Fundaments of Quaternionic Clifford Analysis II: Splitting of Equations

0.27 MB·English

by  Fred Brackx

Fundaments of Quaternionic Clifford Analysis II:
Splitting of Equations
6
1
0 F. Brackx∗, H. De Schepper∗, D. Eelbode∗∗, R. La´viˇcka‡ & V. Souˇcek‡
2
r ∗ Clifford Research Group, Dept. of Math. Analysis, Faculty of Engineering and Architecture,
p Ghent University,Building S22, Galglaan 2, B–9000 Gent,Belgium
A ∗∗ University of Antwerp,Middelheimlaan 2, Antwerpen,Belgium
‡ Charles University in Prague, Faculty of Mathematics and Physics, Mathematical Institute
6
Sokolovska´ 83, 186 75 Praha, Czech Republic

]
V
C Abstract
.
h Quaternionic Clifford analysis is arecent new branch of Clifford analysis, a higherdimen-
t sional function theory which refines harmonic analysis and generalizes to higher dimension
a
the theory of holomorphic functions in the complex plane. So–called quaternionic monogenic
m
functions satisfy a system of first order linear differential equations expressed in terms of
[
four interrelated Dirac operators. The conceptual significance of quaternionic Clifford anal-

2 ysis is unraveled by showing that quaternionic monogenicity can be characterized by means
v of generalized gradients in the sense of Stein and Weiss. At the same time, connections be-
0 tweenquaternionicmonogenicfunctionsandotherbranchesofCliffordanalysis,vizHermitian
4 monogenic and standard or Euclidean monogenic functions are established as well.
4
3
0 1 Introduction
.
1
0 TheDiracoperatorplaysaveryimportantroleinmodernmathematics. Acomprehensivedescrip-
5 tionofitsroleinclassicalgeometrycanbefound,e.g.,inthebookbyLawsonandMichelsohn[19].
1
The Dirac operator can be characterized as the only (up to a multiple) elliptic and conformally
:
v invariant first order system of PDE’s acting on spinor valued functions. A recent monograph [1]
i on the spinorial approach to Riemannian and conformal geometry shows the importance to con-
X
sidering the Dirac operator not only on Riemannian (spin) manifolds but also on manifolds with
r
a a special holonomy. By the Berger-Simons classification, we know that these include K¨ahler and
quaternionic–K¨ahler manifolds. As shown in [1], the spectral properties of the Dirac operator on
these various types of Riemannian manifolds are different.
Standard Clifford analysis, also called Euclidean or orthogonal Clifford analysis, is, in its most
basic form, a higher dimensional generalization of the theory of holomorphic functions in the
complex plane and at the same time a refinement of harmonic analysis, see e.g. [8, 16, 11, 17].
Euclidean space Rm is the flat model for Riemannian (spin) geometry and Clifford analysis is
the function theory for the Dirac operator in this flat model. We refer to this setting as the Eu-
clidean (or orthogonal) case, since the fundamental symmetry group leaving the Dirac operator
∂ = m e ∂ invariant, is the special orthogonal group SO(m), which is doubly covered by
α=1 α Xα
the spin group Spin(m) in the Clifford algebra.
P
The flatmodelfor K¨ahlermanifolds is the EuclideanspaceR2n ofevendimensionwitha chosen
additional datum, a complex structure I, i.e., an SO(2n)–elementsquaring up to E ,E being
2n 2n
−
1
the identity matrix. There are now two invariant first order differential operators in this setting,
namely, the classical Dirac operator ∂ and its image under the action of the complex structure
∂ = 2n I(e )∂ , which are the flat versions of the invariant differential operators defined
I α=1 α Xα
on any K¨ahler manifold. In the books [9, 22] and the series of papers [2, 3, 13, 14, 23] so–called
P
Hermitian Clifford analysis emerged as a refinement of Euclidean Clifford analysis,by considering
functionswhichnowtaketheirvaluesinthecomplexCliffordalgebraC ,orinthecomplexspinor
2n
spaceS. HermitianCliffordanalysisfocusesonthe simultaneousnullsolutionsofbothoperators∂
and ∂ , called Hermitian monogenic functions. The fundamental symmetry group underlying this
I
function theory is the unitary group U(n). It is worth mentioning that the traditional holomor-
phic functionsofseveralcomplexvariablesappearasa specialcaseofHermitianmonogenicity,see
Section 3.
The next step is to go from complex manifolds (in particular K¨ahler ones) to manifolds with
a suitable quaternionic structure. It is possible to consider in general hypercomplex manifolds,
or hyper–Ka¨hler manifolds, or quaternionic–K¨ahler manifolds. The flat model for hypercomplex
manifoldsisthe EuclideanspaceR4p,the dimensionofwhichis assumedtobe afourfold,together
with a choice of three complex structures I, J and K satisfying the usual multiplication relations
forthethreequaternionicunits. Itisclearthat,inthissetting,wehavefourbasicoperatorsacting
on spinor valued functions, namely, ∂, ∂ , ∂ and ∂ . Recently a new branch of Clifford analysis
I J K
arose, focussing at so-called quaternionic monogenic functions (see e.g. [12, 20, 10, 4]). The asso-
ciated function theory is called quaternionic Clifford analysis. The fundamental symmetry group
for quaternionic Clifford analysis is the symplectic group Sp(p).
There is a fundamental problem with this notion of quaternionic monogenic functions, which
was already visible in the case of Hermitian Clifford analysis. In the classicalorthogonalcase, the
Dirac operator acts naturally on spinor valued functions. The basic spinor representation is an
irreducible representation of the symmetry group SO(m). In the Hermitian case, introducing the
complex structure I, the symmetry groupis reduced to a subgroupof SO(2n)isomorphic to U(n).
Consequently, the spinor space S where our functions take their values is not irreducible anymore
butsplits into afinite sumofirreduciblecomponents instead. These componentscanbe described
as the homogeneous components Sr r(W ) of the Grassmann algebra of a maximal isotropic
†
≃ ∧
subspace W of the complexificationC2n. In the quaternionic case the reduction of the symmetry
†
group to Sp(p) leads to a further decomposition of the homogeneous parts Sr. The corresponding
irreducible pieces Sr now depend on two parameters, see Section 5 for a detailed description.
s
A natural approach to invariant first order systems is described by Stein and Weiss in [25], and
it also works equally well on manifolds. The structure of a hypercomplex manifold M is given by
a choice of a principal U(n)-bundle P over M, and invariant operators in the sense of Stein and
Weiss are acting on sections of bundles associated to irreducible U(n)-modules. In our case, their
scheme applies to sections of bundles associated to the irreducible parts (with respect to Sp(p))
of the spinor bundle S, which are induced by the representations Sr. Basic first order invariant
s
operators on hypercomplex manifolds hence can be classified in their turn by the Stein and Weiss
scheme for sections of bundles induced by each representation Sr. The corresponding first order
s
invariant systems are the basic first order PDE’s to be considered on this type of manifolds.
The basic question now is: what is the relation between quaternionic monogenic functions and
the solutions of invariant first order PDE’s stemming from the Stein-Weiss approach? This is the
key problem in establishing quaternionic Clifford analysis, to which the present paper proposes
an answer. First, in Section 5, we recall the decomposition of the spinor space S into its irre-
ducible components Sr. Then, in Section 6, we study a classification of the Stein-Weiss operators
s
– also called generalized gradients – in the quaternionic setting. It leads to an alternative defi-
2
nition of quaternionic monogenic functions. In Theorem 1 we prove that Sr–valued functions are
s
quaternionicmonogenicifandonlyiftheyarecommonnullsolutionsofallStein-Weissgeneralized
gradients. This is the main result of the paper and it shows that both approaches are equivalent.
In order to making the present paper self-contained, the basics of Clifford algebra are recalled
in Section 2. Section 3 shortly outlines the basics of Euclidean and Hermitian Clifford analysis
whileSection4introducesquaternionicCliffordanalysis. InSection7ourresultsareillustratedby
explicitcalculationsofthesystemofequationsforquaternionicmonogenicityinR8. Foradetailed
accounton the constructionof the symplectic cells in spinor space, we refer to [4]. For an account
of the impact on the Fischer decomposition of harmonic polynomials of imposing symmetry with
respect to the symplectic group, we refer to [5].
2 Preliminaries on Clifford algebra
For a detailed description of the structure of Clifford algebras we refer to e.g. [21]. Here we only
recall the necessary basic notions.
The real Clifford algebra R is constructed over the vector space R0,m endowed with a non–
0,m
degeneratequadraticformofsignature(0,m),andgeneratedbytheorthonormalbasis(e ,...,e ).
1 m
The non–commutative Clifford or geometric multiplication in R is governedby the rules
0,m
e e +e e = 2δ , α,β =1,...,m (1)
α β β α αβ
−
AsabasisforR onetakesforanysetA= j ,...,j 1,...,m theelemente =e ...e ,
0,m { 1 h}⊂{ } A j1 jh
with 1 j < j < < j m, together with e = 1, the identity element. The dimension of
1 2 h
R is≤2m. Any Cliff·o··rdnum≤ber a in R may thu∅s be written as a= e a , a R, or still
as0,am= m [a] ,where[a] = e0,am istheso–calledk–vectorpartoAfaA. RAealAnu∈mbersthus
corresponkd=w0ithkthe zero–vkector p|Aa|r=tkofAthAe Clifford numbers. EuclideanPspace R0,m is embedded
P P
in R by identifying (X ,...,X ) with the Clifford 1–vector X = m e X , for which it
0,m 1 m α=1 α α
holds that X2 = X 2 = r2.
−| | − P
When allowing for complex constants, the generators (e ,...,e ), still satisfying (1), produce
1 m
the complex Clifford algebra C = R iR . Any complex Clifford number λ C may
m 0,m 0,m m
⊕ ∈
thus be written as λ = a + ib, a,b R , leading to the definition of the Hermitian conju-
0,m
∈
gation λ = (a + ib) = a ib, where the bar notation stands for the Clifford conjugation in
† †
−
R , i.e. the main anti–involution for which e = e , α = 1,...,m. This Hermitian conjuga-
0,m α α
−
tion leads to a Hermitian inner product on C given by (λ,µ) = [λ µ] and its associated norm
m † 0
λ = [λ λ] =( λ 2)1/2.
| | † 0 A| A|
Thepalgebra of rPeal quaternions is denoted by H. For a quaternion
q =q +q i+q j+q k =(q +q i)+(q +q i)j =z+wj
0 1 2 3 0 1 2 3
its conjugate is given by
q =q q i q j q k=(q q i) j(q q i)=z jw =z wj
0 1 2 3 0 1 2 3
− − − − − − − −
such that
qq =qq = q 2 =q2+q2+q3+q2 = z 2+ w2
| | 0 1 2 3 | | | |
Identifying the quaternion units i,j with the respective basis vectors e ,e , the algebra H is
1 2
isomorphicwiththeCliffordalgebraR . Moreover,itisalsoisomorphicwiththeevensubalgebra
0,2
R+ of the Clifford algebra R , by identifying the quaternion units i,j,k with the respective
0,3 0,3
bivectors e e ,e e ,e e .
2 3 3 1 1 2
3
3 Euclidean and Hermitian Clifford Analysis: the basics
The central notion in standard Clifford analysis is that of a monogenic function. This is a contin-
uously differentiable function defined in an open region of Euclidean space Rm, taking its values
in the Clifford algebra R , or subspaces thereof, and vanishing under the action of the Dirac
0,m
operator ∂ = m e ∂ , i.e. a vector valued first order differential operator, which can be seen
α=1 α Xα
as the Fourier or Fischer dual of the Clifford variable X. Monogenic functions thus are the higher
P
dimensional counterparts of holomorphic functions in the complex plane. As moreover the Dirac
operator factorizes the Laplacian: ∆ = ∂2, the notion of monogenicity can be regarded as a
m
−
refinementofthenotionofharmonicity. ItisimportanttonotethattheDiracoperatorisinvariant
undertheactionoftheSpin(m)–group,whichdoublycoverstheSO(m)–group,whencethissetting
is usually referred to as Euclidean (or orthogonal) Clifford analysis.
TakingthedimensionoftheunderlyingEuclideanvectorspaceRm tobeeven: m=2n,renaming
the variables as:
(X ,...,X )=(x ,y ,x ,y ,...,x ,y )
1 2n 1 1 2 2 n n
and considering the (almost) complex structure I , i.e. the complex linear real SO(2n) matrix
2n
0 1
I =diag
2n 1 0
(cid:18)− (cid:19)
for which I2 = E , E being the identity matrix, we define the rotated vector variable
2n − 2n 2n
n
X =I [X] = I (e x +e y )
I 2n 2n 2k 1 k 2k k
" − #
k=1
X
n n
= I [e ]x +I [e ]y = ( y e +x e )
2n 2k 1 k 2n 2k k k 2k 1 k 2k
− − −
k=1 k=1
X X
and, correspondingly, the rotated Dirac operator
n
∂ =I [∂]= ( ∂ e +∂ e )
I 2n − yk 2k−1 xk 2k
k=1
X
A differentiable function F then is called Hermitian monogenic in some region Ω of R2n, if and
only if in that region F is a solution of the system
∂F =0=∂ F (2)
I
Observethat this notionof Hermitian monogenicity does notinvolve the use of complex numbers,
butcouldbedevelopedasarealfunctiontheoryinstead. Thereishoweveranalternativeapproach
to the concept of Hermitian monogenicity, making use of the projection operators
1
π = ( E +iI )
∓ 2n 2n
2 ∓
andthusinvolvingacomplexification. Inthisapproachwefirstdefine,inthe complexificationC2n
of R2n, the so–called Witt basis vectors
fk =π−[e2k 1] and f†k =π+[e2k 1] (k =1,...,n)
− −
for which also hold the relations
ifk =π−[e2k] and −if†k =π+[e2k] (k =1,...,n)
4
This enables the introduction of the vector variables z and z given by
†
n n
z = π [X] = x π [e ]+ y π [e ]
− k − 2k 1 k − 2k
−
k=1 k=1
X X
n n n
= x f +y (if ) = (x +iy )f = z f
k k k k k k k k k
k=1 k=1 k=1
X X X
n n
z† = π+[X]= xkf†k+yk(−if†k)= zkf†k
k=1 k=1
X X
and, correspondingly, of the Hermitian Dirac operators ∂ and ∂ given by
z z†
n n n
2∂ = π [∂] = f ∂ +if ∂ = f (∂ +i∂ )=2 ∂ f
z† − k xk k yk k xk yk zk k
k=1 k=1 k=1
X X X
n n n
2∂z = π+[∂] = f†k∂xk −if†k∂yk = f†k(∂xk −i∂yk)=2 ∂zkf†k
k=1 k=1 k=1
X X X
As 2(∂ ∂ ) = ∂ and 2(∂ +∂ ) = i∂ , the system (2) is easily seen to be equivalent with the
z − z† z z† I
system
∂ F =0=∂ F (3)
z z†
Note that the –notation corresponds to the Hermitian conjugation in the Clifford algebra C ,
† 2n
·
introduced above.
In order to study Hermitian monogenic functions it suffices to consider functions with values
not in the whole Clifford algebra but in spinor space instead. Indeed, a Clifford algebra may be
decomposed as a direct sum of isomorphic copies of a spinor space S, which, abstractly, may be
defined as a minimal left ideal in the Clifford algebra. A spinor space is an irreducible Spin(2n)
grouprepresentation,andmayberealizedasfollows. TheWittbasisvectorssatisfytheGrassmann
identities
fjfk+fkfj =0, f†jf†k+f†kf†j =0, j,k =1,...,n
which include their isotropy:
f2j =(f†j)2 =0, j =1,...,n
as well as the duality identities
fjf†k+f†kfj =δjk, j,k =1,...,n
The Witt basis vectors (f1,...,fn) and (f†1,...,f†n) respectively span isotropic subspaces W and
W of C2n, such that
†
C2n =W W† (4)
⊕
thosesubspaces being eigenspacesofthe complex structureI with respectiveeigenvalues iand
2n
−
i. They also generate two Grassmann algebras,respectively denoted by C and C †.
n n
With the self-adjoint idempotents V V
1
Ij =fjf†j = 2(1−ie2j−1e2j), j =1,...,n
we compose the primitive self–adjoint idempotent I = I I I leading to the realization of the
1 2 n
···
spinor space S as S=C2nI. Since fjI =0, j =1,...,n, we also have S≃C †nI.
V
5
When decomposing the Grassmann algebra C † into its so–called homogeneous parts
n
Vn (r)
C † = C †
n n
r=0(cid:18) (cid:19)
^ M ^
(r)
where the spaces C † are spanned by all possible products of r Witt basis vectors out of
n
(f†1,...,f†n), the sp(cid:16)inorVsp(cid:17)ace S accordingly decomposes into
n (r)
S= Sr, with Sr C † I (5)
≃ n
r=0 (cid:18) (cid:19)
M ^
These homogeneous parts Sr, r = 0,...,n, provide models for fundamental U(n)–representations
(see [7]) and thus also for fundamental sl(n,C)-representations (see [2], [12]).
Accordingly we can decompose a spinor valued function F :Cn S into its components
−→
n
F = Fr, Fr :Cn Sr, r =0,...,n
−→
r=0
X
Pay attention to the fact that the monogenicity of F does not imply the monogenicity of the
componentsFr,r =0,...,n. However,theHermitianmonogenicityofF doesimplytheHermitian
monogenicity of these components, and vice versa. This is due to the nature of the action of the
Witt basis vectors as (left) multiplication operators,implying that
∂ Fr :Cn Sr+1 and ∂ Fr :Cn Sr 1
z −→ z† −→ −
Moreover,foreachofthecomponentsFr thenotionsofmonogenicityandHermitianmonogenicity
coincide, since
∂ Fr = 0
∂Fr =0 (∂ ∂ )Fr =0 z
⇐⇒ z − z† ⇐⇒( ∂z†Fr = 0
In conclusion we have the following result.
Proposition 1. The function F = n Fr, Fr : Cn Sr, r = 0,...,n, is Hermitian mono-
r=0 −→
genic in some region Ω of Cn if and only if each of its components Fr, r =0,...,n, is monogenic
P
in that region.
The Hermitian Dirac operators ∂ and ∂ are invariant under the action of the group SO (2n),
z z† I
i.e.thesubgroupofSO(2n)consistingofthosematriceswhichcommutewiththecomplexstructure
I . This subgroup SO (2n) inherits a twofold covering by the subgroup Spin (2n) of Spin(2n),
2n I I
consisting of those elements of Spin(2n) that are commuting with
√2
s =s ...s , where s = (1+e e ), j =1,...,n (6)
I 1 n j 2j 1 2j
2 −
This element s itself obviously belongs to Spin (2n) and corresponds, under the double covering,
I I
to the complex structure I . As SO (2n) is isomorphic with the unitary group U(n) and, up to
2n I
this isomorphism, Spin (2n) thus provides a double cover of U(n), we may say that U(n) is the
I
fundamental group underlying the function theory of Hermitian monogenic functions.
6
4 Quaternionic Clifford Analysis
ArefinementofHermitianCliffordanalysisis obtainedby consideringthe hypercomplexstructure
Q = (I ,J ,K ) on R4p C2p Hp, the dimension m = 2n = 4p now being assumed to be
4p 4p 4p
≃ ≃
a multiple of four. This hypercomplex structure arises by introducing, next to the first complex
structure I , a second one, called J , given by
4p 4p
1
1
J4p =diag  1 − 
−
 1 
 
 
Clearly J belongs to SO(4p), with J2 = E , and it anti–commutes with I . Then a third
4p 4p − 4p 4p
complex structure quite naturally arises, namely the SO(4p)–matrix
K =I J = J I
4p 4p 4p 4p 4p
−
for which K2 = E and which anti–commutes with both I and J . It turns out that
4p − 4p 4p 4p
1
−
1
K4p =diag  1 − 
 1 
 
 
TheSO(4p)–matriceswhichcommutewiththe hypercomplexstructureQonR4p formasubgroup
of SO (4p), denoted by SO (4p), which is isomorphic with the symplectic group Sp(p). Recall
I Q
that the symplectic group Sp(p) is the real Lie group of quaternion p p matrices preserving the
×
symplectic inner product
ξ,η =ξ η +ξ η + +ξ η ξ,η Hp
h iH 1 1 2 2 ··· p p ∈
or, equivalently,
Sp(p)= A GL (H):AA =E
p ∗ p
{ ∈ }
Quite naturally, the subgroup SO (4p) of SO(4p) is doubly covered by Spin (4p), the subgroup
Q Q
of Spin(4p) consisting of the Spin(4p)–elements which are commuting with both s and s , where
I J
now s is the Spin(4p)–element corresponding to J . Recall, see (6), that s , corresponding to the
J p I
complex structure I , is given by s = s s , where s = √2 1+e e , j = 1,...,2p.
4p I 1··· 2p j 2 2j−1 2j
Similarly, for s we find
J
(cid:0) (cid:1)
1
s =s s , s = 1+e e 1 e e , j =1,...,p (7)
J 1 p j 4j 3 4j 1 4j 2 4j
··· 2 − − − −
(cid:0) (cid:1)(cid:0) (cid:1)
For the correspondineg picteure atethe level of the Lie algebras we refer to [4].
The introduction of a hypercomplex structure leads to a function theory in the framework of
so–called quaternionic Clifford analysis, where the fundamental invariance will be that of the
symplectic group Sp(p). The most genuine way to introduce the new concept of quaternionic
monogenicity is to directly generalize the system (2) expressing Hermitian monogenicity, now
making use of the hypercomplex structure on R4p and the additional rotated Dirac operators
∂ =J [∂] and ∂ =K [∂]; whence the following definition.
J 4p K 4p
7
Definition 1. A differentiable function F : R4p S is called quaternionic monogenic (q–
−→
monogenic for short) in some region Ω of R4p, if and only if in that region F is a solution of
the system
∂F =0, ∂ F =0, ∂ F =0, ∂ F =0 (8)
I J K
Observethat,inasimilarwayasitwaspossibletointroducethenotionofHermitianmonogenicity
without having to resort to complex numbers, the above Definition 1 expresses the notion of q–
monogenicity without involving quaternions.
Remark 1. The notion of a hypercomplex structure stems from differential geometry where each
tangent bundle of an even dimensional manifold admits the action of the algebra of quaternions,
thequaternionunitsdefiningthreealmostcomplex structures. Accordinglyweshouldinfactcallthe
functionssatisfying system(8)“hypercomplex monogenic”, however this termhasalready beenused
in other contexts in higher dimensional function theory, and moreover, the notion of quaternionic
monogenicity was already introduced in previous papers, whence we stick to the latter terminology.
Thereisanaturalalternativecharacterizationofq–monogenicitypossibleintermsoftheHermi-
tianDiracoperators,yetstillnotinvolvingquaternions. WerecalltheseHermitianDiracoperators
in the current dimension:
2p p
∂z = ∂zkf†k = (∂z2j−1f†2j 1+∂z2jf†2j)
−
k=1 j=1
X X
2p p
∂z† = ∂zkfk = (∂z2j−1f2j−1+∂z2jf2j)
k=1 j=1
X X
and compute their respective images under the action of the complex structure J :
4p
p
∂J = J [∂ ] = (∂ f ∂ f )
z 4p z z2j 2j−1− z2j−1 2j
j=1
X
p
∂z†J = J4p[∂z†] = (∂z2jf†2j−1−∂z2j−1f†2j)
j=1
X
Similarly we can introduce the auxiliary variables
p
zJ = J4p[z] = (z2jf†2j−1−z2j−1f†2j)
j=1
X
p
z J = J [z ] = (z f z f )
† 4p † 2j 2j 1 2j 1 2j
− − −
j=1
X
Here, use has been made of the formulae
J4p[f2j−1]=−f†2j, J4p[f2j]=f†2j−1, J4p[f†2j−1]=−f2j, and J4p[f†2j]=f2j−1
Now the original Dirac operator ∂ and its rotated versions ∂ , ∂ and ∂ may be expressed in
I J K
terms of the Hermitian Dirac operators (∂ ,∂ ) and their J–rotatedversions (∂J,∂ J) as follows:
z z† z z†
1 1
∂z = π+[∂]= 4(1+iI4p)[∂], ∂z† =π−[∂]=−4(1−iI4p)[∂]
1 1
∂J = J [π+[∂]]= (J +iK )[∂], ∂ J =J [π [∂]]= (J iK )[∂]
z 4p 4 4p 4p z† 4p − −4 4p− 4p
8
whence conversely
∂ = 2(∂ ∂ )
z− z†
i∂ = 2(∂ +∂ )
I z z†
∂ = 2(∂J ∂ J)
J z − z†
i∂ = 2(∂J +∂ J)
K z z†
Clearly,thisleadstoanalternativecharacterizationofq–monogenicityasexpressedinthefollowing
proposition.
Proposition 2. A differentiable function F : R4p C2p S is q–monogenic in some region
≃ −→
Ω R4p if and only if F is in Ω a simultaneous null solution of the operators ∂ , ∂ , ∂J and ∂ J.
⊂ z z† z z†
We wantto emphasize that, inSection6,the new operators∂J and∂ J,used, togetherwith the
z z†
known Hermitian Dirac operators ∂ and ∂ , to characterize the concept of q–monogenicity, will
z z†
arise in a natural way as generalized gradients in the sense of Stein & Weiss (see [25]).
As the identification of an underlying symmetry group is necessary for the further development
of the function theory, the following result is crucial.
Proposition 3. The operators ∂ , ∂ , ∂J and ∂ J are invariant under the action of the symplectic
z z† z z†
group Sp(p).
Proof
The action of a Spin(4p)–element s on a spinor valued function F is the so-called L–action given
by L(s)[F(X)]=sF(s 1Xs). The Dirac operator ∂ is invariant under Spin(4p), i.e.
−
[L(s),∂]=0, for all s Spin(4p)
∈
which can be explained by
L(s)∂ F(X)=s∂ F(s 1Xs)=s(s 1∂ s)F(s 1Xs)=∂ L(s)F(X)
X s−1Xs − − X − X
Recallthat Sp(p) is isomorphic to the subgroupSpin (4p) ofSpin(4p), whence the Dirac operator
Q
∂ is, quite trivially, also invariant under the action of Sp(p). The invariance of the operators ∂ ,
z
∂ , ∂J and ∂ J now follows from the fact that their respective definitions only involve projection
z† z z†
operators which are commuting with the Sp(p)–elements. (cid:3)
We will now comment on the behaviour of the notion of q–monogenicity with respect to the
decomposition of a function F = n Fr, Fr : R4p Sr, r = 0,...,n into its homogeneous
r=0 −→
spinor components.
P
IfF is q–monogenic,then so areits components Fr, and vice versa. If Fr is q–monogenic,then,
quite naturally, Fr is Hermitian monogenic with respect to the Hermitian Dirac operators ∂ and
z
∂ — let us call this I–Hermitian monogenicity — but also Hermitian monogenic with respect to
z†
the rotated Hermitian Dirac operators ∂J and ∂ J — let us call this J–Hermitian monogenicity.
z z†
As was already pointed out, for the function Fr, I–Hermitian monogenicity is equivalent with
∂–monogenicity;inthesameorderofideas,theJ–HermitianmonogenicityofFr isequivalentwith
its ∂ –monogenicity. Summarizing, we have the following result.
J
Proposition 4. For the function F = n Fr, Fr : R4p Sr, r = 0,...,n, defined in some
r=0 −→
region Ω of R4p, the following statements are equivalent:
P
(i) F is q–monogenic;
(ii) each of the components Fr is q–monogenic;
(iii) each of the components Fr is simultaneously ∂–monogenic and ∂ –monogenic.
J
9
5 A further decomposition of spinor space
Spinor space S, which was already decomposed into U(2p)–irreducible homogeneous parts Sr, can
further be decomposed into Sp(p)–irreducibles, which we will call symplectic cells. Let us briefly
sketch this decomposition, referring to [4] for a detailed description.
First we introduce the Sp(p)–invariant left multiplication operators
P = f f +f f +...+f f
2 1 4 3 2p 2p 1
−
Q = f†1f†2+f†3f†4+...+f†2p 1f†2p = P†
−
for which P : Sr Sr 2 and Q : Sr Sr+2. Next we define the symplectic cells of spinor space,
−
→ →
which are given by, for r =0,...,p, the subspaces
Srr =KerP|Sr, Sr2p−r =KerQ|S2p−r
and for k =0,...,p r, the subspaces
−
Sr+2k =QkSr, S2p r 2k =PkS2p r
r r r − − r −
Lemma 1. One has
p p
KerP = Sr and KerQ = S2p r
|S r |S r −
r=0 r=0
M M
and, for k =0,1,...,p r 1,
− −
1
Q is an isomorphism Sr+2k Sr+2k+2 with inverse Q 1 = P;
• r −→ r − αk
r
1
P is an isomorphism S2p r 2k S2p r 2k 2 with inverse P 1 = Q
• r − − −→ r − − − − αrp−r−k−1
where the coefficients are given by
αk =(k+1)(p r k)=αp r k 1
r − − r− − −
which implies that the composition of the multiplicative operators P and Q is constant on each
symplectic cell; more specifically one has
P Q=αk on Sr+2k and on S2p r 2k 2
• r r r − − −
QP =αk on Sr+2k+2 and on S2p r 2k
• r r r − −
Proposition 5. One has, for all r =0,...,p,
r r
⌊2⌋ ⌊2⌋
Sr = Srr 2j and S2p−r = Sr2p−2jr
− −
j=0 j=0
M M
andeachofthesymplecticcellsSr intheabovedecompositionsisanirreducibleSp(p)–representation.
s
Theabovedecompositionofthehomogeneousspinorsubspacesintosymplecticcellscanbedepicted
by the following triangular scheme.
10