EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 7, No. 4, 2014, 395-404
ISSN 1307-5543 – www.ejpam.com

Low Dimensional Homology Groups of the Orthosymplectic Lie
Superalgebra osp(1, 2)

Guy Roger Biyogmam

Department of Mathematics, Southwestern Oklahoma State University, Weatherford, OK 73096,
USA

Abstract. We realize the Lie superalgebra osp(1, 2) in terms of first order differential operators and
endow it with the Lie superbracket of vector fields to determine the basis (co)cycles of low dimensional
(co)homology groups of osp(1,2) with trivial coefficients, using the complex introduced by Tanaka [7].
Our calculations agree with the result obtained by Fuks and Leites in [2].

2010 Mathematics Subject Classifications: 17B56, 17B66.

Key Words and Phrases: Lie superalgebras, Homology of Lie superalgebras.

1. Introduction and Generalities

Given a Lie superalgebra g over a field k of characteristic 0, D. Fuks [1] introduced a Koszul
complex associated to g. Using this complex, Fuks and Leites [2] calculated the cohomology
groups with trivial coefficients of the classical Lie superalgebras. In particular, they found that

H∗(osp(1, 2))∼= H∗(sp(2)). (1)

In [7], J. Tanaka introduced another Koszul complex for g. In this work, we use this complex
and take advantage of the small basis of the superalgebra osp(1,2) to calculate low dimensional
(co)homology groups of osp(1, 2) with coefficients in R. The result obtained agrees with (1).
In particular, our calculations provide explicitly three generators of the group H3(osp(1, 2); R)
in terms of the basis of osp(1, 2). Note that in the non trivial case where these (co)homology
groups are non zero, results providing the generators have been limited to the second degree
[3, 4, 6, 8, 9].

Let us recall a few definitions. A Lie superalgebra [5] g is a Z2-graded algebra over a
commutative ring or field such as R or C with a direct sum decomposition

Email address: guy.biyogmam@swosu.edu

http://www.ejpam.com 395 c© 2014 EJPAM All rights reserved.



G. Biyogmam / Eur. J. Pure Appl. Math, 7 (2014), 395-404 396

g= g0 ⊕ g1,

together with a bilinear operation [−, −] : g× g→ g such that [gi ,gi] ⊆ gi+ j , and satisfying

i) [X , Y ] + (−1)|X ||Y |[Y, X ] = 0 (super antisymmetry)

ii) [X , [Y, Z]] = [[X , Y ], Z] + (−1)|Y ||Z |[Y [X , Z]] (super Jacobi identity).

The elements X and Y are said to be homogeneous and the parity |X | of an homogeneous
element X is 0 or 1 according to whether it is in g0 or g1, in which case X is said to be even or
odd respectively. The Grassmann algebra

∧∗(g) is defined as the quotient of the tensor algebra
⊗∗(g) by the two-sided ideal of g⊗2 generated by

�

X ⊗ Y + (−1)|X ||Y |Y ⊗ X , X , Y ∈ g
	

.

Let Vect(g) be the superspace of vector fields on g. The superbracket of two vector fields X
and Y is bilinear and defined for two homogeneous vector fields by:

[X , Y ] = X ◦ Y − (−1)|X ||Y |Y ◦ X . (2)

2. Lie Superalgebra Homology

For any Lie superalgebra g over a ring k and V any g-module, J. Tanaka [7] defined the
Lie algebra homology of g with coefficients in V , written H∗(g; V ), as the homology of the
complex ∧∗(g)⊗ V , namely

0
0
←− V

d
←− g∧

1
⊗ V

d
←− g∧

2
⊗ V

d
←− . . .

d
←− g∧

n−1
⊗ V

d
←− g∧

n
⊗ V ← . . .

where g∧
n

is the nth exterior power (as defined above) of g over k, and where

d(g1 ∧ . . .∧ gn ⊗ v) =
∑

1≤ j≤n

(−1) j+ζ
′
i g1 ∧ . . .Òg j . . .∧ gn ⊗ [g j , v]

+
∑

1≤i< j≤n

(−1)i+ j+ζi+ζ j+εiε j [gi , g j]∧ g1 ∧ . . .Ògi . . .Òg j . . .∧ gn ⊗ v,

where εi = |X i|, ζ′i = εi(εi+1 + . . . + εn), ζi = εi(ε1 + . . . + εi−1), and bgi means that the
variable gi is deleted. In particular if V = k the trivial module, we identify g1 ∧ . . .∧ gn with
g1 ∧ . . .∧ gn ⊗ 1 and have

d(g1 ∧ . . .∧ gn) =
∑

1≤i< j≤n

(−1)i+ j+ζi+ζ j+εiε j [gi , g j]∧ g1 ∧ . . . ĝi . . . ĝ j . . .∧ gn.

Note that this complex is infinite since for g ∈ g1 (the odd part of g), g ∧ g is not always 0 as
in the Lie algebra case. The standard Koszul complex for homology of Lie superalgebras is the
complex introduced by D. Fuks [1]. For trivial coefficients, it is defined as follows:



G. Biyogmam / Eur. J. Pure Appl. Math, 7 (2014), 395-404 397

0
0
←− k

d
←− C1(g)

d
←− C2(g)

d
←− . . .

d
←− Cn−1(g)

d
←− Cn(g)← . . .

where Cn(g) =
⊕

p+q=n∧p(g0)⊗Sq(g1) and where the differentials Cn(g)
dn−→ Cn−1(g) are given

by

dn

�

(g1 ∧ . . .∧ gn)⊗ (h1 . . . hq)
�

=
∑

1≤i< j≤p

(−1)i+ j
�

[gi , g j]∧ g1 ∧ . . . bgi . . . bg j . . .∧ gp

�

⊗
�

h1 . . . hq

�

+
∑

1≤i≤n

(−1)i−1
�

g1 ∧ . . . bgi . . .∧ gp

�

⊗
�

gi .(h1 . . . . . . hq

�

)

+
∑

1≤i< j≤q

�

g1 ∧ . . . . . .∧ gp

�

⊗
�

h1 . . .bhi . . .bh j . . . hq

�

,

for n ≥ 2, x i ∈ g0, y j ∈ g1. In the following subsection, we calculate H∗(osp(1,2); R), using
J. Tanaka’s definition.

3. Lie Superalgebra Homology of osp(1,2)

Throughout this section, we assume that k = R. Recall that osp(1, 2n) consists of matrices
of the form

M =





0 A1 A2
At

2 B C
−At

1 D −B t





where A1 and A2 are (1× n)-matrices, B is a (n× n)-matrix, C and D are symmetric (n× n)-
matrices. Let ei, j be matrices whose entries are 1 for i = j and 0 else. Then for n ≥ 2, the
following forms a basis of osp(1,2n) :

B={ei,i − ei+n,i+n, ei,i+n, ei+n,i , ei, j − e j+n,i+n, ei, j+n + e j,i+n,

en+i, j + en+ j,i , e1, j − en+ j,1, e1, j+n + e j,1; 2≤ i, j ≤ n, i < j}.

Assume that Rn is given the coordinates
�

x1, x2, . . . , xn

�

, and let ∂
∂ x i , be the unit vector fields

parallel to the x i axes respectively. It is easy to show in the case osp(1,2) that the Lie super-
algebra generated by the family B below of vector fields (endowed with the superbracket of
vector fields) is isomorphic to the orthosymplectic Lie superalgebra osp(1,2):

B= {E23, e23, e32, o23, o32}

where

E23 :=x2
∂

∂ x2
− x3

∂

∂ x3
,

e23 :=x2
∂

∂ x3
,

e32 :=x3
∂

∂ x2
,



G. Biyogmam / Eur. J. Pure Appl. Math, 7 (2014), 395-404 398

o23 :=x1
∂

∂ x2
− x3

∂

∂ x1
,

o32 :=x1
∂

∂ x3
+ x2

∂

∂ x1
.

The remaining of this section details the proof that there are isomorphisms of super vector
space

Hr(osp(1,2); R)∼=



























R, for r = 0

0, for r = 1, 2



E23 ∧ e23 ∧ e32

�

=



E23 ∧ o23 ∧ o32

�

=



e23 ∧ o23 ∧ o23 − e32 ∧ o32 ∧ o32

�

, for r = 3

0, for r = 4.

3.1. Zero and First Homology Groups

Notice that in the Tanaka complex, we have the boundary maps d0 : R→ 0 an d1 : osp(1, 2)→ R
with d1(b) = 0 for all b ∈ osp(1,2). So ker d0 = R, Imd1 = 0. So

H0(osp(1,2) ; R) =
ker d0

Imd1
=
R
0
= R.

Now using the identity (2) and the basis of osp(1, 2) provided above, we obtain the following
superbrackets:

[E23, e23] = 2e23
[E23, e32] = −2e32
[e23, e32] = E23
[E23, o23] = −o23

[E23, o32] = o32
[e23, o23] = −o32
[e23, o32] = 0
[e32, o23] = 0

[e32, o32] = −o23
[o23, o32] = E23
[o23, o23] = −2e32
[o32, o32] = 2e23

Remark 1. The set {E23, e23, e32} constitutes the even part of osp(1,2) and generates the Lie
algebra sl(2) i.e., osp0̄(1, 2) ∼= sl(2). The set {o23, o32} constitutes the odd part of osp(1, 2) and
is isomorphic to a 2-dimensional standard representation of sl(2).

To calculate the first homology group, notice that from the boundary map d1 above, ker d1 =
osp(1,2). Now by definition of Tanaka’s complex, the boundary map d2 is given by:

d(E23 ∧ e23) = −[E23, e23] = −2e23
d(E23 ∧ e32) = −[E23, e32] = 2e32
d(e23 ∧ e32) = −[e23, e32] = −E23
d(E23 ∧ o23) = −[E23, o23] = o23
d(E23 ∧ o32) = −[E23, o32] = −o32
d(e23 ∧ o23) = −[e23, o23] = o32

d(e23 ∧ o32) = −[e23, o32] = 0
d(e32 ∧ o23) = −[e32, o23] = 0
d(e32 ∧ o32) = −[e32, o32] = o23
d(o23 ∧ o32) = −[o23, o32] = −E23
d(o23 ∧ o23) = −[o23, o23] = 2e32
d(o32 ∧ o32) = −[e32, o32] = −2e23.

From these formulas, it is clear that Imd2 = osp(1, 2). So

H1(osp(1, 2) ; R) =
kerd1

Imd2
=

osp(1,2)
osp(1,2)

= 0



G. Biyogmam / Eur. J. Pure Appl. Math, 7 (2014), 395-404 399

3.2. Second Homology Group

From the boundary map d2 above, we have

ker d2 =<E23 ∧ o32 + e23 ∧ o23, e23 ∧ o32, e32 ∧ o23

E23 ∧ e23 − o32 ∧ o32, e23 ∧ e32 − o23 ∧ o32

E23 ∧ e32 − o23 ∧ o23, E23 ∧ o23 − e32 ∧ o32 > .

Now by definition of Tanaka’s complex, the boundary map d3 is given by:

d(E23 ∧ e23 ∧ e32) = 0

d(E23 ∧ e23 ∧ o23) = −e23 ∧ o23 − E23 ∧ o32 (3)

d(E23 ∧ e23 ∧ o32) = −3e23 ∧ o32 (4)

d(E23 ∧ e32 ∧ o23) = 3e32 ∧ o23 (5)

d(E23 ∧ e32 ∧ o32) = e32 ∧ o32 − E23 ∧ o23 (6)

d(e23 ∧ e32 ∧ o23) = −E23 ∧ o23 + e32 ∧ o32

d(e23 ∧ e32 ∧ o32) = −E23 ∧ o32 − e23 ∧ o23

d(E23 ∧ o23 ∧ o32) = 0

d(e23 ∧ o23 ∧ o32) = o32 ∧ o32 − E23 ∧ e23

d(e32 ∧ o23 ∧ o32) = −E23 ∧ e32 + o23 ∧ o23 (7)

d(E23 ∧ o23 ∧ o23) = 2o23 ∧ o23 + 2e32 ∧ E23

d(E23 ∧ o32 ∧ o32) = −2o32 ∧ o32 + 2E23 ∧ e23 (8)

d(e23 ∧ o23 ∧ o23) = 2o23 ∧ o32 − 2e23 ∧ e32 (9)

d(e23 ∧ o32 ∧ o32) = 0

d(e32 ∧ o23 ∧ o23) = 0

d(e32 ∧ o32 ∧ o32) = 2o23 ∧ o32 − 2e23 ∧ e32

d(o23 ∧ o23 ∧ o23) = 6e32 ∧ o23

d(o23 ∧ o23 ∧ o32) = 2e32 ∧ o32 − 2E23 ∧ o23

d(o23 ∧ o32 ∧ o32) = −2e23 ∧ o23 − 2E23 ∧ o32

d(o32 ∧ o32 ∧ o32) = −6e23 ∧ o32.

From the formulas (3)–(9) above, it is clear that Imd3 = ker d2. Therefore

H2(osp(1,2) ; R) = 0.

3.3. Third Homology Group

From the boundary map d3 above, we have

ker d3 =<E23 ∧ e23 ∧ e32, E23 ∧ o23 ∧ o32, e23 ∧ o32 ∧ o32, e32 ∧ o23 ∧ o23,



G. Biyogmam / Eur. J. Pure Appl. Math, 7 (2014), 395-404 400

2E23 ∧ e23 ∧ o32 − o32 ∧ o32 ∧ o32, 2E23 ∧ e32 ∧ o23 − o23 ∧ o23 ∧ o23,

2E23 ∧ e23 ∧ o23 − o23 ∧ o32 ∧ o32, 2E23 ∧ e32 ∧ o32 − o23 ∧ o23 ∧ o32,

E23 ∧ e23 ∧ o23 − e23 ∧ e32 ∧ o32, E23 ∧ e32 ∧ o32 − e23 ∧ e32 ∧ o23,

E23 ∧ o23 ∧ o23 − 2e32 ∧ o23 ∧ o32, E23 ∧ o32 ∧ o32 + 2e23 ∧ o23 ∧ o32,

e23 ∧ o23 ∧ o23 − e32 ∧ o32 ∧ o32 > .

Now by definition of Tanaka’s complex, the boundary map d4 is given by :

d(E23 ∧ e23 ∧ e32 ∧ o23) = e23 ∧ e32 ∧ o23 − E23 ∧ e32 ∧ o32 (10)

d(E23 ∧ e23 ∧ e32 ∧ o32) = −e23 ∧ e32 ∧ o32 + E23 ∧ e23 ∧ o23 (11)

d(E23 ∧ e23 ∧ o23 ∧ o32) = −E23 ∧ o32 ∧ o32 − 2e23 ∧ o23 ∧ o32 (12)

d(E23 ∧ e32 ∧ o23 ∧ o32) = 2e32 ∧ o23 ∧ o32 − E23 ∧ o23 ∧ o23 (13)

d(e23 ∧ e32 ∧ o23 ∧ o32) = −E23 ∧ o23 ∧ o32 + e32 ∧ o32 ∧ o32 (14)

− e23 ∧ o23 ∧ o23 − E23 ∧ e23 ∧ e32 (15)

d(E23 ∧ e23 ∧ o23 ∧ o23) = −2E23 ∧ o23 ∧ o32 + 2E23 ∧ e23 ∧ e32 (16)

d(E23 ∧ e23 ∧ o32 ∧ o32) = −4e23 ∧ o32 ∧ o32 (17)

d(E23 ∧ e32 ∧ o23 ∧ o23) = 4e32 ∧ o23 ∧ o23 (18)

d(E23 ∧ e32 ∧ o32 ∧ o32) = −2E23 ∧ o23 ∧ o32 + 2E23 ∧ e23 ∧ e32

d(e23 ∧ e32 ∧ o23 ∧ o23) = −E23 ∧ o23 ∧ o23 + 2e32 ∧ o23 ∧ o32

d(e23 ∧ e32 ∧ o32 ∧ o32) = −E23 ∧ o32 ∧ o32 − 2e23 ∧ o23 ∧ o32

d(E23 ∧ o23 ∧ o23 ∧ o23) = 3o23 ∧ o23 ∧ o23 − 6E23 ∧ e32 ∧ o23 (19)

d(E23 ∧ o23 ∧ o23 ∧ o32) = o23 ∧ o23 ∧ o32 − 2E23 ∧ e32 ∧ o32 (20)

d(E23 ∧ o23 ∧ o32 ∧ o32) = −o23 ∧ o32 ∧ o32 + 2E23 ∧ e23 ∧ o23 (21)

d(E23 ∧ o32 ∧ o32 ∧ o32) = −3o32 ∧ o32 ∧ o32 + 6E23 ∧ e23 ∧ o32 (22)

d(e23 ∧ o23 ∧ o23 ∧ o23) = 3o23 ∧ o23 ∧ o32 − 6e23 ∧ e32 ∧ o23

d(e23 ∧ o23 ∧ o23 ∧ o32) = 2o23 ∧ o32 ∧ o32 − 2e23 ∧ e32 ∧ o32 − 2E23 ∧ e23 ∧ o23

d(e23 ∧ o23 ∧ o32 ∧ o32) = o32 ∧ o32 ∧ o32 − 2E23 ∧ e23 ∧ o32

d(e23 ∧ o32 ∧ o32 ∧ o32) = 0

d(e32 ∧ o23 ∧ o23 ∧ o23) = 0

d(e32 ∧ o23 ∧ o23 ∧ o32) = o23 ∧ o23 ∧ o23 − 2E23 ∧ e32 ∧ o23

d(e32 ∧ o23 ∧ o32 ∧ o32) = 2o23 ∧ o23 ∧ o32 − 2E23 ∧ e32 ∧ o32 − 2e23 ∧ e32 ∧ o23

d(e32 ∧ o32 ∧ o32 ∧ o32) = 3o23 ∧ o32 ∧ o32 − 6e23 ∧ e32 ∧ o32

d(o23 ∧ o23 ∧ o23 ∧ o23) = 12e32 ∧ o23 ∧ o23

d(o23 ∧ o23 ∧ o23 ∧ o32) = 6e32 ∧ o23 ∧ o32 − 3E23 ∧ o23 ∧ o23

d(o23 ∧ o23 ∧ o32 ∧ o32) = 2e32 ∧ o32 ∧ o32 − 4E23 ∧ o23 ∧ o32 − 2e23 ∧ o23 ∧ o23 (23)

d(o23 ∧ o32 ∧ o32 ∧ o32) = −3E23 ∧ o32 ∧ o32 − 6e23 ∧ o23 ∧ o32

d(o32 ∧ o32 ∧ o32 ∧ o32) = −12e23 ∧ o32 ∧ o32.



G. Biyogmam / Eur. J. Pure Appl. Math, 7 (2014), 395-404 401

From the formulas (10)–(13), (17)–(25) above, it is clear that all the cycles but
E23 ∧ e23 ∧ e32, E23 ∧ o23 ∧ o32, e23 ∧ o23 ∧ o23 − e32 ∧ o32 ∧ o32 are boundaries, so are zero
in homology. By (15) and (16) or (26), these remaining three cycles differ by a boundary, so
they generate the same homology class. Therefore

H3(osp(1, 2) ; R) =



E23 ∧ e23 ∧ e32

�

=



E23 ∧ o23 ∧ o32

�

=



e23 ∧ o23 ∧ o23 − e32 ∧ o32 ∧ o32

�

.

3.4. Fourth Homology Group

In this subsection, e∧k stands for e ∧ e ∧ . . .∧ e
︸ ︷︷ ︸

k-times

. From the boundary map d4 above, we have

ker d4 =



e23 ∧ o∧3
32 , e32 ∧ o∧3

23 , E23 ∧ e23 ∧ o∧2
23 − E23 ∧ e32 ∧ o∧2

32 ,

3E23 ∧ e23 ∧ o23 ∧ o32 − o23 ∧ o∧3
32 , 3E23 ∧ e32 ∧ o23 ∧ o32 − o∧3

23 ∧ o32,

3E23 ∧ e23 ∧ o∧2
32 − o32 ∧ o∧3

32 , 3E23 ∧ e32 ∧ o∧2
23 − o∧4

23 ,

E23 ∧ o∧3
23 − 3e32 ∧ o∧2

23 ∧ o32, E23 ∧ o∧3
32 + 3e23 ∧ o23 ∧ o∧2

32 ,

E23 ∧ e23 ∧ o23 ∧ o32 − e23 ∧ e32 ∧ o∧2
32 , E23 ∧ e32 ∧ o23 ∧ o32 − e23 ∧ e32 ∧ o∧2

23

�

.

Now by definition of Tanaka’s complex, the boundary map d5 is given by:

d(E23 ∧ e23 ∧ e32 ∧ o23 ∧ o32) = E23 ∧ e23 ∧ o23 ∧ o23 − E23 ∧ e32 ∧ o32 ∧ o32 (24)

d(E23 ∧ e23 ∧ e32 ∧ o∧2
23 ) = 2e23 ∧ e32 ∧ o23 ∧ o23 − 2E23 ∧ e32 ∧ o23 ∧ o32 (25)

d(E23 ∧ e23 ∧ e32 ∧ o∧2
32 ) = 2E23 ∧ e23 ∧ o23 ∧ o32 − 2e23 ∧ e32 ∧ o32 ∧ o32 (26)

d(E23 ∧ e23 ∧ o∧3
23 ) = e23 ∧ o∧3

23 − 3E23 ∧ o∧2
23 ∧ o32 + 6E23 ∧ e23 ∧ e32 ∧ o23

d(E23 ∧ e23 ∧ o∧2
23 ∧ o32) = −2E23 ∧ o23 ∧ o∧2

32 + 2E23 ∧ e23 ∧ e32 ∧ o32

− e23 ∧ o∧2
23 ∧ o32

d(E23 ∧ e23 ∧ o23 ∧ o∧2
32 ) = E23 ∧ o∧3

32 − 3e23 ∧ o23 ∧ o∧2
32

d(E23 ∧ e23 ∧ o∧3
32 ) = −5e23 ∧ o∧3

32 (27)

d(E23 ∧ e32 ∧ o∧3
23 ) = 5e32 ∧ o∧3

23 (28)

d(E23 ∧ e32 ∧ o∧2
23 ∧ o32) = −E23 ∧ o∧3

23 + 3e32 ∧ o∧2
23 ∧ o32 (29)

d(E23 ∧ e32 ∧ o23 ∧ o∧2
32 ) = 2E23 ∧ e23 ∧ e32 ∧ o23 − 2E23 ∧ o∧2

23 ∧ o32

+ e32 ∧ o23 ∧ o∧2
32

d(E23 ∧ e32 ∧ o∧3
32 ) = 6E23 ∧ e23 ∧ e32 ∧ o32 − 3E23 ∧ o23 ∧ o∧2

32 + e32 ∧ o∧3
32

d(e23 ∧ e32 ∧ o∧3
23 ) = −E23 ∧ o∧3

23 + 3e32 ∧ o∧2
23 ∧ o32

d(e23 ∧ e32 ∧ o∧2
23 ∧ o32) = −E23 ∧ o∧2

23 ∧ o32 − 2E23 ∧ e23 ∧ e32 ∧ o23

+ 2e32 ∧ o23 ∧ o∧2
32 − e23 ∧ o∧3

23

d(e23 ∧ e32 ∧ o23 ∧ o∧2
32 ) = −E23 ∧ o23 ∧ o∧2

32 − 2E23 ∧ e23 ∧ e32 ∧ o32

− 2e23 ∧ o∧2
23 ∧ o32 + e32 ∧ o∧3

32



G. Biyogmam / Eur. J. Pure Appl. Math, 7 (2014), 395-404 402

d(e23 ∧ e32 ∧ o∧3
32 ) = −E23 ∧ o∧3

32 − 3e23 ∧ o23 ∧ o∧2
32 (30)

d(E23 ∧ o∧4
23 ) = 6E23 ∧ e32 ∧ o∧2

23 + 4o∧4
23

d(E23 ∧ o∧3
23 ∧ o32) = −6E23 ∧ e32 ∧ o23 ∧ o32 + 2o∧3

23 ∧ o32 (31)

d(E23 ∧ o∧2
23 ∧ o∧2

32 ) = −2E23 ∧ e32 ∧ o∧2
32 + 2E23 ∧ e23 ∧ o∧2

23

d(E23 ∧ o23 ∧ o∧3
32 ) = 6E23 ∧ e23 ∧ o23 ∧ o32 − 2o23 ∧ o∧3

32 (32)

d(E23 ∧ o∧4
32 ) = 6E23 ∧ e23 ∧ o∧2

32 − 4o∧4
32

d(e23 ∧ o∧4
23 ) = −6e23 ∧ e32 ∧ o∧2

23 + 4o∧3
23 ∧ o32

d(e23 ∧ o∧3
23 ∧ o32) = 3o∧2

23 ∧ o∧2
32 − 6e23 ∧ e32 ∧ o23 ∧ o32

+ 3E23 ∧ e23 ∧ o∧2
23

d(e23 ∧ o∧2
23 ∧ o∧2

32 ) = −4E23 ∧ e23 ∧ o23 ∧ o32 − 2e23 ∧ e32 ∧ o∧2
32

+ 2o23 ∧ o∧3
32 (33)

d(e23 ∧ o23 ∧ o∧3
32 ) = 3E23 ∧ e23 ∧ o∧2

32 + o∧4
32 (34)

d(e23 ∧ o∧4
32 ) = 0

d(e32 ∧ o∧4
23 ) = 0

d(e32 ∧ o∧3
23 ∧ o32) = 3E23 ∧ e32 ∧ o∧2

23 + o∧4
23 (35)

d(e32 ∧ o∧2
23 ∧ o∧2

32 ) = 4E23 ∧ e32 ∧ o23 ∧ o32 + 2e23 ∧ e32 ∧ o∧2
23

− 2o∧3
23 ∧ o32 (36)

d(e32 ∧ o23 ∧ o∧3
32 ) = 3o∧2

23 ∧ o∧2
32 + 6e23 ∧ e32 ∧ o23 ∧ o32 + 3E23 ∧ e32 ∧ o∧2

32

d(e32 ∧ o∧4
32 ) = 6e23 ∧ e32 ∧ o∧2

32 + 4o23 ∧ o∧3
32

d(o∧5
23 ) = 20e32 ∧ o∧3

23

d(o∧4
23 ∧ o32) = 12e32 ∧ o∧2

23 ∧ o32 − 4E23 ∧ o∧3
23

d(o∧3
23 ∧ o∧2

32 ) = 6e32 ∧ o23 ∧ o∧2
32 − 6E23 ∧ o∧2

23 ∧ o32 − 2e23 ∧ o∧3
23

d(o∧2
23 ∧ o∧3

32 ) = −6e23 ∧ o∧2
23 ∧ o32 − 6E23 ∧ o23 ∧ o∧2

32 + 2e32 ∧ o∧3
32

d(o23 ∧ o∧4
32 ) = −12e23 ∧ o23 ∧ o∧2

32 − 4E23 ∧ o∧3
32

d(o∧5
32 ) = −20e23 ∧ o∧3

32 .

From the formulas (24)–(32), (34), (35) above, it is clear that all the cycles but

E23 ∧ e23 ∧ o23 ∧ o32 − e23 ∧ e32 ∧ o32 ∧ o32,

E23 ∧ e32 ∧ o23 ∧ o32 − e23 ∧ e32 ∧ o23 ∧ o23

are boundaries, so are zero in homology. For these remaining two cycles, combining (31) and
(36) shows that the first is a boundary. Similarly, combining (32) and (33) shows that the
second is also boundary. Therefore

H4(osp(1,2) ; R) = 0.



REFERENCES 403

In the following section, we provide the low dimensional cohomology groups with trivial
coefficients of osp(1, 2).

4. Lie Superalgebra Cohomology of osp(1, 2)

Theorem 1. There are isomorphisms of super vector spaces

H r(osp(1, 2); R)∼=



























R for r = 0,

0 for r = 1, 2,



E∗23 ∧ e∗23 ∧ e∗32

�

=



E∗23 ∧ o∗23 ∧ o∗32

�

=



e∗23 ∧ o∗23 ∧ o∗23 − e∗32 ∧ o∗32 ∧ o∗32

�

, for r = 3

0, for r = 4.

Proof. We use the super vector space isomorphism (see [8, lemma 1.7])

H∗(osp(1, 2); R)∼= Hom(H∗(osp(1, 2); R), R)

and the dual basis

E∗23 := x2d x2 − x3d x3

e∗23 := x2d x3
e∗32 := x3d x2

o∗23 := x1d x2 − x3d x1
o∗32 := x1d x3 + x2d x1

where d x i is the dual of ∂
∂ x i with respect to the basis of osp(1,2) given in section 2.

References

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[3] K Iohara and Y Koga. Central Extensions of Lie Superalgebras, Commentarii Mathematici
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[4] K Iohara and Y Koga. Second homology of Lie Superalgebras, Mathematische Nachrichten,
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[5] V G Kac. Lie Superalgebras, Advances in Mathematics, 26, (1977), 8-96.

[6] Y Y Kochetkov. Homology of Nilpotent Subalgebras of the Lie Superalgebra K(1,1), Mathe-
matical Notes, 73 2:218-227, 2003.

[7] J Tanaka. On Homology and Cohomology of Lie Superalgebras with Coefficients in Their
Finite-Dimensional Representations, Proceedings of the Japan Academy, 71, Ser. A:51-53,
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[8] J Tanaka. Homology and Cohomology of Lie Superalgebras sl(2, 1) with Coefficients in The
Space of Finite-Dimensional Irreducible Representations, Journal of Mathematics of Kyoto
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