
0(x) - 2f12(x)g
\prime
0(x) + f2(x)g
\prime
1(x) - f1(x)g
\prime
2(x)
\bigr)
+
+\mu \theta 2
\bigl(
g12(x)f
\prime
2(x) - g2(x)f
\prime
12(x) - f \prime
1(x)g
\prime
0(x) + f \prime
0(x)g
\prime
1(x)+
+2f2(x)g
\prime
12(x) + 2g1(x)f
\prime \prime
0 (x) - 2f1(x)g
\prime \prime
0(x)
\bigr)
+ \mu \theta 1
\bigl(
g12(x)f
\prime
1(x) - g1(x)f
\prime
12(x)+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
1334 O. BASDOURI, A. BRAGHTHA, S. HAMMAMI
+f \prime
2(x)g
\prime
0(x) - 2f12(x)g
\prime
1(x) - f \prime
0(x)g
\prime
2(x) + f1(x)g
\prime
12(x) - g2(x)f
\prime \prime
0 (x) + f2(x)g
\prime \prime
0(x)
\bigr)
+
+\mu \theta 1\theta 2
\bigl(
2f \prime
1(x)g
\prime
2(x) + 2f \prime
2(x)g
\prime
1(x) + 2g1(x)f
\prime \prime
1 (x)+
+2g2(x)f
\prime \prime
2 (x) + 2f1(x)g
\prime \prime
1(x) + 2f2(x)g
\prime \prime
2(x)
\bigr)
.
Hence, we deduce that the cohomology space is one-dimensional since the equation \Omega - \delta b = 0
has no solutions. So, the 2-cocycle is nontrivial and \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{H}2
diff
\bigl(
\scrK (2);\frakF 2
1
\bigr)
= \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{Z}2
diff
\bigl(
\scrK (2);\frakF 2
1
\bigr)
.
For \lambda = 3, the equation (3.5) has a single solution \Omega . It is now easy to check that the equa-
tion \Omega - \delta b = 0 has no solutions. So the 2-cocycle is nontrivial and \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{H}2
diff
\bigl(
\scrK (2);\frakF 2
3
\bigr)
=
= \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{Z}2
diff
\bigl(
\scrK (2);\frakF 2
3
\bigr)
= 1.
For \lambda = 5, the equation (3.5) has no solutions. Then
\mathrm{H}2
diff
\bigl(
\scrK (2),\frakF 2
5
\bigr)
\simeq 0.
By using “MATHEMATICA”, that the condition of 2-cocycle has solutions, we deduce the ex-
pressions of \Omega . To be more precise, we get
\Omega =
\left\{
(\=\eta 1(F )\=\eta 2(G) - \=\eta 2(F )\=\eta 1(G)) \theta 1\theta 2, if \lambda = 0,
(F \=\eta 1\=\eta 2(G) - \=\eta 1\=\eta 2(F )G+ \=\eta 1(F )\=\eta 2(G) + \=\eta 2(F )\=\eta 1(G)) \theta 1\theta 2, if \lambda = 1,\bigl(
( - 1)| F | (M(F,G)) + 2(N(F,G))
\bigr)
, if \lambda = 3,
where
M(F,G) = \=\eta 1
\bigl(
F \prime \prime \bigr) \=\eta 1 \bigl( G\prime \prime \bigr) + \=\eta 2
\bigl(
F \prime \prime \bigr) \=\eta 2 \bigl( G\prime \prime \bigr) ,
N(F,G) = \=\eta 1\=\eta 2
\bigl(
F \prime \bigr) \=\eta 1\=\eta 2 \bigl( G\prime \prime \bigr) - \=\eta 1\=\eta 2
\bigl(
F \prime \prime \bigr) \=\eta 1\=\eta 2 \bigl( G\prime \bigr) .
Theorem 3.1 is proved.
References
1. B. Agrebaoui, I. Basdouri, M. Boujelben, The second cohomology spaces of \scrK (1) with coefficient in the superspace
of weighted densities and deformation of the superspace of symbols on S1| 1 , Submitted, hal-01699198, v1.
2. I. Basdouri, On osp(1| 2)-relative cohomology on S1| 1 , Commun. Algebra, 42, № 4, 1698 – 1710 (2014).
3. I. Basdouri, M. Ben Ammar, Cohomology of osp(1| 2) acting on linear differential operators on the supercercle S1| 1 ,
Lett. Math. Phys., 81, 239 – 251 (2007).
4. V. Ovsienko, C. Roger, Extension of Virasoro group and Virasoro algebra by modules of tensor densities on S1 ,
Funct. Anal. and Appl., 30, № 4, 290 – 291 (1996).
5. V. Ovsienko, P. Marcel, Extension of the Virasoro and Neveu – Schwartz algebras and generalized Sturm – Liouville
operators, Lett. Math. Phys., 40, 31 – 39 (1997).
Received 16.10.17,
after revision — 12.11.18
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 10
|
| id | umjimathkievua-article-6030 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:31Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/33/ef55276be48bf09c91171b86b8fe1833.pdf |
| spelling | umjimathkievua-article-60302025-03-31T08:49:43Z The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities Basdouri, O. Braghtha , A. Hammami , S. O. A. S. Basdouri, O. Braghtha , A. Hammami , S. Cohomologie Lie superalgebra weighted densities Cohomologie Lie superalgebra weighted densities UDC 515.1 Over the $(1,2)$-dimensional supercircle, we investigate the second cohomology space associated the lie superalgebra $\mathcal{K}(2)$ of vector fields on the supercircle $S^{1|2}$ with coefficients in the space of weighted densities. We explicitly give 2-cocycle spanning these cohomology spaces. &nbsp; УДК 515.1Простори другої когомологiї $\mathcal{K}(2)$ з коефiцiєнтами, що належать до суперпростору зважених щiльностей Над $(1,2)$-вимірним суперколом вивчаються простори другої когомології, які пов'язані з супералгеброю Лі $\mathcal{K}(2)$ векторних полів на суперколі $S^{1|2}$ з коефіцієнтами у просторі зважених щільностей. Ми явно отримали 2-коцикл, що охоплює ці простори когомології. Institute of Mathematics, NAS of Ukraine 2020-10-08 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6030 10.37863/umzh.v72i10.6030 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 10 (2020); 1323-1334 Український математичний журнал; Том 72 № 10 (2020); 1323-1334 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6030/8758 |
| spellingShingle | Basdouri, O. Braghtha , A. Hammami , S. O. A. S. Basdouri, O. Braghtha , A. Hammami , S. The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities |
| title | The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities |
| title_alt | The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities |
| title_full | The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities |
| title_fullStr | The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities |
| title_full_unstemmed | The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities |
| title_short | The second cohomology spaces $\mathcal{K}(2)$ with coefficents in the superspace of weighted densities |
| title_sort | second cohomology spaces $\mathcal{k}(2)$ with coefficents in the superspace of weighted densities |
| topic_facet | Cohomologie Lie superalgebra weighted densities Cohomologie Lie superalgebra weighted densities |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6030 |
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