The Spin Group in Superspace
Hennie De Schepper, Alí Guzmán Adán, Frank Sommen
Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering and
Architecture, Ghent University, Krijgslaan 281, 9000 Gent, Belgium.

arXiv:1804.00963v1 [] 3 Apr 2018

Abstract
There are two well-known ways of describing elements of the rotation group SO(m). First,
according to the Cartan-Dieudonné theorem, every rotation matrix can be written as an even
number of reflections. And second, they can also be expressed as the exponential of some
anti-symmetric matrix.
In this paper, we study similar descriptions of the corresponding extension of SO(m) to
superspace. The setting is natural to describe the behavior of bosonic and fermionic particles.
This group of super-rotations SO0 is also an extension of the symplectic group. While still
being connected, it is thus no longer compact. As a consequence, it cannot be fully described
by just one action of the exponential map on its Lie algebra. Instead, we obtain an Iwasawatype decomposition for this group in terms of three exponentials acting on three direct
summands of the corresponding Lie algebra of supermatrices.
At the same time, SO0 strictly contains the group generated by super-vector reflections.
Therefore, its Lie algebra is isomorphic to a certain extension of the algebra of superbivectors.
This means that the Spin group in superspace has to be seen as the group generated by the
exponentials of the so-called extended superbivectors in order to cover SO0 . We also study
the actions of this Spin group on supervectors and provide a proper subset of it that is a
double cover of SO0 . Finally, we show that every fractional Fourier transform in n bosonic
dimensions can be seen as an element of the spin group in superspace.
Keywords. Spin groups, symplectic groups, Clifford analysis, superspace
Mathematics Subject Classification (2010). 30G35 (22E60)

1

Introduction

Mathematical analysis on superspace (super analysis) is a very important tool in the study of several
branches of contemporary theoretical physics such as supergravity or superstring theories. It was introduced by Berezin, see [1], and possesses a huge mathematical impact since it breaks several traditional
patterns of classical analysis, such as commutation among coordinate variables and positive dimensions.
Indeed, elements in superspace are defined by means of bosonic (commuting) variables xj and fermionic
(anti-commuting) variables x`j , where, in a natural way, the bosonic variables describe positive dimensions
while the fermionic ones describe negative dimensions.
Generalizations of some objects from classical analysis appear in superspace with a wider nature.
For example, the notion of a supervector variable is introduced as a variable that takes values in some
Grassmann envelope and, through the corresponding definitions of bosonic and fermionic partial derivatives ∂xj and ∂x`j , the supervector derivative (supergradient or super Dirac operator) can be introduced
as well. These analogies lead to the introduction of Clifford analysis in superspace as it was done in the
papers [2, 3, 13, 14, 16].
A proper extension of Clifford analysis to this setting needs a set of rules (or axioms) that guarantee
the preservation of the Clifford nature in the new objects, although they can have a very different behavior
compared to the classical ones. This set of rules is provided by the notion of a radial algebra, which is an
abstract setting that algebraically describes all main properties which any Clifford environment should

1

satisfy, see [3, 5, 12, 15]. This approach does not depend on an a priori defined vector space or signature.
In particular, the main exigence is that the anticommutator {x, y} of every pair of vector variables is a
commuting element, leading to the definition of an abstract inner product by the formula
1
hx, yi = − {x, y}.
2

(1)

However this abstract description leaves some issues unclarified. In particular, it is not possible to describe
the whole set of linear transformations leaving the above bilinear form invariant and in consequence, there
is no proper definition for the spin group in this general abstract setting. We can, of course, define vector
reflections by ψ(w)[x] = wxw where w, x are abstract vector variables. Yet this doesn’t enable us to
prove that all linear transformations leaving (1) invariant can be written as the composition of vector
reflections. In order to find the set of those transformations it is necessary to work with a concrete
representation of the radial algebra, i.e., an underlying vector space V where the vector variables are
defined endowed with a fixed bilinear form and signature.
For example, in the Clifford polynomial representation of the radial algebra (see [5, 12]), we have the
Euclidean space V = Rm where the most important group leaving the inner product (1) invariant is the
set of rotations SO(m). The spin group appears in this setting as a double cover of SO(m) given by


2k

Y
Spin(m) :=
wj : k ∈ N, wj ∈ Sm−1 ,


j=1

where Sm−1 = {w ∈ Rm : w2 = −1} denotes the unit sphere in Rm . The relation between Spin(m) and
SO(m) is easily seen through the Lie group representation h : Spin(m) → SO(m)
h(s)[x] = sxs,

s ∈ Spin(m), x ∈ Rm ,

which describes the action of every element of SO(m) in terms of Clifford multiplication. Such a description of the spin group follows from the Cartan-Dieudonné theorem which states that every orthogonal
transformation in an m-dimensional symmetric bilinear space can be written as the composition of at
most m reflections.
The situation may however not be the same in another representation of the radial algebra where
the Cartan-Dieudonné theorem is no longer valid. In this paper we deal with one of those cases: the
representation in superspace. Our main goal is to properly define the (super) spin group as a set of
elements describing every superrotation through Clifford multiplication in superspace. To that end, we
consider vector variables taking values in the Grassmann envelope Rm|2n (ΛN ). This makes it possible
to study the group of supermatrices leaving (1) invariant and to define in a proper way the spin group
in superspace. It is worth noticing that the superstructures are absorbed by the Grassmann algebras
leading to classical Lie groups and Lie algebras instead of supergroups or superalgebras.
The paper is organized as follows. We start with some preliminaries on Grassmann algebras, Grassmann envelopes and supermatrices in Section 2. In particular, we carefully recall the notion of an
exponential map for Grassmann numbers and supermatrices as elements of finite dimensional associative
algebras. In Section 3 we briefly describe the Clifford setting in superspace leading to the introduction of
the Lie algebra of superbivectors. An extension of this algebra plays an important rôle in the description
of the super spin group. The use of the exponential map in such an extension (which takes us out of the
radial algebra) necessitates the introduction of the corresponding tensor algebra. Section 4 is devoted to
the study of the invariance of the “inner product” (1) in this setting. There, we study several groups of
supermatrices and in particular, the group of superrotations SO0 and its Lie algebra so0 , which combine
both orthogonal and symplectic structures. We prove that every superrotation can be uniquely decomposed as the product of three exponentials acting in some special subspaces of so0 . Finally, in section 5
we study the problem of defining the spin group in this setting and its differences with the classical case.
We show that the compositions of even numbers of vector reflections are not enough to fully describe
SO0 since they only show an orthogonal structure and don’t include the symplectic part of SO0 . Next
we propose an alternative, by defining the spin group through the exponential of extended superbivectors
and show that they indeed cover the whole set of superrotations. In particular, we explicitly describe a
subset S which is a double covering of SO0 and contains in particular every fractional Fourier transform.

2

2

Preliminaries

2.1

Grassmann algebras and Grassmann envelopes

Let ΛN be the Grassmann algebra of order N ∈ N over the field K (K = R or C) with canonical generators
f1 , . . . , fN which are subject to the multiplication rules
fj fk + fk fj = 0,

(2)

implying in particular that fj2 = 0. Every element of ΛN can be written in the form
X
a=
aA fA ,

(3)

A⊂{1,...,N }

where aA ∈ K and fA = fj1 · · · fjk , for A = {j1 , . . . , jk }, with 1 ≤ j1 ≤ . . . ≤ jk ≤ N ; here we put f∅ = 1.
(k)
We define the space of homogeneous elements of degree k by ΛN = spanK {fA : |A| = k}, where in
(k)
particular ΛN = {0} for k > N . It then easily follows from (2)-(3) that
ΛN =

N
M

(k)

(k)

(`)

(k+`)

ΛN and ΛN ΛN ⊂ ΛN

k=0

P
(k)
The projection of ΛN on its k-homogeneous part is denoted by [·]k : ΛN → ΛN , i.e. [a]k = |A|=k aA fA .
In particular we denote [a]0 = a∅ =: a0 . It is well known that ΛN shows a natural Z2 -grading. In
L
L
(2k)
(2k+1)
as the spaces of homogeneous even and odd
fact, defining ΛN,0 = k≥0 ΛN and ΛN,1 = k≥0 ΛN
elements respectively, we obtain the superalgebra structure ΛN = ΛN,0 ⊕ ΛN,1 where ΛN,j ΛN,k ⊂ ΛN,j+k
for j, k ∈ Z2 . We recall that ΛN is graded commutative in the sense that
vw = wv,

vw`= w`v,

v`w`= −w`v`,

v, w ∈ ΛN,0 ,

v`, w`∈ ΛN,1 .

(0)

Any element a ∈ ΛN is the sum of a number a0 ∈ ΛN = K and a nilpotent element. In fact, every
LN
(k)
N +1
a ∈ Λ+
= 0. It is easily seen that the projection [·]0 : ΛN → K is an algebra
N :=
k=1 ΛN satisfies a
homomorphism, i.e. [ab]0 = a0 b0 for a, b ∈ ΛN . In particular the following property holds.
Lemma 1. Let a ∈ ΛN such that a2 ∈ K \ {0}. Then a ∈ K.
It sometimes is useful to consider ΛN embedded in a Grassmann algebra of higher order; in particular,
we consider ΛN ⊂ ΛN +1 := AlgK {f1 , . . . , fN , fN +1 } being fN +1 a new canonical Grassmann generator.
(k)
(k)
This embedding preserves all grading structures since ΛN ⊂ ΛN +1 and ΛN,j ⊂ ΛN +1,j for j ∈ Z2 .
The algebra ΛN is a K-vector space of dimension 2N . As every finite dimensional K-vector space,
ΛN becomes a Banach space with the introduction of an arbitrary norm, all norms being equivalent. In
particular, the following result is obtained through straightforward computation.
P
Lemma 2. The norm k · k1 defined on ΛN by kak1 = A⊂{1,...,N } |aA | satisfies
kabk1 ≤ kak1 kbk1 ,

for every a, b ∈ ΛN .

(4)

The exponential of a ∈ ΛN , denoted by ea or exp(a), is defined by the power series
ea =

∞
X
aj
j=0

(5)

j!

Proposition 1. The series (5) converges for every a ∈ ΛN and ea is a continuous function in ΛN .
Proof.
From (4) it follows that
∞
X
kaj k1
j=0

j!

≤

∞
X
kakj

1

j=0

3

j!

= ekak1

whence (5) (absolutely) converges in the Banach space ΛN . Now consider BR := {a ∈ ΛN : kak1 ≤ R}
for some R > 0, where it holds that
kaj k1
kakj1
Rj
≤
≤
j!
j!
j!
P∞ j
P∞ Rj
and j=0 j! converges, whence from the Weierstrass M -criterium we have that j=0 aj! uniformly cona
a
verges in BR and e thus is continuous in BR . Then, e is continuous in ΛN .

Now consider the graded vector space Kp|q with standard homogeneous basis e1 , . . . , ep , e`1 , . . . , e`q , i.e.
K = Kp|0 ⊕ K0|q where {e1 , . . . , ep } is a basis for Kp|0 and {e`1 , . . . , e`q } is a basis for K0|q . Elements in
Kp|0 are called even homogeneous elements while elements in K0|q are called odd homogeneous elements.
The Grassmann envelope Kp|q (ΛN ) is defined as the set of formal linear combinations of the form
p|q

x = x + x`=

p
X

xj ej +

j=1

q
X

where xj ∈ ΛN,0 ,

x`j e`j ,

x`j ∈ ΛN,1 .

(6)

j=1

Then Kp|q (ΛN ) is a K-vector space of dimension 2N −1 (p + q), inheriting the Z2 -grading of Kp|q , i.e.
Kp|q (ΛN ) = Kp|0 (ΛN ) ⊕ K0|q (ΛN ), where Kp|0 (ΛN ) denotes the subspace of vectors of the form (6) with
x`j = 0, and K0|q (ΛN ) denotes the subspace of vectors of the form (6) with xj = 0. The subspaces
Kp|0 (ΛN ) and K0|q (ΛN ) are called the Grassmann envelopes of Kp|0 and K0|q , respectively.
In Kp|q (ΛN ),Pthere exists a subspace which is naturally isomorphic to Kp|0 . It consists of
(6)
Pvectors
m
p
of the form x = j=1 xj ej with xj ∈ K. The map [·]0 : Kp|q (ΛN ) → Kp|0 defined by [x]0 = j=1 [xj ]0 ej
will be useful.
T
The standard basis of Kp|q can be represented by the columns ej = (0, . . . , 1, . . . , 0) (1 on the j-th
T
place from the left) and e`j = (0, . . . , 1, . . . , 0) (1 on the (p + j)-th place from the left). In this basis,
T
elements of Kp|q (ΛN ) take the form x = (x1 , . . . , xp , x`1 , . . . , x`q ) .

2.2

Supermatrices


The Z2 -grading of Kp|q yields the Z2 -grading of the space End Kp|q of endomorphisms on Kp|q . This
space is isomorphic to the space Mat(p|q) of block matrices of the form

 
 

A B`
A 0
0 B`
M=
=
+
(7)
0 D
C` D
C` 0
where1 A ∈ Kp×p , B` ∈ Kp×q , C` ∈ Kq×p and D ∈ Kq×q . The first term in (7) is the even part of M and
the second term is the odd one. The Grassmann envelope of Mat(p|q) is denoted by Mat(p|q)(ΛN ) and
consists of matrices of the form (7) but with entries in ΛN (namely, A, D with even entries and B`, C`
with odd entries). Elements in Mat(p|q)(ΛN ) are called supermatrices.
The Z2 -grading of Mat(p|q)(ΛN ), inherited from Mat(p|q), together with the usual matrix multiplication, provides a superalgebra structure on this Grassmann envelope. More precisely, for any k ∈ N let
(2k)
(k)
Mat(p|q)(ΛN ) be the space of all homogeneous supermatrices of degree k. This is, Mat(p|q)(ΛN ) con(2k)
(2k+1)
sists of all diagonal block matrices with entries in ΛN , and Mat(p|q)(ΛN
) consists of all off-diagonal
(2k+1)
block matrices with entries in ΛN
. These subspaces define a grading in Mat(p|q)(ΛN ) by
Mat(p|q)(ΛN ) =

N
M

(k)

Mat(p|q)(ΛN ),

and

(k)

(`)

(k+`)

Mat(p|q)(ΛN ) Mat(p|q)(ΛN ) ⊂ Mat(p|q)(ΛN

).

k=0

Then, it is clear that every supermatrix M can be written as the sum of a numeric matrix M0 ∈
LN
(0)
(k)
Mat(p|q)(ΛN ) and a nilpotent supermatrix M ∈ Mat(p|q)(Λ+
N ) :=
k=1 Mat(p|q)(ΛN ). In accordance
with the general ideas valid for Grassmann algebras and Grassmann envelopes we define the algebra
(0)
homomorphism [·]0 : Mat(p|q)(ΛN ) → Mat(p|q)(ΛN ) as the projection:




A B`
A0 0
M=
−→
= M0 = [M ]0
0 D0
C` D
1

Given a set S, we use the notation Sp×q to refer to the set of matrices of order p × q with entries in S.

4

where A0 and D0 are the numeric projections of A and D on Kp×p and Kq×q respectively. We recall that
(0)
Mat(p|q)(ΛN ) is equal to the even subalgebra of Mat(p|q). Given a set of supermatrices S we define
[S]0 = {[M ]0 : M ∈ S}.
Every supermatrix M defines a linear operator over Kp|q (ΛN ) which acts on a vector x = x + x`,
x ∈ Kp|0 (ΛN ), x`∈ K0|q (ΛN ) as


 

A B`
x
Ax + B`x`
Mx =
=
∈ Kp|q (ΛN ).
x`
C` D
C`x + Dx`
However, the supermatrix M ∈ Mat(p|q)(ΛN ) defining an endomorphism of Kp|q (ΛN ) is not unique. In
fact, the zero endomorphism is defined by every supermatrix on the ideal HN ⊂ Mat(p|q)(ΛN ) given by








0 0
0 B`
(N )
(N )
HN = L =
: L ∈ Mat(p|q)(ΛN ) or HN = L =
: L ∈ Mat(p|q)(ΛN ) ,
0 D
0 0
for N odd or even respectively. Hence, all supermatrices in M + HN define the same endomorphism over
Kp|q (ΛN ) as M . But this situation changes if we consider endomorphisms over Kp|q (ΛN +1 ) defined by
supermatrices with entries in ΛN .
Lemma 3. Let E be an endomorphism over Kp|q (ΛN +1 ). If E admits a supermatrix representation in
Mat(p|q)(ΛN ), then this representation is unique in Mat(p|q)(ΛN ).
This result is easily proved using the introduction of a new canonical Grassmann generator fN +1 to
show that an element a ∈ ΛN satisfies afN +1 = 0 if and only if a = 0.
To study group structures in Mat(p|q)(ΛN ) we start from the Lie group GL(p|q)(ΛN ) of all invertible
elements of Mat(p|q)(ΛN ). The following is a well-known characterization of this group, see [1].


A B`
Theorem 1. Let M =
∈ Mat(p|q)(ΛN ). Then the following statements are equivalent.
C` D
(i) M ∈ GL(p|q)(ΛN ).
(ii) A, D are invertible.
(iii) A0 , D0 are invertible.
In addition, for every M ∈ GL(p|q)(ΛN ) its inverse is
M

−1

=

−1
A − B`D−1 C`
−1
−D−1 C` A − B`D−1 C`

−1 !
−A−1 B` D − C`A−1 B`
−1
.
D − C`A−1 B`

The usual definitions of transpose, trace and determinant of a matrix are not appropriate in the
graded case. For example, although the transpose
 T

A
C`T
MT =
B`T DT
of a supermatrix M is a well defined element of Mat(p|q)(ΛN ), the usual property (M L)T 6= LT M T will
not hold in general. This problem is fixed by introducing the supertranspose by


AT
C`T
M ST =
.
−B`T DT
The transpose and supertranspose operations satisfy the following relations, see [1].
p×q
and C` ∈ Λq×p
. Then,
Proposition 2. Let M, L ∈ Mat(p|q)(ΛN ), x ∈ Kp|q (ΛN ), B` ∈ ΛN,1
N,1

(i)

T
B`C` = −C`T B`T ,
ST

(ii) (M L)

= LST M ST ,

5

T

(iii) (M x) = xT M ST ,




A −B`
Ip
ST ST
2
(iv) M
=
= SM S, where S =
0
−C` D


ST
−1
(v) M −1
= M ST
for every M ∈ GL(p|q)(ΛN ).

0
−Iq


,

The situation for the trace is similar. The usual trace tr(M ) of an element M ∈ Mat(p|q)(ΛN ) is well
defined, but in general, tr(M L) 6= tr(LM ) for M, L ∈ Mat(p|q)(ΛN ). The notion of supertrace provides
a solution to this problem; it is defined as the map str : Mat(p|q)(ΛN ) → ΛN,0 given by

str

A B`
C` D


= tr(A) − tr(D).

The following properties easily follow from the above definition, see [1].
q×p
Proposition 3. Let M, L ∈ Mat(p|q)(ΛN ), B` ∈ Λp×q
and C` ∈ ΛN,1
. Then
N,1


(i) tr B`C` = − tr C`B` ,

(ii) str(M L) = str(LM ),

(iii) str M ST = str(M ).
The superdeterminant or Berezinian is a function from GL(p|q)(ΛN ) to ΛN,0 defined by:
sdet(M ) =

det(A)
det(A − B`D−1 C`)
=
.
det(D)
det(D − C`A−1 B`)

Some of its most basic properties are given in the following proposition, see [1].
Proposition 4. Let M, L ∈ GL(p|q)(ΛN ), then
i) sdet(M L) = sdet(M ) sdet(L),

ii) sdet M ST = sdet(M ).
Since Mat(p|q)(ΛN ) is a finite dimensional vector space, every two norms in this space are equivalent,
whence, without loss of generality, we can define for M = {mj,k }j,k=1,...,p+q ∈ Mat(p|q)(ΛN ) the norm
Pp+q
kM k = j,k=1 kmj,k k1 , satisfying kM Lk ≤ kM kkLk for every pair M, L ∈ Mat(p|q)(ΛN ).
Similarly to Proposition 1 the following result can be proven.
Proposition 5. For every M ∈ Mat(p|q)(ΛN ) the exponential defined by
eM =

∞
X
Mj
j=0

j!

absolutely converges. In consequence, eM is a continuous function in Mat(p|q)(ΛN ).
Also the supertranspose, the supertrace and the superdeterminant are continuous functions.
Proposition 6. Let M, L ∈ Mat(p|q)(ΛN ). Then
(i) e0 = Ip+q ;
ST
ST
(ii) eM
= eM ;
(iii) If M L = LM then eM +L = eM eL ;
−1
(iv) eM ∈ GL(p|q)(ΛN ) and eM
= e−M ;
2

Ik denotes the identity matrix in Rk×k .

6

(v) e(a+b)M = eaM ebM for every pair a, b ∈ ΛN,0 ;
(vi) eCM C

−1

= CeM C −1 for every C ∈ GL(p|q)(ΛN );
d tM
dt e

(vii) etM (t ∈ R) is a smooth curve in Mat(p|q)(ΛN ),

(viii) sdet eM = estr(M ) .

= M etM = etM M and

d tM
dt e
t=0

= M;

Remark 2.1. The proofs of (i)-(vii) are straightforward computations. A detailed proof for (viii) can be
found in [1]. Similar properties to (i) and (iii)-(vii) can be obtained for the exponential map in ΛN .
We also can define the notion of logarithm for a supermatrix M ∈ Mat(p|q)(ΛN ) by
ln(M ) =

∞
X

(−1)j+1

j=1

(M − Ip+q )j
,
j

(8)

wherever it converges.
Proposition 7.
(i) The series (8) converges and yields a continuous function near Ip+q .
(ii) In Mat(p|q)(ΛN ), let U be a neighbourhood of Ip+q on which ln is defined and let V be a neighbourhood of 0 such that exp(V ) := {eM : M ∈ V } ⊂ U . Then eln(M ) = M , ∀M ∈ U , and ln(eL ) = L,
∀L ∈ V .
Proof.
(i) Observe that
∞
X
(M − Ip+q )j
j
j=1

≤

∞
j
X
kM − Ip+q k
.
j
j=1

Whence, since the radius of convergence of the last series is 1, (8) absolutely converges and defines
a continuous function in the ball kM − Ip+q k < 1.
(ii) The statement immediately follows from the absolutely convergence of the series for exp and ln,
and from the formal identities eln x = ln(ex ) = x in the indeterminate x.

It is worth mentioning that the same procedure can be repeated in ΛN . With the above definitions
of the exponential and logarithmic maps, it is possible to obtain all classical results known for Lie groups
and Lie algebras of real and complex matrices .
The exponential of a nilpotent matrix M ∈ Mat(p|q)(Λ+
N ) clearly is given by a finite sum, which
yields the bijective mapping:
+
exp : Mat(p|q)(Λ+
N ) → Ip+q + Mat(p|q)(ΛN )

with inverse
+
ln : Ip+q + Mat(p|q)(Λ+
N ) → Mat(p|q)(ΛN ),

since both expansions have a finite number of non-zero terms, whence problems of convergence do not
arise.
We recall that a supermatrix M belongs to GL(p|q)(ΛN ) if and only if its numeric projection M0 has
an inverse. Then, M = M0 (Ip+q + M0−1 M) = M0 exp(L), for some unique L ∈ Mat(p|q)(Λ+
N ).

7

3
3.1

The superspace framework with Grassmann coefficients
The Clifford setting

In order to set up the Clifford analysis framework in superspace, take p = m, q = 2n (m, n ∈ N)
and K = R. The canonical homogeneous basis e1 , . . . , em , e`1 , . . . , e`2n of Rm|2n can be endowed with an
orthogonal and a symplectic structure by the multiplication rules
ej ek + ek ej = −2δj,k ,

ej e`k + e`k ej = 0,

e`j e`k − e`k e`j = gj,k ,

where the symplectic form gj,k is defined by
g2j,2k = g2j−1,2k−1 = 0,

g2j−1,2k = −g2k,2j−1 = δj,k ,

j, k = 1, . . . , n.

Following these relations, elements in Rm|2n generate an infinite dimensional algebra denoted by Cm,2n .
A similar approach in the Grassmann envelope Rm|2n (ΛN ) leads to the definition of the algebra
Am,2n (ΛN ) = AlgR (f1 , . . . , fN , e1 , . . . , em , e`1 , . . . , e`2n ) = ΛN ⊗ Cm,2n
where it is assumed that elements of ΛN commute with elements of Cm,2n .
α2n
Every element in Am,2n (ΛN ) can be written as a finite sum of terms of the form aej1 · · · ejk e`1α1 · · · e`2n
2n
where a ∈ ΛN , 1 ≤ j1 ≤ . . . ≤ jk ≤ m and (α1 , . . . , α2n ) ∈ (N ∪ {0}) is a multi-index. In this algebra
we consider the corresponding generalization of the projection [·]0 which now goes from Am,2n (ΛN ) to
α2n
α2n
.
]0 = [a]0 ej1 · · · ejk e`1α1 · · · e`2n
Cm,2n and is defined by [aej1 · · · ejk e`1α1 · · · e`2n
Developing a proper Clifford analysis in this framework requires a suitable realization of the radial
algebra in Am,2n (ΛN ). For a detailed treatment of the radial algebra setting we refer the reader to
[4, 5, 12, 15]. This realization in Am,2n (ΛN ) is seen through a set of vector variables taking values in the
Grassmann envelope Rm|2n (ΛN ), i.e., supervectors of the form:
x = x + x`=

m
X

xj ej +

j=1

2n
X

x`j e`j ,

xj ∈ ΛN,0 ,

x`j ∈ ΛN,1 ,

j=1

which satisfy the fundamental axiom of radial algebra, i.e. the anti-commutator of every pair of them is
a commuting element. Indeed, for every pair x, y ∈ Rm|2n (ΛN )
{x, y} = −2

m
X
j=1

xj yj +

n
X
(x`2j−1 y`2j − x`2j y`2j−1 ) ∈ ΛN,0 .

(9)

j=1

The algebra generated by all elements in Rm|2n (ΛN ) is called the radial algebra embedded in Am,2n (ΛN )
and will be denoted by Rm|2n (ΛN ). This algebra is generated by the set of elements
{fA ej : A ⊂ {1, . . . , N }, |A| even , j = 1, . . . , m} ∪ {fA e`j : A ⊂ {1, . . . , N }, |A| odd , j = 1, . . . , 2n} ,
which turns Rm|2n (ΛN ) into a finite dimensional real space. For more details about this kind of realization
of the radial algebra we refer the reader to [3, 4, 13, 14].

3.2

Superbivectors

Superbivectors play a very important rôle in this work. Following the radial algebra approach, the space
of bivectors can be defined as the space of all ΛN,0 -linear combinations of wedge products of supervectors,
i.e.
X
X
X
1
x ∧ y = [x, y] =
(xj yk − xk yj )ej ek +
(xj y`k − x`k yj )ej e`k +
(x`j y`k + x`k y`j ) e`j e`k ,
2
1≤j≤m
1≤j<k≤m

where e`j

1≤j≤k≤2n

1≤k≤2n

(2)

e`k = 12 {e`j , e`k }. Hence, the space Rm|2n (ΛN ) of superbivectors consists of elements of the form
X
X
X
B=
bj,k ej ek +
b`j,k ej e`k +
Bj,k e`j e`k ,
(10)
1≤j<k≤m

1≤j≤k≤2n

1≤j≤m
1≤k≤2n

8

where bj,k ∈ ΛN,0 , b`j,k ∈ ΛN,1 and Bj,k ∈ ΛN,0 ∩ Λ+
N, . Observe that the coefficients Bj,k are commuting
but nilpotent since they are generated by elements of the form x`j y`k + x`k y`j that belong to Λ+
N, . This
(2)

constitutes an important limitation for the space of superbivectors because it means that Rm|2n (ΛN )
doesn’t allow for any other structure than the orthogonal one. In
P fact, the real projection [B]0 of
every superbivector B is just the classical Clifford bivector: [B]0 = 1≤j<k≤m [bj,k ]0 ej ek . Thence it is
(2)E

(2)

necessary to introduce an extension Rm|2n (ΛN ) of Rm|2n (ΛN ) containing elements B of the form (10)
but with Bj,k ∈ ΛN,0 . This extension allows to consider two different structures in the same element B:
the orthogonal and the symplectic one. In fact, in this case we have
X
X
[B]0 =
[bj,k ]0 ej ek +
[Bj,k ]0 e`j e`k .
1≤j<k≤m

1≤j≤k≤2n

(2)

(2)E

Remark 3.1. Observe that Rm|2n (ΛN ) and Rm|2n (ΛN ) are finite dimensional real vector spaces with

m(m − 1)
+ 2N −1 2mn + 2N −1 − 1 n(2n + 1),
2
m(m
− 1)
(2)E
+ 2N −1 2mn + 2N −1 n(2n + 1).
dim Rm|2n (ΛN ) = 2N −1
2
(2)

dim Rm|2n (ΛN ) = 2N −1

(2)E

The extension Rm|2n (ΛN ) of the superbivector space clearly lies out of the radial algebra Rm|2n (ΛN )
and generates an infinite dimensional algebra using the multiplication rules defined in Am,2n (ΛN ). Ele(2)E
ments in Rm|2n (ΛN ) are called extended superbivectors. Both superbivectors and extended superbivectors
preserve several properties of classical bivectors.
(2)E

(2)

Proposition 8. The space Rm|2n (ΛN ) is a Lie algebra. In addition, Rm|2n (ΛN ) is a Lie subalgebra of
(2)E

Rm|2n (ΛN ).
Proof.
We only need to check that the Lie bracket defined by the commutator in the associative algebra
(2)E
(2)
Am,2n (ΛN ) is an internal binary operation in Rm|2n (ΛN ) and Rm|2n (ΛN ). Direct computation shows
that for a, b ∈ ΛN,0 and a`, b`∈ ΛN,1 we get:
[aej ek , ber es ] = ab (2δj,s er ek − 2δs,k er ej + 2δr,j ek es − 2δr,k ej es ),
[aej ek , b`er e`s ] = ab`(2δr,j ek e`s − 2δr,k ej e`s ),
[aej ek , be`r

e`s ] = 0,
e`s + (1 − δj,r )gs,k ej er ),

[a`ej e`k , b`er e`s ] = a`b`(2δr,j e`k
[a`ej e`k , be`r
[ae`j

e`k , be`r

e`s ] = a`b (gk,s ej e`r + gk,r ej e`s ),
e`s ] = ab (gj,s e`r

e`k + gk,s e`r

e`j + gj,r e`k

e`s + gk,r e`j

e`s ).


It is well known from the radial algebra framework that the commutator of a bivector with a vector
always is a linear combination of vectors with coefficients in the scalar subalgebra. Indeed, for vector
variables x, y, z we obtain
1
1
[x ∧ y, z] = [[x, y], z] = [2xy − {x, y}, z] = [xy, z] = {y, z}x − {x, z}y.
2
2
(2)E

This property can be easily generalized to Rm|2n (ΛN ) by straightforward computation. In particular, the
following result holds.
Proposition 9. Let x ∈ Rm|2n (ΛN ), let {b1 , . . . , b2N −1 } be a basis for ΛN,0 and let {b`1 , . . . , b`2N −1 } be a
basis for ΛN,1 . Then,
[br ej ek , x] = 2br (xj ek − xk ej ),
[b`r ej e`2k−1 , x] = b`r (2xj e`2k−1 + x`2k ej ),
[b`r ej e`2k , x] = b`r (2xj e`2k − x`2k−1 ej ),

[br e`2j
[br e`2j−1
[br e`2j−1

e`2k , x] = −br (x`2j−1 e`2k + x`2k−1 e`2j ),
e`2k−1 , x] = br (x`2j e`2k−1 + x`2k e`2j−1 ),
e`2k , x] = br (x`2j e`2k − x`2k−1 e`2j−1 ).

The above computations also are valid for supervectors x in any extension of Rm|2n (ΛN ) such as Rm|2n (ΛN +1 ).

9

3.3

Tensor algebra and exponential map

Since Am,2n (ΛN ) is infinite dimensional, the definition of the exponential map by means of the power
P∞ j
series ex = j=0 xj! is not as straightforward as it was for the algebras ΛN or Mat(p|q)(ΛN ). A correct
definition of the exponential map in Am,2n (ΛN ) requires the introduction of the tensor algebra. More
details about the general theory of tensor algebras can be found several basic references, see e.g. [6, 11, 17].
Let T (V )L
be the tensor algebra of the vector space V with BV = {f1 , . . . , fN , e1 , . . . , em , e`1 , . . . , e`2n },
∞
i.e., T (V ) = j=0 T j (V ) where T j (V ) = spanR {v1 ⊗ · · · ⊗ vj : v` ∈ BV } is the j-fold tensor product of
V with itself. Then Am,2n (ΛN ) can be seen as a subalgebra of T (V )/I where I ⊂ T (V ) is the twosided
ideal generated by the elements:
fj ⊗ fk + fk ⊗ fj ,

fj ⊗ ek − ek ⊗ fj ,

fj ⊗ e`k − e`k ⊗ fj ,

ej ⊗ ek + ek ⊗ ej + 2δj,k ,

ej ⊗ e`k + e`k ⊗ ej ,

e`j ⊗ e`k − e`k ⊗ e`j − gj,k .

Indeed, T (V )/I is isomorphic to the extension of Am,2n (ΛN ) which also contains infinite sums of
α2n
where a ∈ ΛN , 1 ≤ j1 ≤ . . . ≤ jk ≤ m and
arbitrary terms of the form aej1 · · · ejk e`1α1 · · · e`2n
2n
(α1 , . . . , α2n ) ∈ (N ∪ {0}) is a multi-index.
P∞ j
The exponential map exp(x) = ex = j=0 xj! is known to be well defined in the tensor algebra T (V ),
see e.g. [6], whence it also is well defined in T (V )/I. It has the following mapping properties:
exp : Am,2n (ΛN ) → T (V )/I,

exp : Rm|2n (ΛN ) → Rm|2n (ΛN ).

The first statement directly follows from the definition of T (V )/I, while the second one can be obtained following the standard procedure established for ΛN and Mat(p|q)(ΛN ), since Rm|2n (ΛN ) is finite
dimensional.

The ortho symplectic structure in Rm|2n (ΛN )

4
4.1

Invariance of the inner product

The ΛN,0 -valued function given by the anti-commutator of supervectors (9) leads to the definition of the
symmetric bilinear form h·, ·i : Rm|2n (ΛN ) × Rm|2n (ΛN ) −→ ΛN,0 given by
m

n

X
1X
1
xj yj −
(x`2j−1 y`2j − x`2j y`2j−1 ) ∈ ΛN,0 .
hx, yi = − {x, y} =
2
2 j=1
j=1
which we will use as a generalized inner product. It can be easily written in terms of supermatrices, since




Im
0
0 1
T
hx, yi = x Qy
where Q =
, and J2n = diag
.
−1 0
0 − 21 J2n
Pm
When N = 1, the inner product hx, yi = j=1 xj yj coincides with the Euclidean inner product in Rm .
So from now on, we assume N > 1.
In order to find all the supermatrices M ∈ Mat(m|2n)(ΛN ) the corresponding linear operators of
which leave the inner product h·, ·i invariant, observe that

T
hM x, M yi = hx, yi ⇐⇒ (M x) QM y = xT Qy ⇐⇒ xT M ST QM − Q y = 0.
We thus have to identify those supermatrices L ∈ Mat(m|2n)(ΛN ) for which
xT L y = 0

∀x, y ∈ Rm|2n (ΛN ).

(11)

Lemma 4. A supermatrix L ∈ Mat(m|2n)(ΛN ) satisfies (11) if and only if L ∈ WN where WN is a
twosided ideal of Mat(m|2n)(ΛN ) defined as follows:
• if N = 2` + 1, W2`+1 is generated by the set of supermatrices



 



0 B`
0 0
(N )
(N −1)
L=
:
L
∈
Mat(m|2n)(Λ
)
∪
L
=
:
L
∈
Mat(m|2n)(Λ
)
;
N
N
0 D
C` 0

10

• if N = 2`, W2` is generated by the set of supermatrices




0 0
(N )
L=
: L ∈ Mat(m|2n)(ΛN ) .
0 D
Proof.
We can easily check by straightforward computations that the two subspaces defined above are two-sided
ideals of Mat(m|2n)(ΛN ) i.e., M WN ⊂ WN and WN M ⊂ WN for every M ∈ Mat(m|2n)(ΛN ). Let


A B`
L=
C` D
2n×2n
, B` = {b`j,k } ∈ Λm×2n
, C` = {c`j,k } ∈ Λ2n×m
and D = {dj,k } ∈ ΛN,0
.
with A = {aj,k } ∈ Λm×m
N,1
N,1
N,0
Choosing for x, y all possible coordinate supervectors we obtain that L satisfies (11) if and only if its
entries are such that

xj aj,k yk = 0

∀xj , yk ∈ ΛN,0 ,

xj b`j,k y`k = 0

∀xj ∈ ΛN,0 , y`k ∈ ΛN,1 ,

x`j c`j,k yk = 0

∀x`j ∈ ΛN,1 , yk ∈ ΛN,0 ,

x`j dj,k y`k = 0

∀x`j , y`k ∈ ΛN,1 .

Taking in the first equation xj = yk = 1 we obtain aj,k = 0 for every j, k = 1, . . . , m. For the remaining
coefficients we need to distinguish between two cases.
(N )
Case: N = 2` + 1. We first prove that b`j,k , c`j,k ∈ spanR {f1 · · · f2`+1 }. If b`j,k ∈ ΛN , it is clear that
b`j,k y`k = 0, for every y`k ∈ ΛN,1 . It then easily follows that xj b`j,k y`k = 0 for every pair xj ∈ ΛN,0 , y`k ∈
(N )

ΛN,1 . If on the contrary, b`j,k ∈
/ ΛN , there would be at least one of the generators fr missing in one of
the canonical terms of b`j,k . Then, taking xj = 1 and y`k = fr we will have xj b`j,k y`k 6= 0. Hence,
xj b`j,k y`k = 0

∀xj ∈ ΛN,0 , y`k ∈ ΛN,1

if and only if b`j,k ∈ spanR {f1 · · · f2`+1 }.

The same holds for c`j,k . Next we prove that dj,k ∈ spanR {f1 · · · fr−1 fr+1 · · · f2`+1 : 1 ≤ r ≤ 2` + 1}. In
(N −1)
fact, if dj,k ∈ ΛN
it is clear that x`j dj,k y`k = 0, since x`j dj,k y`k is the sum of homogeneous terms of
(N −1)
degree at least N + 1. If on the contrary, dj,k ∈
/ ΛN
, there would be at least two different Grassmann
generators fr , fs missing in one of the canonical terms of dj,k . Then, taking x`j = fr and y`k = fs we have
that x`j dj,k y`k 6= 0. Hence,
x`j dj,k y`k = 0

∀x`j , y`k ∈ ΛN,1

if and only if dj,k ∈ spanR {f1 · · · fr−1 fr+1 · · · f2`+1 : 1 ≤ r ≤ 2` + 1}.

Case: N = 2`. We first prove that b`j,k = c`j,k = 0. If b`j,k 6= 0, there will be always at least one
Grassmann generator fr missing in one of the terms of b`j,k since N is an even number. Then, taking
xj = 1 and y`k = fr we will have xj b`j,k y`k 6= 0. Hence,
xj b`j,k y`k = 0

∀xj ∈ ΛN,0 , y`k ∈ ΛN,1

if and only if b`j,k = 0.
(N )

The same holds for c`j,k . We now prove that dj,k ∈ spanR {f1 · · · f2` }. Indeed, if dj,k ∈ ΛN it is clear that
(N )
x`j dj,k y`k = 0. If on the contrary, dj,k ∈
/ ΛN , there would be at least two different Grassmann generators
fr , fs missing in one of the canonical terms of dj,k . Then, taking x`j = fr and y`k = fs we have that
x`j dj,k y`k 6= 0. Hence,
x`j dj,k y`k = 0

∀x`j , y`k ∈ ΛN,1

if and only if dj,k ∈ spanR {f1 · · · f2` }.


As a direct consequence of Lemma 4 we have the following result.
Corollary 1. The set O(m|2n)(ΛN ) of supermatrices in Mat(m|2n)(ΛN ) leaving the inner product h·, ·i
invariant is characterized by
O(m|2n)(ΛN ) = {M ∈ Mat(m|2n)(ΛN ) : M ST QM − Q ∈ WN }.

11

Remark 4.1. The form L = M ST QM − Q suggests that we do not need the whole ideal WN to describe
O(m|2n)(ΛN ). In fact, supermatrices of the previous form satisfy
LST = M ST QST M ST

ST


− QST = M ST QS (SM S) − QS = M ST QM − Q S = LS,

whence the subspace WN∗ = {L ∈ WN : LST = LS} can be considered in the above definition. However
the use of WN∗ is not important for the purposes of this paper, whence we will continue working with WN .
We will now study the algebraic structure of O(m|2n)(ΛN ).
Theorem 2. The following statements hold:
(i) O(m|2n)(ΛN ) ⊂ GL(m|2n)(ΛN ).
(ii) O(m|2n)(ΛN ) is a group under the usual matrix multiplication.
(iii) O(m|2n)(ΛN ) is a closed subgroup of GL(m|2n)(ΛN ).
Proof.
(i) To prove that every supermatrix in O(m|2n)(ΛN ) is invertible, first note that the real projection
of a supermatrix in WN (N > 1) is zero. Then for every M ∈ O(m|2n)(ΛN ) we have


 ST 
A0 0
T
M
Q[M
]
−
Q
=
[M
]
Q[M
]
−
Q
=
0,
where
[M
]
=
.
0
0
0
0
0
0 D0
This can be rewritten in terms of the real blocks A0 and D0 as AT0 A0 = Im and D0T J2n D0 = J2n ,
implying that A0 ∈ O(m) and D0 ∈ Sp(2n). On account of Theorem 1, M thus is invertible.
(ii) Here we only need to prove that matrix inversion and matrix multiplication are internal operations in O(m|2n)(ΛN ). For the inversion, the condition M ST QM − Q = L ∈ WN implies that
ST
ST
M −1
QM −1 − Q = − M −1
LM −1 ∈ WN since WN is an ideal. Hence, M ∈ O(m|2n)(ΛN )
−1
implies M
∈ O(m|2n)(ΛN ). For the multiplication, let M1 , M2 ∈ O(m|2n)(ΛN ) and L1 , L2 ∈
WN such that MjST QMj − Q = Lj , j = 1, 2. Then:
(M1 M2 )

ST


Q (M1 M2 ) − Q = M2ST M1ST QM1 M2 − Q = M2ST (Q + L1 ) M2 − Q
= L2 + M2ST L1 M2 .

Since WN is an ideal, L2 + M2ST L1 M2 ∈ WN . Hence, M1 M2 ∈ O(m|2n)(ΛN ).
(iii) Let {Mj }j∈N ⊂ O(m|2n)(ΛN ) be a sequence that converges to a supermatrix M ∈ Mat(m|2n)(ΛN ).
Since algebraic operations are continuous in Mat(m|2n)(ΛN ) we have that
lim MjST QMj − Q = M ST QM − Q.

j→∞

But MjST QMj − Q ∈ WN for every j ∈ N and, since WN is a finite dimensional subspace of
Mat(m|2n)(ΛN ), it is closed. Whence the limit M ST QM − Q belongs to WN .

Remark 4.2. The above theorem states that O(m|2n)(ΛN ) is a Lie group.
The group O(m|2n)(ΛN ) can be partitioned in a natural way into the classes
OL = OL (m|2n)(ΛN ) := {M ∈ O(m|2n)(ΛN ) : M ST QM − Q = L},

L ∈ WN .

Not for every L ∈ WN the set OL is non empty. In particular, OL = ∅ for every L ∈ WN \ WN∗ .
Proposition 10. The following statements hold:
(i) Let L1 , L2 ∈ WN and M1 ∈ OL1 , M2 ∈ OL2 . Then
M1 M2 ∈ OM2ST L1 M2 +L2

and

M1−1 ∈ O−(M −1 )ST L
1

12

−1
1 M1

.

(ii) OL is a subgroup of O(m|2n)(ΛN ) if and only if L = 0.
(iii) The binary relation:
R = {(M1 , M2 ) ∈ O(m|2n)(ΛN )2 : M1 , M2 ∈ OL ; L ∈ WN }
= {(M1 , M2 ) ∈ O(m|2n)(ΛN )2 : M1 M2−1 ∈ O0 }.
is an equivalence relation.
Proof.
(i) This was already proven in the proof of Theorem 2 (ii).
(ii) If OL is a subgroup then Im+2n ∈ OL and L = Im+2n QIm+2n − Q = 0. To prove now that O0
is a subgroup we just have to consider L1 = L2 = 0 in (i). Then M1 , M2 ∈ O0 directly implies
M1 M2 ∈ O0 and M1−1 ∈ O0 .
ST
(iii) Let M1 ∈ OL1 and M2 ∈ OL2 . From (i) we get that M1 M2−1 ∈ OL where L = M2−1
(L1 − L2 ) M2−1 .
−1
Thence, M1 , M2 ∈ OL for some L ∈ WN if and only if M1 M2 ∈ O0 . Since O0 is a subgroup of
O(m|2n)(ΛN ) it now easily follows that R is an equivalence relation.

Remark 4.3. The subgroup O0 is a closed subgroup of GL(m|2n)(ΛN ), whence it is a Lie group. It
plays a crucial rôle in the study of O(m|2n)(ΛN ) since given any representative element M of OL we can
describe the whole set OL by means of the relation OL = O0 M . For that reason, we will, from now on,
focus our attention to O0 .
Proposition 11. The following statements hold:
(i) A supermatrix M ∈ Mat(m|2n)(ΛN ) belongs to O0 if and only if

1 `T
`
T

A A − 2 C J2n C = Im ,
1 `T
T `
A B − 2 C J2n D = 0,

 `T ` 1 T
B B + 2 D J2n D = 12 J2n .

(12)

(ii) sdet(M ) = ±1 for every M ∈ O0 .
(iii) [O(m|2n)(ΛN )]0 = [O0 ]0 = O(m) × Sp(2n).
Proof.
i) The relation M ST QM = Q can be written in terms of A, B`, C`, D as:

 
Im
AT A − 12 C`T J2n C`
AT B` − 21 C`T J2n D
=
0
−B`T A − 12 DT J2n C` −B`T B` − 21 DT J2n D

0
− 12 J2n


.

(ii) On account of Proposition 4, M ST QM = Q implies that sdet(M )2 sdet(Q) = sdet(Q), whence
sdet(M )2 = 1. The statement then follows from Lemma 1.
(iii) See the proof of Theorem 2 i).

4.2



Group of superrotations SO0 .

As in the classical way, we can introduce now the set of superrotations by
SO0 = SO0 (m|2n)(ΛN ) = {M ∈ O0 : sdet(M ) = 1}.
This is easily seen to be a Lie subgroup of O0 with real projection equal to SO(m) × Sp(2n). In fact,
the conditions M ST QM = Q and sdet(M ) = 1 imply that M0T QM0 = Q and sdet(M0 ) = 1. This means
that


A0 0
M0 =
0 D0
with AT0 A0 = Im , D0T J2n D0 = J2n and det(A0 ) = det(D0 ). But D0 ∈ Sp(2n) implies det(D0 ) = 1. Then
det(A0 ) = 1 and A0 ∈ SO(m).
The following proposition states that, as in the classical case, SO0 is connected and in consequence,
it is the identity component of O0 .

13

Proposition 12. SO0 is a connected Lie group.
Proof.
Since the real projection SO(m) × Sp(2n) of SO0 is a connected group, it suffices to prove that for
every M ∈ SO0 there exist a continuous path inside SO0 connecting M with its real projection M0 . To
PN
(j)
that end, let us write M = j=0 [M ]j , where [M ]j is the projection of M on Mat(m|2n)(ΛN ) for each
j = 0, 1, . . . , N . Then, observe that

 



N
N
N
k
X
X
X
X





M ST QM = Q ⇐⇒ 
M ST j  Q  [M ]j  = Q ⇐⇒
M ST j Q[M ]k−j  = Q
j=0

j=0

⇐⇒ M0T QM0 = Q,

and

j=0

k=0

k
X
 ST 
M
Q[M ]k−j = 0,
j

k = 1, . . . , N.

j=0

PN
Let us now take the path M (t) = j=0 tj [M ]j . For t ∈ [0, 1] this is a continuous path with M (0) = M0
and M (1) = M . In addition, M (t)T0 QM (t)0 = M0T QM0 = Q and for every k = 1, . . . , N we have,
k
k
X
X


 ST 
Q[M ]k−j = 0.
M (t)ST j Q[M (t)]k−j = tk
M
j
j=0

j=0

Hence, M (t)ST QM (t) = Q, t ∈ [0, 1]. Finally, observe that sdet(M (t)) = 1 for every t ∈ [0, 1], since
sdet(M (t)0 ) = sdet(M0 ) = 1.

We will now investigate the corresponding Lie algebras of O(m|2n)(ΛN ), O0 and SO0 .
Theorem 3.
(i) The Lie algebra so(m|2n)(ΛN ) of O(m|2n)(ΛN ) is given by
so(m|2n)(ΛN ) = {X ∈ Mat(m|2n)(ΛN ) : X ST Q + QX ∈ WN }.
(ii) The Lie algebra so0 = so0 (m|2n)(ΛN ) of O0 coincides with the Lie algebra of SO0 and is given by
the space of all "super anti-symmetric" supermatrices
so0 = {X ∈ Mat(m|2n)(ΛN ) : X ST Q + QX = 0}.


A B`
(iii) A supermatrix X =
∈ Mat(m|2n)(ΛN ) belongs to so0 if and only if
C` D
AT + A = 0,

1
B` − C`T J2n = 0
2

and DT J2n + J2n D = 0.

(iv) [so(m|2n)(ΛN )]0 = [so0 ]0 = so(m) ⊕ sp(2n).
Proof.
(i) If X ∈ Mat(p|q)(ΛN ) is in the Lie algebra of O(m|2n)(ΛN ) then etX ∈ O(m|2n)(ΛN ) for every
ST
t ∈ R, i.e. etX QetX − Q = L(t) ∈ WN . Differentiating at t = 0 we obtain X ST Q + QX = L0 (0),
0
where L (0) ∈ WN since WN is a closed subspace. On the other hand, if X ∈ Mat(p|q)(ΛN ) satisfies
X ST Q + QX = L ∈ WN , then X ST = L1 − QXQ−1 where L1 = LQ−1 ∈ WN . Computing the
exponential of tX ST we obtain
j
j
∞
∞
X
X
Q(−tX)Q−1 + tL1
Q(−tX)Q−1
ST
=
+ L2 (t) = Qe−tX Q−1 + L2 (t),
etX =
j!
j!
j=0
j=0
where L2 (t) is a convergent infinite sum of products that contain the factor tL1 at least one time.
Then, since WN is a two-sided ideal, L2 (t) ∈ WN for every t ∈ R, whence
etX

ST

QetX − Q = L2 (t)QetX ∈ WN

14

∀t ∈ R.

(ii) To prove that so0 is the Lie algebra of O0 it suffices to repeat the above reasoning with L = L1 =
L2 = 0. From Proposition 6 it easily follows that the Lie algebra of SO0 is
{X ∈ Mat(p|q)(ΛN ) : X ST Q + QX = 0, str(X) = 0}.
But X ST Q + QX = 0 implies str(X) = 0. In fact, the condition X ST = −QXQ−1 implies that
str(X ST ) = − str(QXQ−1 ) = − str(X), yielding str(X) = str(X ST ) = − str(X) and str(X) = 0.
Hence, the Lie algebra of SO0 is so0 .
(iii) Observe that the relation X ST Q + QX = 0 can be written in terms of A, B`, C`, D as follows:


AT + A
− 12 C`T J2n + B`
= 0.
−B`T − 12 J2n C` − 12 DT J2n − 12 J2n D
(iv) As the real
we have [so(m|2n)(ΛN )]0 = [so0 ]0 .
 projection
 of every element in WN (N > 1) is zero,

A B`
A0 0
Let X =
∈ so0 , then X0 = [X]0 =
satisfies X0ST Q+QX0 = 0. Using (iii)
0 D0
C` D
we obtain AT0 + A0 = 0 and D0T J2n + J2n D0 = 0 which implies that A0 ∈ so(m) and D0 ∈ sp(2n).

Remark 4.4. As in the Remark 4.1, the supermatrices of the form L = X ST Q + QX satisfy LST =
QXS + X ST QS = LS. Then, the subspace WN∗ can replace WN in the above definition of so(m|2n)(ΛN ).
The connectedness of SO0 allows to write any of its elements as a finite product of exponentials of
supermatrices in so0 , see [8]. In the classical case, a single exponential suffices for such a description since
SO(m) is compact and in consequence exp : so(m) → SO(m) is surjective. This property, however, does
not hold in the group of superrotations SO0 , since the exponential map from sp(2n) to the non-compact
Lie group Sp(2n) ∼
= {Im } × Sp(2n) ⊂ SO0 is not surjective, whence not every element in SO0 can be
written as a single exponential of a supermatrix in so0 . Nevertheless, it is possible to find a decomposition
for elements of SO0 in terms of a fixed number of exponentials of so0 elements.
Every supermatrix M ∈ SO0 has a unique decomposition M = M0 + M = M0 (Im+2n + L) where
−1
M0 is its real projection, M ∈ Mat(m|2n)(Λ+
N ) its nilpotent projection and L = M0 M. We will now
separately study the decompositions for M0 ∈ SO(m) × Sp(2n) and Im+2n + L ∈ SO0 .
First consider M0 ∈ SO(m)×Sp(2n). We already mentioned that exp : so(m) → SO(m) is surjective,
while exp : sp(2n) → Sp(2n) is not. However, it can be proved that Sp(2n) = exp(sp(2n)) · exp(sp(2n)),
invoking the following polar decomposition for real algebraic Lie groups, see Proposition 4.3.3 in [9].
Proposition 13. Let G ⊂ GL(p) be an algebraic Lie group such that G = GT and g its Lie algebra.
Then every A ∈ G can be uniquely written as A = ReX , R ∈ G ∩ O(p), X ∈ g ∩ Sym(p), where Sym(p)
is the subspace of all symmetric matrices in Rp×p .
Taking p = 2n and G = Sp(2n) in the above proposition we get that every symplectic matrix D0 can
be uniquely written as D0 = R0 eZ0 with R0 ∈ Sp(2n) ∩ O(2n) and Z0 ∈ sp(2n) ∩ Sym(2n). But the group
Sp(2n) ∩ O(2n) is isomorphic to U (n) which is connected and compact. Then the exponential map from
the Lie algebra sp(2n) ∩ so(2n) ∼
= u(n) is surjective on Sp(2n) ∩ O(2n). This means that D0 ∈ Sp(2n) can
be written as D0 = eY0 eZ0 with Y0 ∈ sp(2n) ∩ so(2n) and Z0 ∈ sp(2n) ∩ Sym(2n). Hence, the supermatrix
M0 ∈ SO(m) × Sp(2n) can be decomposed as
 X
  X


Im
0
e 0
0
e 0
0
M0 =
=
= eX eY ,
0 eZ0
0
e Y0 e Z 0
0
e Y0




X0 0
0 0
where X =
∈ so(m) × [sp(2n) ∩ so(2n)] and Y =
∈ {0m } × [sp(2n) ∩ Sym(2n)].
0 Y0
0 Z0
Now consider the element Im+2n + L ∈ SO0 . As shown at the end of Section 2, the function
+
exp : Mat(m|2n)(Λ+
N ) → Im+2n + Mat(m|2n)(ΛN ) is a bijection with the logarithmic function defined in
(8) as its inverse. Then the supermatrix Z = ln(Im+2n + L) satisfies eZ = Im+2n + L and is nilpotent.
Those properties suffice for proving that Z ∈ so0 . From now on we will denote the set so0 ∩Mat(m|2n)(Λ+
N)
by so0 (m|2n)(Λ+
).
N

15

Z
Proposition 14. Let Z ∈ Mat(m|2n)(Λ+
N ) such that e ∈ SO0 . Then Z ∈ so0 .

Proof.
ST
It suffices to prove that etZ ∈ SO0 for every t ∈ R. The expression etZ QetZ − Q can be written as the
following polynomial in the real variable t.
 "

#
N
N
k k
j
ST
j
X
X
t
Z
t
(Z
)
tZST
tZ
Q
−Q
P (t) = e
Qe − Q = 
j!
k!
j=0
k=0


N X
k
k  
N
N
j
ST j
k−j k−j
k X
X
X
X
t (Z )
t Z
k
tk
t 
=
Q
=
(ZST )j QZk−j  =
Pk (Z),
j!
(k − j)!
k! j=0 j
k!
j=0
k=1

k=1

k=1


Pk
where Pk (Z) = j=0 kj (ZST )j QZk−j . If P (t) is not identically zero, i.e. not all the Pk (Z) are 0, we can
take k0 ∈ {1, 2, . . . , N } to be the largest subindex for which Pk0 (Z) 6= 0. Then,
lim

1

t→∞ tk0

P (t) =

Pk0 (Z)
6= 0,
k0 !

contradicting that P (Z) = {0}. So P (t) identically vanishes, yielding etZ ∈ SO0 for every t ∈ R.



This way, we have proven the following result.
Theorem 4. Every supermatrix in SO0 can be written as M = eX eY eZ with X ∈ so(m) × [sp(2n) ∩
so(2n)], Y ∈ {0m } × [sp(2n) ∩ Sym(2n)] and Z ∈ so0 (m|2n)(Λ+
N ). Moreover, the elements Y and Z are
unique.

4.3

Relation with superbivectors.

Theorem 3 allows to compute the dimension of so0 as a real vector space.


+
2mn
+
n(2n
+
1)
.
Corollary 2. The dimension of the real Lie algebra so0 is 2N −1 m(m−1)
2
Proof.
Since so0 is the direct sum of the corresponding subspaces of block components A, B`, C` and D respectively,
it suffices to compute the dimension of each one of them. According to Theorem 3 (iii) we have:


A
0



0
C`



0
0

V1 =
V2 =
V3 =

0
0



m×m
: AT = −A, A ∈ ΛN,0

1 `T
C J2n
2





∼
= ΛN,0 ⊗ so(m),



2n×m
∼
,
= ΛN,1 ⊗ R
0


0
∼
: DT J2n + J2n D = 0, D ∈ Λ2n×2n
= ΛN,0 ⊗ sp(2n),
N,0
D
2n×m
: C` ∈ ΛN,1

then dim V1 = 2N −1

m(m − 1)
,
2

then dim V2 = 2N −1 m2n,
then dim V3 = 2N −1 n(2n + 1).


(2)E
Comparing this result with the one in Remark 3.1 we obtain that dim Rm|2n (ΛN ) = dim so0 . This
means that both vector spaces are isomorphic. This isomorphism also holds on the Lie algebra level.
Following the classical Clifford approach, the commutator
[B, x]

(2)E

∀ B ∈ Rm|2n (ΛN ), x ∈ Rm|2n (ΛN ),

(13)
(2)E

should be the key for the Lie algebra isomorphism. Proposition 9 shows that for every B ∈ Rm|2n (ΛN )
the commutator (13) defines an endomorphism over Rm|2n (ΛN ) that can be represented by a supermatrix
in Mat(m|2n)(ΛN ). But as it was explained in Section 2, that supermatrix is not unique. This issue can
be solved with the natural extension of the linear operator defined in (13) to Rm|2n (ΛN +1 ).

16

(2)E

Lemma 5. The map φ : Rm|2n (ΛN ) → Mat(m|2n)(ΛN ) defined by
φ(B)x = [B, x]

(2)E

∀ B ∈ Rm|2n (ΛN ), x ∈ Rm|2n (ΛN +1 ),

(14)

takes values in so0 . In particular, if we consider {b1 , . . . , b2N −1 } and {b`1 , . . . , b`2N −1 } to be the canonical
basis of ΛN,0 and ΛN,1 respectively, we obtain the following basis for so0 .


Ek,j − Ej,k 0
,
φ(br ej ek ) = 2br
0
0


0
Ej,2k
,
φ(b`r ej e`2k−1 ) = b`r
2E2k−1,j
0


0
−Ej,2k−1
φ(b`r ej e`2k ) = b`r
,
2E2k,j
0


0
0
φ(br e`2j e`2k ) = −br
,
0 E2j,2k−1 + E2k,2j−1


0
0
φ(br e`2j−1 e`2k−1 ) = br
,
0 E2j−1,2k + E2k−1,2j


0
0
,
φ(br e`2j−1 e`2k ) = br
0 E2k,2j − E2j−1,2k−1


0
0
,
φ(br e`2j e`2k−1 ) = br
0 E2j,2k − E2k−1,2j−1

1 ≤ r ≤ 2N −1 , 1 ≤ j < k ≤ m,
1 ≤ r ≤ 2N −1 , 1 ≤ j ≤ m, 1 ≤ k ≤ n,
1 ≤ r ≤ 2N −1 , 1 ≤ j ≤ m, 1 ≤ k ≤ n,
1 ≤ r ≤ 2N −1 , 1 ≤ j ≤ k ≤ n,
1 ≤ r ≤ 2N −1 , 1 ≤ j ≤ k ≤ n,
1 ≤ r ≤ 2N −1 , 1 ≤ j ≤ k ≤ n,
1 ≤ r ≤ 2N −1 , 1 ≤ j < k ≤ n,

where Ej,k denotes the matrix with all entries equal 0, except the one in the j-th row and k-th column
which is equal to 1, and the order of Ej,k should be deduced from the context.
Proof.
The above equalities can be directly obtained from Proposition 9, whence we should only check that all
supermatrices obtained above form a basis for so0 . The matrices Ej,k satisfy the relations
T
Ej,k
= Ek,j , Ej,2k−1 J2n = Ej,2k , Ej,2k J2n = −Ej,2k−1 , J2n E2j,k = E2j−1,k , J2n E2j−1,k = −E2j,k .

Then
• for φ(br ej ek ) we have A = 2br (Ek,j − Ej,k ), B` = 0, C` = 0 and D = 0, whence
AT = 2br (Ej,k − Ek,j ) = −A;
• for φ(b`r ej e`2k−1 ) we have A = 0, B` = b`r Ej,2k , C` = 2b`r E2k−1,j and D = 0, whence
1 `T
C J2n = b`r Ej,2k−1 J2n = b`r Ej,2k = B`;
2
• for φ(b`r ej e`2k ) we have A = 0, B` = −b`r Ej,2k−1 , C` = 2b`r E2k,j and D = 0, whence
1 `T
C J2n = b`r Ej,2k J2n = −b`r Ej,2k−1 = B`;
2
• for φ(br e`2j

e`2k ) we have A = 0, B` = 0, C` = 0 and D = −br (E2j,2k−1 + E2k,2j−1 ), whence

DT J2n + J2n D = −br (E2k−1,2j J2n + E2j−1,2k J2n + J2n E2j,2k−1 + J2n E2k,2j−1 )
= −br (−E2k−1,2j−1 − E2j−1,2k−1 + E2j−1,2k−1 + E2k−1,2j−1 ) = 0;
• for φ(br e`2j−1

e`2k−1 ) we have A = 0, B` = 0, C` = 0 and D = br (E2j−1,2k + E2k−1,2j ), whence

DT J2n + J2n D = br (E2k,2j−1 J2n + E2j,2k−1 J2n + J2n E2j−1,2k + J2n E2k−1,2j )
= br (E2k,2j + E2j,2k − E2j,2k − E2k,2j ) = 0;

17

• for φ(br e`2j−1

e`2k ) we have A = 0, B` = 0, C` = 0 and D = br (E2k,2j − E2j−1,2k−1 ), whence

DT J2n + J2n D = br (E2j,2k J2n − E2k−1,2j−1 J2n + J2n E2k,2j − J2n E2j−1,2k−1 )
= br (−E2j,2k−1 − E2k−1,2j + E2k−1,2j + E2j,2k−1 ) = 0.
The above computations show that all supermatrices obtained belong to so0 . Direct verification shows
that they form a set of 2N −1 m(m−1)
+ 2N −1 2mn + 2N −1 n(2n + 1) linear independent elements, i.e. a
2
basis of so0 .

(2)E

Theorem 5. The map φ : Rm|2n (ΛN ) → so0 defined in (14) is a Lie algebra isomorphism.
Proof.
From Lemma 5 follows that φ is a vector space isomorphism. In addition, due to the Jacoby identity in
(2)E
the associative algebra Am,2n (ΛN ) we have for every B1 , B2 ∈ Rm|2n (ΛN ) and x ∈ Rm|2n (ΛN +1 ) that
[φ(B1 ), φ(B2 )] x = φ(B1 )φ(B2 )x − φ(B2 )φ(B1 )x
= [B1 , [B2 , x]] + [B2 , [x, B1 ]] = [[B1 , B2 ] , x] = φ ([B1 , B2 ]) x.
implying, by Lemma 3, that [φ(B1 ), φ(B2 )] = φ ([B1 , B2 ]), i.e., φ is a Lie algebra isomorphism.

5



The Spin group in Superspace

So far we have seen that the Lie algebra so0 of the Lie group of superrotations SO0 has a realization
in Am,2n (ΛN ) as the Lie algebra of extended superbivectors. In this section, we discuss the proper way
of defining the corresponding realization of SO0 in T (V )/I, i.e., the analogue of the Spin group in the
Clifford superspace framework.

5.1

Supervector reflections

The group generated by the supervector reflections was briefly introduced in [14] using the notion of
the unit super-sphere defined as the super-surface S(m|2n)(ΛN ) = {w ∈ Rm|2n (ΛN ) : w2 = −1}. The
reflection associated to the supervector w ∈ S(m|2n)(ΛN ) is defined by ψ(w)[x] = wxw, x ∈ Rm|2n (ΛN ).
It is known from the radial algebra setting that ψ(w) maps vectors into vectors. Indeed,
wxw = {x, w}w − w2 x = {x, w}w + x = x − 2hx, wiw.
Every supervector reflection can be identified with a unique matrix in O(m|2n)(ΛN ). On account of
the results in Section 2, an option for doing this is extending the operator ψ(w) to Rm|2n (ΛN +1 ) in the
natural way, i.e.
ψ(w)[x] = wxw
x ∈ Rm|2n (ΛN +1 ).
(15)
Pm
P2n
Lemma 6. Let w = w + w`= j=1 wj ej + j=1 w`j e`j ∈ S(m|2n)(ΛN ). Then, the endomorphism (15)
can be represented by an unique supermatrix


A(w) B`(w)
ψ(w) =
∈ O(m|2n)(ΛN )
C`(w) D(w)
with A(w) = −2Dw Em×m Dw + Im , B`(w) = Dw Em×2n Dw`J2n , C`(w) = −2Dw`E2n×m Dw , and finally
D(w) = Dw`E2n×2n Dw`J2n + I2n , where






w1
w`1
1 1 ... 1




 . .
. 
p×q
..
..
Dw = 
 , Dw` = 
 and Ep×q =  .. .. . . . ..  ∈ R .
.
.
wm

w`2n

18

1

1

...

1

Proof.
Pm
P2n
Observe that ψ(w)[x] = wxw = {x, w}w + x = k=1 yk ek + k=1 y`j e`k , where


m
n
X
X
yk = −2 
wj wk xj  + xk +
w`2j−1 wk x`2j − w`2j wk x`2j−1 ,
j=1

j=1



m
n
X
X
y`k = −2 
wj w`k xj  + x`k +
−w`2j−1 w`k x`2j + w`2j w`k x`2j−1 .
j=1


Then, ψ(w)x =

A(w) B`(w)
C`(w) D(w)




A(w) = −2 


w12
w1 w2
..
.
w1 wm



w2 w1
w22
..
.
w2 wm

j=1

x
x`
...
...
..
.
...


where,
wm w1
wm w2
..
.
2
wm




 + Im = −2Dw Em×m Dw + Im ,


−w`2 w1 w`1 w1 . . . −w`2n w1
 −w`2 w2 w`1 w2 . . . −w`2n w2

B`(w) = 
..
..
..
..

.
.
.
.
−w`2 wm w`1 wm . . . −w`2n wm

w1 w`1
w2 w`1 . . . wm w`1
 w1 w`2
w2 w`2 . . . wm w`2

C`(w) = −2 
..
..
..
..

.
.
.
.
w1 w`2n w2 w`2n . . . wm w`2n

w`2 w`1
−w`1 w`1 . . . w`2n w`1
 w`2 w`2
−w`1 w`2 . . . w`2n w`2

D(w) = 
..
..
..
..

.
.
.
.
w`2 w`2n −w`1 w`2n . . . w`2n w`2n


w`2n−1 w1
w`2n−1 w2
..
.
w`2n−1 wm





 = Dw Em×2n Dw`J2n ,




 = −2Dw`E2n×m Dw ,

−w`2n−1 w`1
−w`2n−1 w`2
..
.
−w`2n−1 w`2n




 + I2n = Dw`E2n×2n Dw`J2n + I2n .


The uniqueness of such a supermatrix in Mat(m|2n)(ΛN ) is guaranteed by Lemma 3 and it is easily seen
that ψ(w) belongs to O(m|2n)(ΛN ) since
1
1
1
hψ(w)[x], ψ(w)[y]i = − {wxw, wyw} = w{x, y}w = − {x, y} = hx, yi.
2
2
2

Algebraic operations with the matrices A(w), B`(w), C`(w), D(w) are easy since




m
n
X
X
2
Ep×m Dw
Em×q = 
wj2  Ep×q ,
Ep×2n Dw`J2n Dw`E2n×q = 2 
w`2j−1 w`2j  Ep×q .
j=1

j=1

We can now define the bosonic Pin group in superspace as
Pinb (m|2n)(ΛN ) = {w1 · · · wk : wj ∈ S(m|2n)(ΛN ), k ∈ N},
and extend the map ψ to a Lie group homomorphism ψ : Pinb (m|2n)(ΛN ) → O(m|2n)(ΛN ) by
ψ(w1 · · · wk )[x] = w1 · · · wk x wk · · · w1 = ψ(w1 ) ◦ · · · ◦ ψ(wk )[x].
Proposition 15. Let w ∈ S(m|2n)(ΛN ). Then ψ(w) ∈ O0 and sdet (ψ(w)) = −1.
Proof.

19

(16)

`
To prove that ψ(w) ∈ O0 it suffices to prove thatPA(w), B`(w),
PnC (w), D(w) satisfy (12). This can be easily
m
2
2
done using (16) and the identity −1 = w = − j=1 wj + j=1 w`2j−1 w`2j . In fact, we have
2
Em×m Dw − 4Dw Em×m Dw + Im
A(w)T A(w) = 4Dw Em×m Dw


m
X
= 4
wj2  Dw Em×m Dw − 4Dw Em×m Dw + Im ,
j=1



n
X
C`(w)T J2n C`(w) = 4Dw Em×2n Dw`J2n Dw`E2n×m Dw = 8 
w`2j−1 w`2j  Dw Em×m Dw .
j=1

Then, A(w)T A(w) − 21 C`(w)T J2n C`(w) = 4
Also,

hP
m

2
j=1 wj − 1 −

i
w
`
w
`
j=1 2j−1 2j Dw Em×m Dw + Im = Im .

Pn

2
Em×2n Dw`J2n + Dw Em×2n Dw`J2n
A(w)T B`(w) = −2Dw Em×m Dw


m
X
= −2 
wj2  Dw Em×2n Dw`J2n + Dw Em×2n Dw`J2n ,
j=1

C`(w)T J2n D(w) = −2Dw Em×2n Dw`J2n Dw`E2n×2n Dw`J2n − 2Dw Em×2n Dw`J2n


n
X
= −4 
w`2j−1 w`2j  Dw Em×2n Dw`J2n − 2Dw Em×2n Dw`J2n .
j=1

h P
i
Pn
m
Hence, A(w)T B`(w)− 12 C`(w)T J2n D(w) = 2 − j=1 wj2 + 1 + j=1 w`2j−1 w`2j Dw Em×2n Dw`J2n = 0. In
the same way we have


m
X
2
Em×2n Dw`J2n = − 
wj2  J2n Dw`E2n×2n Dw`J2n ,
B`(w)T B`(w) = −J2n Dw`E2n×m Dw
j=1
T

D(w) J2n D(w) = J2n Dw`E2n×2n Dw`J2n Dw`E2n×2n Dw`J2n + 2J2n Dw`E2n×2n Dw`J2n + J2n


n
X
= 2
w`2j−1 w`2j + 1 J2n Dw`E2n×2n Dw`J2n + J2n ,
j=1

whence


m
n
X
X
1
1
1
B`(w)T B`(w) + DT (w)J2n D(w) = −
wj2 + 1 +
w`2j−1 w`2j  J2n Dw`E2n×2n Dw`J2n + J2n = J2n .
2
2
2
j=1
j=1
Then, A(w), B`(w), C`(w), D(w) satisfy (12) and in consequence, ψ(w) ∈ O0 . To prove that sdet(ψ(w)) =
−1, first observe that ψ(w) = ψ(w)−1 since ψ(w) ◦ ψ(w)[x] = wwxww = x. Hence, due to Theorem 1 we
−1
obtain A(w) = A(w) − B`(w)D(w)−1 C`(w)
, yielding


det A(w) − B`(w)D(w)−1 C`(w)
1
=
.
sdet(ψ(w)) =
det[D(w)]
det[A(w)] det[D(w)]
We will compute det[D(w)] using the formula det[D(w)] = exp(tr ln D(w)) and the fact that D(w) − I2n
is a nilpotent matrix. Observe that
∞
∞
j
X
X
Dw`E2n×2n Dw`J2n
j+1 (D(w) − I2n )
ln D(w) =
(−1)
=
(−1)j+1
j
j
j=1
j=1

20

j
.

It follows from (16) that

Dw`E2n×2n Dw`J2n

j


j−1
n
X
= 2j−1 
w`2j−1 w`2j 
Dw`E2n×2n Dw`J2n .
j=1

Then,

P
j−1 
n
j−1
∞
2
w
`
w
`
j=1 2j−1 2j

X
ln D(w) =  (−1)j+1
 Dw`E2n×2n Dw`J2n
j
j=1
and in consequence,

tr ln D(w) = −

∞
X

(−1)j+1

2j

P

n
j=1

w`2j−1 w`2j

j

= − ln 1 + 2

j

j=1



n
X


w`2j−1 w`2j  .

j=1

Hence

1

det(D(w)) =
1+2

n
P

w`2j−1 w`2j

j=1

Similar computations yield det(A(w)) = 1 − 2

m
P
j=1

wj2 which shows that sdet(ψ(w)) = −1.



The above proposition states that the Lie group homomorphism ψ takes values in O0 , and its restriction to the bosonic spin group, defined as
Spinb (m|2n)(ΛN ) = {w1 · · · w2k : wj ∈ S(m|2n)(ΛN ), k ∈ N},
takes values in the subgroup SO0 .
In the classical case, the Pin and the Spin groups are a double covering of the groups O(m) and
SO(m) respectively. A natural question in this setting is whether Pinb (m|2n)(ΛN ) and Spinb (m|2n)(ΛN )
cover the groups O0 and SO0 . The answer to this question is negative and the main reason is that the
real projection of every vector w ∈ S(m|2n)(ΛN ) is in the unitary sphere Sm−1 of Rm , i.e.,
[w]0 =

m
X

[wj ]0 ej

and

[w]20 = −1.

j=1

Then, the real projection of ψ (Pinb (m|2n)(ΛN )) is just O(m), while [O0 ]0 = O(m) × Sp(2n). This means
that these bosonic versions of Pin and Spin do not describe the symplectic parts of O0 and SO0 . This
phenomenon is due to the natural structure of supervectors: their real projections belong to a space
with an orthogonal structure while the symplectic structure plays no rôle. Up to a nilpotent vector, they
are classical Clifford vectors, whence it is impossible to generate by this approach the real symplectic
geometry that is also present in the structure of O0 and SO0 . That is why we have chosen the name
of "bosonic" Pin and "bosonic" Spin groups. This also explains why we had to extend the space of
superbivectors in section 3. The ordinary superbivectors are generated over ΛN,0 by the wedge product
of supervectors. Then, they can only describe so(m) and not sp(2n) and in consequence, they do not
cover so0 .
As in the classical setting (see [7]), it is possible to obtain the following result that shows, from
another point of view, that Pinb (m|2n)(ΛN ) cannot completely describe O0 .
(2)

Proposition 16. The Lie algebra of Pinb (m|2n)(ΛN ) is included in Rm|2n (ΛN ).
Proof.
Let γ(t) = w1 (t) · · · wk (t) be a path in Pinb (m|2n)(ΛN ) with wj (t) ∈ S(m|2n)(ΛN ) for every t ∈ R and
Pk
0
γ(0) = 1. The tangent to γ at t = 0 is dγ
j=1 w1 (0) · · · wj (0) · · · wk (0). We will show that each
dt t=0 =
summand of

dγ
dt t=0

(2)

belongs to Rm|2n (ΛN ).

21

For j = 1 we have w10 (0)w2 (0) · · · wk (0) = −w10 (0)w1 (0). But w1 (t)w1 (t) ≡ −1 implies
w10 (0)w1 (0) + w1 (0)w10 (0) = 0

⇒

hw10 (0), w1 (0)i = 0.
(2)

Then w10 (0)w1 (0) = −hw10 (0), w1 (0)i + w10 (0) ∧ w1 (0) = w10 (0) ∧ w1 (0) ∈ Rm|2n (ΛN ). For j = 2,
w1 (0)w20 (0) · · · wk (0) = w1 (0)w20 (0)w2 (0)w1 (0) = − [w1 (0)w20 (0)w1 (0)] · [w1 (0)w2 (0)w1 (0)]
= −ψ(w1 (0))[w20 (0)] · ψ(w1 (0))[w2 (0)].
But ψ(w1 (0)) preserves the inner product, so w1 (0)w20 (0) · · · wk (0) = ψ(w1 (0))[w20 (0)] ∧ ψ(w1 (0))[w2 (0)]
(2)
∈ Rm|2n (ΛN ). We can proceed similarly for every j = 3, . . . , k.


5.2

A proper definition for the group Spin(m|2n)(ΛN )

The above approach shows that the radial algebra setting does not contain a suitable realization of SO0
(2)E
in the Clifford superspace framework. Observe that the Clifford realization of so0 given by Rm|2n (ΛN )
lies outside of the radial algebra Rm|2n (ΛN ), which suggests that something similar should happen with
the Clifford realization of the Lie group SO0 . In this case, a proper definition for the Spin group would
(2)E
be generated by the exponentials (in general contained in T (V )/I) of all the elements in Rm|2n (ΛN ), i.e.
n
o
(2)E
Spin(m|2n)(ΛN ) := eB1 · · · eBk : B1 , . . . , Bk ∈ Rm|2n (ΛN ), k ∈ N ,
and the action of this group on Rm|2n (ΛN ) is given by the group homomorphism h : Spin(m|2n)(ΛN ) →
SO0 defined by the restriction to Rm|2n (ΛN ) of
h(eB )[x] = eB xe−B ,

(2)E

B ∈ Rm|2n (ΛN ), x ∈ Rm|2n (ΛN +1 ).

In fact, for every extended superbivector B, h(eB ) maps supervectors into supervectors and admits a
supermatrix representation in Mat(m|2n)(ΛN ) belonging to SO0 . This is summarized below.
(2)E

Proposition 17. Let B ∈ Rm|2n (ΛN ). Then, h(eB )[x] = eφ(B) x for every x ∈ Rm|2n (ΛN +1 ).
Proof.
k  
X
k
B j x(−B)k−j holds. Then,
In every associative algebra, the identity [B, [B . . . [B, x] . . .]] =
{z
}
|
j
j=0
k

h(eB )[x] = eB xe−B =

∞
X
Bk
k=0

k!

!
x

∞
X
(−B)k
k=0

!
=

k!

∞
X
k=0



k
j
k−j
X
B
(−B)


x
j!
(k − j)!
j=0



∞
k  
∞
∞
X
X
X
1
φ(B)k x
1 X k
B j x(−B)k−j  =
[B, [B . . . [B, x] . . .]] =
= eφ(B) x.
=
{z
}
k! j=0 j
k! |
k!
k=0

k=0

k

k=0


(2)E

Remark 5.1. The above proposition means that the Lie algebra isomorphism φ : Rm|2n (ΛN ) → so0 is
the derivative at the origin of the Lie group homomorphism h : Spin(m|2n)(ΛN ) → SO0 , i.e.,
etφ(B) = h(etB )

(2)E

∀t ∈ R, B ∈ Rm|2n (ΛN ).

On account of the connectedness of SO0 it can be shown that the group Spin(m|2n)(ΛN ) is a realization of SO0 in T (V )/I through the representation h.
Theorem 6. For every M ∈ SO0 there exist s ∈ Spin(m|2n)(ΛN ) such that h(s) = M .

22

Proof.
Since SO0 is a connected Lie group (Proposition 12), for every supermatrix M ∈ SO0 there exist
(2)E
X1 , . . . , Xk ∈ so0 such that eX1 · · · eXk = M , see Corollary 3.47 in [8]. Taking B1 , . . . , Bk ∈ Rm|2n (ΛN )
such that φ(Bj ) = Xj , j = 1, . . . , k, we obtain
M x = eX1 · · · eXk x = eφ(B1 ) · · · eφ(Bk ) x = h(eB1 ) ◦ · · · ◦ h(eBk )[x] = h(eB1 · · · eBk )[x].
The above equality is valid for every x ∈ Rm|2n (ΛN +1 ). Then, since h(eB1 · · · eBk ) and M belong to
Mat(m|2n)(ΛN ), Lemma 3 guarantees that s = eB1 · · · eBk ∈ Spin(m|2n)(ΛN ) satisfies h(s) = M .

The decomposition of SO0 given in Theorem 4 provides the exact number of exponentials of extended
superbivectors to be considered in Spin(m|2n)(ΛN ) in order to cover the whole group SO0 . If we consider
(2)E
the subspaces S1 , S2 , S3 of Rm|2n (ΛN ) given by
m(m − 1)
+ n2 ,
2
dim S2 = n2 + n,


S1 = φ−1 so(m) × [sp(2n) ∩ so(2n)] ,

S2 = φ−1 {0m } × [sp(2n) ∩ Sym(2n)] ,

S3 = φ−1 so0 (m|2n)(Λ+
N)

dim S1 =

dim S3 = dim so0 −

m(m − 1)
− n(2n + 1),
2

(2)E

we get the decomposition Rm|2n (ΛN ) = S1 ⊕S2 ⊕S3 , leading to the subset S = exp(S1 ) exp(S2 ) exp(S3 ) ⊂
Spin(m|2n)(ΛN ) which suffices for describing SO0 . Indeed, from Theorem 4 it follows that the restriction
h : S → SO0 is surjective. We now investigate the explicit form of the superbivectors in each one of the
subspaces S1 , S2 and S3 .
Proposition 18. The following statements hold.


1 ≤ j < k ≤ m,
ej ek ,
(i) A basis for S1 is e`2j−1 e`2k−1 + e`2j e`2k , 1 ≤ j ≤ k ≤ n,


e`2j−1 e`2k − e`2j e`2k−1 , 1 ≤ j < k ≤ n.


1 ≤ j ≤ n,
e`2j−1 e`2j ,
(ii) A basis for S2 is: e`2j−1 e`2k−1 − e`2j e`2k , 1 ≤ j ≤ k ≤ n,


e`2j−1 e`2k + e`2j e`2k−1 , 1 ≤ j < k ≤ n.
(iii) S3 consists of all elements of the form (10) with bj,k , Bj,k ∈ ΛN,0 ∩ Λ+
N, and b`j,k ∈ ΛN,1 .
Proof.
We first recall that a basis for the Lie algebra sp(2n) is given by the elements
Aj,k := E2j,2k−1 + E2k,2j−1 , 1 ≤ j ≤ k ≤ n

Bj,k := E2j−1,2k + E2k−1,2j , 1 ≤ j ≤ k ≤ n,

Cj,k := E2k,2j − E2j−1,2k−1 , 1 ≤ j ≤ k ≤ n,

Dj,k := E2j,2k − E2k−1,2j−1 , 1 ≤ j < k ≤ n,

where the matrices Ej,k ∈ Rn×n are defined as in Lemma 5. It holds that ATj,k = Bj,k for 1 ≤ j ≤ k ≤ n,
T
T
Cj,k
= Dj,k for 1 ≤ j < k ≤ n and Cj,j
= Cj,j for 1 ≤ j ≤ n. Hence, for every matrix D0 ∈ sp(2n) we
have
X
X
D0 =
(aj,k Aj,k + bj,k Bj,k + cj,k Cj,k ) +
dj,k Dj,k ,
1≤j≤k≤n

D0T =

X

1≤j<k≤n

(aj,k Bj,k + bj,k Aj,k ) +

1≤j≤k≤n

X
1≤j<k≤n

where aj,k , bj,k , cj,k , dj,k ∈ R.

23

(cj,k Dj,k + dj,k Cj,k ) +

n
X
j=1

cj,j Cj,j ,

(i) From the previous equalities we get that D0T = −D0 if and only if
X
X
D0 =
aj,k (Aj,k − Bj,k ) +
cj,k (Cj,k − Dj,k ) .
1≤j≤k≤n

1≤j<k≤n

Then, a basis for sp(2n) ∩ so(2n) is {Aj,k − Bj,k : 1 ≤ j ≤ k ≤ n} ∪ {Cj,k − Dj,k : 1 ≤ j < k ≤ n}.
The rest of the proof directly follows from Lemma 5.
(ii) In this case we have that D0T = D0 if and only if
X

D0 =

X

aj,k (Aj,k + Bj,k ) +

1≤j≤k≤n

cj,k (Cj,k + Dj,k ) +

n
X

cj,j Cj,j ,

j=1

1≤j<k≤n

whence a basis for sp(2n) ∩ Sym(2n) is
{Aj,k + Bj,k : 1 ≤ j ≤ k ≤ n} ∪ {Cj,j : 1 ≤ j ≤ n} ∪ {Cj,k + Dj,k : 1 ≤ j < k ≤ n}.
The rest of the proof directly follows from Lemma 5.


iii) This trivially follows from Lemma 5.

5.3

Spin covering of the group SO0

It is a natural question in this setting whether the spin group still is a double covering of the group of
rotations, as it is in classical Clifford analysis. In order to answer this question, we investigate how many
times S ⊂ Spin(m|2n)(ΛN ) covers SO0 , or more precisely, the cardinal of the set {s ∈ S : h(s) = M }
given a certain fixed element M ∈ SO0 .
From Proposition 17 we have that the representation h of an element s = eB1 eB2 eB3 ∈ S, Bj ∈ Sj ,
has the form h(s) = eφ(B1 ) eφ(B2 ) eφ(B3 ) . Following the decomposition M = eX eY eZ given in Theorem 4
for M ∈ SO0 , we get that h(s) = M if and only if eφ(B1 ) = eX , B2 = φ−1 (Y ) and B3 = φ−1 (Z). Then,
the cardinal of {s ∈ S : h(s) = M } only depends on the number of extended superbivectors B1 ∈ S1 that
satisfy eφ(B1 ) = eX . It reduces our analysis to finding the kernel of the restriction h|exp(S1 ) : exp(S1 ) →
SO(m) × [Sp(2n) ∩ SO(2n)] of the Lie group homomorphism h to exp(S1 ). This kernel is given by
ker h|exp(S1 ) = {eB : eφ(B) = Im+2n , B ∈ S1 }.
(2)

We recall, from Proposition 18, that B ∈ S1 may be written as B = Bo + Bs where Bo ∈ R0,m is a
classical real bivector and Bs ∈ φ−1 ({0m } × [sp(2n) ∩ so(2n)]). The components Bo , Bs commute and in
consequence, eB = eBo eBs . Consider the projections φo and φs of φ over the algebra of classical bivectors
(2)
R0,m and φ−1 ({0m } × [sp(2n) ∩ so(2n)]) respectively, i.e.
(2)

φs : φ−1 ({0m } × [sp(2n) ∩ so(2n)]) → sp(2n) ∩ so(2n),

φo : R0,m → so(m),
where


φ(B) =

φo (Bo )
0
0
φs (Bs )



or equivalently: φo (Bo )[x] = [Bo , x] for x ∈ Rm|0 (ΛN +1 ), and φs (Bs )[x] = [Bs , x], for x ∈ R0|2n (ΛN +1 ).
Hence eφ(B) = Im+2n if and only if eφo (Bo ) = Im and eφs (Bs ) = I2n . For the first condition, we know
(2)
from classical Clifford analysis that Spin(m) = {eB : B ∈ R0,m } is a double covering of SO(m) and in
φo (Bo )
B0
consequence, e
= Im implies e = ±1. Let us now compute all possible values for eBs for which
φs (Bs )
e
= I2n . To that end, we need the following linear algebra result.
Proposition 19. Every matrix D0 ∈ so(2n) ∩ sp(2n) can be written in the form D0 = RΣRT where
R ∈ SO(2n) ∩ Sp(2n) and




Σ=



0
−θ1



θ1
0
..

.
0
−θn

θn
0

24




,



θj ∈ R,

j = 1, . . . , n.

(17)

Proof.
c
The map Ψ(D0 ) = 12 QD0 QT , where



Q=


1
0
..
.

i
0
..
.

0
1
..
.

0
i
..
.

...
...
..
.

0
0
..
.

0
0
..
.

0

0

0

0

...

1

i




 ∈ Cn×2n ,


is a Lie group isomorphism between SO(2n) ∩ Sp(2n) and U (n). In addition, Ψ is its own infinitesimal representation on the Lie algebra level, and in consequence, a Lie algebra isomorphism
 between
c
so(2n) ∩ sp(2n) and u(n). The inverse of Ψ is given by Ψ−1 (L) = 21 QT L Q + QT Lc Qc . For every
D0 ∈ so(2n) ∩ sp(2n), let us consider the skew-Hermitian matrix L = Ψ(D0 ) ∈ u(n). It is know that
every skew-Hermitian matrix is unitarily diagonalizable and all its eigenvalues are purely imaginary, see
c
[10]. Hence, L = Ψ(D0 ) can be written as L = U Ξ U T where U ∈ U (n) and Ξ = diag(−iθ1 , . . . , −iθn ),
−1
T
−1
θj ∈ R. Then, D0 = Ψ (L) = RΣR where R = Ψ (U ) ∈ SO(2n) ∩ Sp(2n) and Σ = Ψ−1 (Ξ) has the
form (17).


e

Since φs (Bs ) ∈ so(2n) ∩ sp(2n), we have φs (Bs ) = RΣRT as in the previous proposition. Hence,
= ReΣ RT where eΣ is the block-diagonal matrix

φs (Bs )

eΣ = diag(eθ1 J2 , . . . , eθn J2 ) with eθj J2 = cos θj I2 + sin θj J2 .
Hence eφs (Bs ) = I2n if and only if eΣ = I2n , which is seen to be equivalent to θj = 2kj π, kj ∈ Z
(j = 1, . . . , n), or to
Σ=

n
X

2kj π (E2j−1,2j − E2j,2j−1 ) ,

kj ∈ Z (j = 1, . . . , n).

j=1

Now, SO(2n) ∩ Sp(2n) being connected and compact, there exists BR ∈ φ−1 (so(2n) ∩ sp(2n)) such
φ(BR )
that R
. We recall that the h-action leaves any multivector structure invariant, in particular,
=e
(2)E
(2)E
(2)E
B
h[e ] Rm|2n (ΛN ) ⊂ Rm|2n (ΛN ) for every B ∈ Rm|2n (ΛN ). Then, using that φ is the derivative at the
origin of h, we get that the extended superbivector h(eBR )[φ−1 (Σ)] = eBR φ−1 (Σ)e−BR is such that

φ eBR φ−1 (Σ)e−BR = eφ(BR ) Σe−φ(BR ) = RΣRT = φ(Bs ),
−1

implying that Bs = eBR φ−1 (Σ)e−BR . Then, in order to compute eBs = eBR eφ
−1
to compute eφ (Σ) . Following the correspondences given in Lemma 5 we get
φ−1 (Σ) =

n
X

2kj π φ−1 (E2j−1,2j − E2j,2j−1 ) =

j=1

n
X

(Σ) −BR

e

, we first have


2
2
kj π e`2j−1
+ e`2j
.

j=1

and, in consequence

−1

eφ

(Σ)

= exp 

n
X


n
Y



2
2 
2
2
kj π e`2j−1
+ e`2j
=
exp kj π e`2j−1
+ e`2j
.

j=1

(18)

j=1



2
2
Let us compute exp π e`2j−1
+ e`2j
, j ∈ {1, . . . , n}. Consider x = e`2j−1 −ie`2j , y = e`2j−1 +ie`2j ; where i is
2
2
2
2
the usual imaginary unit in C. It is clear that xy = e`2j−1
+e`2j
+i(e`2j−1 e`2j −e`2j e`2j−1 ) = e`2j−1
+e`2j
+i and


2
2
[x, y] = 2i which is a commuting element. Then, exp π e`2j−1 + e`2j = exp (π xy − iπ) = − exp (π xy).
In order to compute exp (π xy) we first prove the following results.
Lemma 7. For every k ∈ N the following relations hold.


(i) yk x , xyk = −2ik yk−1 ,
(ii) xk yk xy = xk+1 yk+1 − 2ik xk yk .

25

Proof.
(i) We proceed by induction. For k = 1 we get [y, x] = −2i which obviously is true. Now assume that
(i) is true for k ≥ 1, then for k + 1 we get

yk+1 x = y yk x = yxyk − 2ik yk = (xy − 2i)yk − 2ik yk = xyk+1 − 2i(k + 1)yk .

(ii) From (i) we get xk yk xy = xk xyk − 2ik yk−1 y = xk+1 yk+1 − 2ik xk yk .

Pk
k
k−j
Lemma 8. For every k ∈ N it holds that (xy) =
S(k, j) xj yj , where S(n, j) is the
j=1 (−2i)
Stirling number of the second kind corresponding to k and j.
Remark 5.2. The Stirling number of the second kind S(k, j) is the number of ways of partitioning a
set of k elements into j non empty subsets. Among the properties of the Stirling numbers we recall the
following ones:
S(k, 1) = S(k, k) = 1,

∞
X

S(k + 1, j + 1) = S(k, j) + (j + 1)S(k, j + 1),

k=j

j

S(k, j)

(ex − 1)
xk
=
.
k!
j!

Proof of Lemma 8.
We proceed by induction. For k = 1 the statement clearly is true. Now assume it to be true for k ≥ 1.
Using Lemma 7, we have for k + 1 that
(xy)k+1 =

k
k
X
X
(−2i)k−j S(k, j) xj yj xy =
(−2i)k−j S(k, j)xj+1 yj+1 + (−2i)k+1−j j S(k, j) xj yj
j=1

j=1
k

= (−2i) xy +

k−1
X

!
(−2i)

k−j

[S(k, j) + (j + 1)S(k, j + 1)] x

j+1

y

j+1

+ xk+1 yk+1

j=1

=

k+1
X

(−2i)k+1−j S(k + 1, j) xj yj .

j=1


Then we obtain
eπxy =

∞
X
πk
k=0

=1+

k!

(xy)k = 1 +

k=1 j=1

k!

(−2i)k−j S(k, j) xj yj



∞
k
X
(−2πi)
(−2i)−j 
(−2i)k−j S(k, j) xj yj = 1 +
S(k, j) xj yj
k!
k!
j=1
∞
X

∞ X
∞
X
πk
j=1 k=j

=1+

∞ X
k
X
πk

∞
X

k=j

j
−1
xj yj = 1.
j!

−2πi

(−2i)−j

e

j=1



2
2
from which we conclude that exp π e`2j−1
+ e`2j
= − exp (π xy) = −1.
Remark 5.3. Within the algebra AlgR {e`1 , . . . , e`2n } the elements e`2j−1 , e`2j may be identified with the opeπ
π
rators e 4 i ∂aj , e− 4 i aj respectively, the aj ’s being real variables. Indeed, these identifications immediately
lead to the Weyl algebra defining relations
π

π

π

π

e 4 i ∂aj e− 4 i ak − e− 4 i ak e 4 i ∂aj = ∂aj ak − ak ∂aj = δj,k .


2
2
Hence e`2j−1
+ e`2j
may be identified with the harmonic oscillator i ∂a2j − a2j and in consequence, the
h 
i


2
2
element exp π e`2j−1
+ e`2j
corresponds to exp πi ∂a2j − a2j . We recall that the classical Fourier
transform in one variable can be written as an operator exponential
π 

π 
F[f ] = exp
i exp
i ∂a2j − a2j [f ].
4
4
h 
i
Hence, exp πi ∂a2j − a2j = −F 4 = −id, where id denotes the identity operator.

26

P

kj
Qn
−1
2
2
= (−1) kj , whence eBs = ±1. Then,
Going back to (18) we have eφ (Σ) = j=1 exp π e`2j−1
+ e`2j
for B = Bo + Bs ∈ S1 such that eφ(B) = Im+2n , we have eB = eBo eBs = ±1, i.e. ker h|exp(S1 ) = {−1, 1}.

Theorem 7. The set S = exp(S1 ) exp(S2 ) exp(S3 ) is a double covering of SO0 .

Pn θ
2
2
,
Remark 5.4. As shown before, every extended superbivector of the form B = j=1 2j π e`2j−1
+ e`2j
θj ∈ R, belongs to S1 . Then, though the identifications made in remark 5.3 we can see all the operators


n
n
n
i Y
h π

X
Y
θ
π 
j
exp 
πi(∂a2j − a2j ) =
exp θj i(∂a2j − a2j ) =
exp −θj i Fa2θj j ,
2
2
2
j=1
j=1
j=1
2θ

as elements of the Spin group in superspace. Here, Faj j denotes the one-dimensional fractional Fourier
transform of order 2θj in the variable aj .

6

Conclusions and future work

In this paper we have shown that vector reflections in superspace are not enough to describe the set of
linear transformations leaving the inner product invariant. This constitutes a very important difference
with the classical case in which the algebra of bivectors x ∧ y is isomorphic to the special orthogonal
algebra so(m). Such a property is no longer fulfilled in this setting. The real projection of the algebra of
(2)
superbivectors Rm|2n (ΛN ) does not include the symplectic algebra structure which is present in the Lie
algebra of supermatrices so0 , corresponding to the group of super rotations.
That fact has an major impact on the definition of the Spin group in this setting. The set of
elements defined through the multiplication of an even number of unit vectors in superspace does not
suffice for describing Spin(m|2n)(ΛN ). A suitable alternative, in this case, is to define the (super) spin
elements as products of exponentials of extended superbivectors. Such an extension of the Lie algebra
of superbivectors contains, through the corresponding identifications, harmonic oscillators. This way, we
obtain the Spin group as a cover of the set of superrotations SO0 through the usual representation h. In
addition, every fractional Fourier transform can be identified with a spin element.
In forthcoming work, we will prove the invariance of the (super) Dirac operator ∂x under the corresponding actions of this (super) Spin group. We will also study the invariance of the Hermitian system
under the action of the corresponding Spin subgroup in superspace.

Acknowledgements
Alí Guzmán Adán is supported by a BOF-doctoral grant from Ghent University with grant number
01D06014.

References
[1] F. A. Berezin. Introduction to Super Analysis. D. Reidel Publishing Co., Inc., New York, NY, USA,
1987.
[2] H. De Bie and F. Sommen.
322(12):2978–2993, 2007.

A clifford analysis approach to superspace.

Annals of Physics,

[3] H. De Bie and F. Sommen. Correct rules for clifford calculus on superspace. Advances in Applied
Clifford Algebras, 17(3):357–382, 2007.
[4] H. De Schepper, A. Guzman Adan, and F. Sommen. Hermitian clifford analysis on superspace.
Submitted for publication, 2016.
[5] Hennie De Schepper, Alí Guzmán Adán, and Frank Sommen. The radial algebra as an abstract
framework for orthogonal and hermitian clifford analysis. Complex Analysis and Operator Theory,
pages 1–34, 2016.

27

[6] P. DIENES. The exponential function in linear algebras. The Quarterly Journal of Mathematics,
os-1(1):300–309, 1930.
[7] Thomas Friedrich. Dirac operators in Riemannian geometry, volume 25 of Graduate Studies in
Mathematics. American Mathematical Society, Providence, RI, 2000. Translated from the 1997
German original by Andreas Nestke.
[8] Brian Hall. Lie groups, Lie algebras, and representations: An elementary introduction, volume 222
of Graduate Texts in Mathematics. Springer, Cham, second edition, 2015.
[9] Joachim Hilgert and Karl-Hermann Neeb. Structure and geometry of Lie groups. Springer Monographs in Mathematics. Springer, New York, 2012.
[10] Roger A. Horn and Charles R. Johnson. Matrix analysis. Cambridge University Press, Cambridge,
second edition, 2013.
[11] Anthony W. Knapp. Lie groups beyond an introduction, volume 140 of Progress in Mathematics.
Birkhäuser Boston, Inc., Boston, MA, second edition, 2002.
[12] F. Sommen. An algebra of abstract vector variables. In Portugaliae Mathematica 54 (3) (1997),
pages 287–310, 1997.
[13] F. Sommen. An extension of clifford analysis towards super-symmetry. In Clifford algebras and their
applications in mathematical physics, pages 199–224. Springer, 2000.
[14] F. Sommen. Clifford analysis on super-space. Advances in Applied Clifford Algebras, 11(1):291–304,
2001.
[15] F. Sommen. Analysis Using Abstract Vector Variables, pages 119–128. Birkhäuser Boston, Boston,
MA, 2002.
[16] F. Sommen. Clifford analysis on super-space. ii. Progress in analysis, 1:383–405, 2003.
[17] Takeo Yokonuma. Tensor spaces and exterior algebra, volume 108 of Translations of Mathematical
Monographs. American Mathematical Society, Providence, RI, 1992. Translated from the 1977
Japanese edition by the author.

Hennie de Schepper
Clifford Research Group, Department of Mathematical Analysis, Ghent University,
Krijgslaan 281, 9000 Gent, Belgium.
e-mail: hennie.deschepper@ugent.be

Alí Guzmán Adán
Clifford Research Group, Department of Mathematical Analysis, Ghent University,
Krijgslaan 281, 9000 Gent, Belgium.
e-mail: ali.guzmanadan@ugent.be

Frank Sommen
Clifford Research Group, Department of Mathematical Analysis, Ghent University,
Krijgslaan 281, 9000 Gent, Belgium.
e-mail: franciscus.sommen@ugent.be

28

