The Spingroup and its actions in discrete Clifford analysis

0.25 MB·

by  Hilde De Ridder

The Spingroup and its actions in discrete Clifford analysis
H. De Ridder∗, F. Sommen†
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Abstract
n
a Recently, it has been established that the discrete star Laplace and the discrete
J
Diracoperator,i.e. thediscreteversionsoftheircontinuouscounterpartswhenworking
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onthe standardgrid,are rotation-invariant. This was done startingfromthe Lie alge-
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braso(m,C) correspondingto the specialorthogonalLie groupSO(m); consideringits

] representationinthe discreteCliffordalgebrasettingandprovingthatthese operators
h
are symmetries of the Dirac and Laplaceoperators. This set-up showedin an abstract
p
way that representation-theoretically the discrete setting mirrors the Euclidean Clif-
-
h fordanalysissetting. Howeverfromapracticalpointofview,thegroup-actionremains
t indispensable for actual calculations. In this paper, we define the discrete Spingroup,
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m which is a double cover of SO(m), and consider its actions on discrete functions. We
[ showthatthisgroup-actionmakesthespacesHkandMkintoSpin(m)-representations.
We will often consider the compliance of our results to the results under the so(m,C)-

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action.
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4 Keywords: discrete Dirac operator, Clifford analysis, Spingroup, rotation
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8 MSC(2010): 43A65, 47A67, 11E88, 15A66, 30G25, 39A12, 44A55
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.
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1 Introduction
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From an application point of view, one has always been interested in discrete complex
:
v analysis and, more recently, in higher-dimensional function theories both generalizing dis-
i
X crete complex analysis and refining discrete harmonic analysis. This interest has been
r even further sparked by the increase in computational power and the potential of quickly
a
applying even higher-dimensional function-theoretical results. Pioneering work on discrete
holomorphic functions on a complex grid was done in [15, 19] and research on these dis-
crete holomorphic functions on (more general grids) was continued on in amongst others
[20,21]. WhenconsideringadiscreteversionofEuclideanCliffordanalysis(seeforexample
∗Ghent University, Department of Mathematical Analysis, Building S8, Krijgslaan 281, 9000 Gent,
Belgium, fax: [phone], phone: [phone], email: [email protected]
†Ghent University, Department of Mathematical Analysis, Building S8, Krijgslaan 281, 9000 Gent,
Belgium, fax: [phone], phone: [phone], email: [email protected]
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[1, 3, 17]), foundations were laid in [13, 18, 14], although these works often differ in terms
of the chosen discrete Dirac operator and/or on the chosen graph on which functions are
defined. In this paper, we will restrict ourself to the ‘split’ discrete Clifford algebra; a basic
framework established in [14, 11] that uses both forward and backward differences.
The key notion of discrete Clifford analysis is a discrete Dirac operator, factorising the
discrete Laplace operator, leading to a refinement of harmonic analysis. The (massless)
Dirac operator finds its origin in particle physics, from the study of elementary particles
with spin number one half [12, 22]. It is well known that both the continuous Laplace
and Dirac operator are rotation invariant operators, i.e. invariant under the groups SO(m)
and Spin(m) respectively, or equivalently, their mutual Lie algebra so(m). The space
of C-valued harmonic polynomials homogeneous of degree k is in fact a model for an
irreducible SO(m,C)-representation with highest weight (k,0,...,0) [17, 2]. A similar
result is true for spinor-valued monogenic polynomials, homogeneous of degree k, where
the highest weight of the irreducible representation is given by k+ 1, 1,..., 1 in the case
2 2 2
of an odd dimension. Since the space of Dirac spinors S decomposes as a direct sum of
(cid:0) (cid:1)
positive and negative Weyl spinors S+⊕S− in even dimension, the space of spinor-valued
monogenic polynomials homogeneous of degree k decomposes in even dimension in a sum
of exactly two irreducible SO(m,C)-representations with highest weights k+ 1, 1,...,1
2 2 2
and k+ 1,1,..., 1,−1 .
2 2 2 2 (cid:0) (cid:1)
Very(cid:0)recently, the repres(cid:1)entation-theoretical aspects underlyingthe discrete counterpart of
this function theory, including the rotational invariance of the star-Laplacian and discrete
Dirac operator, have been studied. It has been established in recent papers [7, 10, 5,
6] that the spaces H and M of discrete harmonic, respectively discrete monogenic k-
k k
homogeneous polynomials are invariant under the action of the special orthogonal Lie
algebra so(m,C). However, up till now we were always restricted to the use of the Lie
algebra so(m,C) as the (action of the) the special orthogonal Lie group SO(m) (or its
double cover the Spingroup) was not yet defined. In this paper, the aim is to do just that,
define and consider a discrete Spingroup which is a double cover of the special orthogonal
Lie group. We considered the Spingroups action on spaces of discrete harmonic resp.
monogenic polynomials. Although it may abstractly be seen as ‘just’ another realisation
of the Spingroup, it is novel as the definition of the Spingroup does not use vectors in the
discrete vector variables, as one would expect, but vectors in some recently defined (see
[10]) operators R . The fact that there is a discrete Spingroup with similar actions as in
j
Euclidean Clifford analysis makes it clear that, although we are restricted to the points to
the grid, rotations are also inherently present in the discrete Clifford analysis setting.
In Section 2, we give a short overview of the necessary definitions and operators of discrete
Clifford analysis. In Section 3 we introducethe definitions of discrete Spingroupsand show
that they are double covers of the special orthogonal group SO(m). In Section 4 we define
several Spingroup actions on the space of discrete polynomials and, extending by means of
the Taylor series, the space of all discrete functions. We conclude this section by making
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the connection to the corresponding Lie algebra. In Section 5 we consider the first non-
trivial example, i.e. the two-dimensional case; we give explicit examples and compare to
earlier results. In Section 6 we extend our Spingroup action to discrete distributions and
consideredsomebasicexamplesintwodimensions. Finally,inSections7and8,weconsider
irreducible representations of integer and half-integer highest weights by constructing the
corresponding highest weight functions.
2 Preliminaries
Let Rm bethem-dimensional Euclidean spacewith orthonormalbasis e , j = 1,...,m and
j
consider the Clifford algebra R over Rm, i.e. the multiplication of two basis elements
m,0
must satisfy the anti-commutator rule e e +e e = 2δ . Passing to the so-called ‘split’
i j j i ij
discrete Clifford setting, see e.g. [11, 4], we embed theClifford algebra R into thebigger
m,0
complex one C and introduce forward and backward basis elements e± by splitting the
2m,0 j
basis elements e = e++e− in forward and backward basis elements e± which sum up to
j j j j
the original basis elements. These e± satisfy the following anti-commutator rules
j
e+,e+ = e−,e− = 0, e+,e− = δ , j,k = 1,...,m,
j k j k j k jk
n o n o n o
which follow from the principles of dimensional equivalence and reflection invariance [11].
We denote furthermore e⊥ = e+ −e−, then e⊥e = e+e− −e−e+.
j j j j j j j j j
Now consider the standard equidistant lattice Zm. The partial derivatives ∂ used in
xj
EuclideanCliffordanalysis(seee.g. [1,3])arereplacedbyforwardandbackwarddifferences
∆±, j = 1,...,m, acting on discrete Clifford-valued functions f as follows:
j
∆+[f](x) = f(x+e )−f(x), ∆−[f](x) = f(x)−f(x−e ), x ∈ Zm.
j j j j
An appropriate definition of a discrete Dirac operator ∂ factorizing the discrete Laplace
operator ∆, i.e. satisfying ∂2 = ∆, is obtained by combining the forward and backward
basis elements with the corresponding forward and backward differences, more precisely
m m
∂ = e+∆++e−∆− = ∂ .
j j j j j
Xj=1(cid:16) (cid:17) Xj=1
The discrete Dirac operator is complemented with a vector variable operator ξ of the form
ξ = m e+X− +e−X+ = m ξ and a discrete Euler operator E to generate an
j=1 j j j j j=1 j
osp(1|2)-rea(cid:16)lisation, cf. [4].(cid:17)This means that they satisfy the usual intertwining relations
P P
{∂,ξ} = 2E+m, [∂,E] = ∂, [ξ,E]= −ξ.
On the co-ordinate level, this is expressed by means of the relations ∂ ξ −ξ ∂ = 1 and
j j j j
{∂ ,ξ } = {ξ ,ξ } = {∂ ,∂ } = 0, j 6= k.
j k j k j k
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Definition 1. A discrete (Clifford-algebra valued) function is discrete harmonic (resp.
(left) discrete monogenic) in a domain Ω ⊂ Zm if ∆f(x) = 0 (resp. ∂f(x) = 0), for all
x ∈ Ω.
The space of all discrete Clifford-algebra valued harmonic (resp. monogenic) polynomials
is denoted H (resp. M) while the space of discrete Clifford-algebra valued harmonic (resp.
monogenic) homogeneous polynomials of degree k is denoted H (resp. M ).
k k
The natural powers ξk[1] of the operator ξ acting on the ground state 1 are the basic
j j
discretehomogeneous polynomialsofdegreek inthevariablex ,replacingthebasicpowers
j
xk in the continuous setting and constituting a basis for all discrete polynomials, cf. [9].
j
The skew-Weyl relations imply that ∂ ξk[1] = δ kξk−1[1]. An explicit formula for the
ℓ j j,ℓ j
polynomials ξk[1] is given in [4]. An important property of these polynomials is the fact
ξk[1](x ) = 0 for k > 2|x |+1 which implies the absolute convergence of the Taylor series
j j j
of any discrete function. Every discrete function, definedon Zm, can beexpressed in terms
of these basis discrete homogeneous polynomials by means of its Taylor series expansion
around the origin, cf. [8].
In [10], we defined the mutually anti-commuting vector-valued operators R , satisfying
j
R [1] = e , {R ,ξ } = 2R ξ δ , {R ,∂ } = 2R ∂ δ .
j j j k j j j,k j k j j j,k
Note that in combination with the operators ξ and ∂ , we obtain mutually commuting
j j
operators ξ R and ∂ R , j = 1,...,m, for which one can easily check that they also
j j j j
generate an osp(1|2)-realisation:
[ξ R ,ξ R ] = [ξ R ,∂ R ] = 0,
j j k k j j k k
[∂ R ,ξ R ] = δ .
j j k k j,k
These operators allowed us to define so(m,C)-generators L resp. dR(e ) within the
a,b a,b
discrete Clifford setting, which are symmetries of the discrete Laplace operator ∆ resp.
discrete Dirac operator ∂.
Definition 2. For a 6= b, we define
L = R R (ξ ∂ +ξ ∂ ),
a,b b a a b b a
1 1
dR(e ) = R R ξ ∂ +ξ ∂ − = L − R R .
a,b b a a b b a a,b b a
2 2
(cid:18) (cid:19)
For a = b let L = dR(e )= 0.
a,a a,a
ThespacesofdiscretesphericalharmonicsH ofdegreekanddiscretesphericalmonogenics
k
M of degree k are (not irreducible) representations of so(m,C), their decomposition into
k
irreducible parts was recently considered in [5, 6].
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Asecondsetofvector-valuedoperatorsS e⊥ wasobtainedin[10],whereS nowdenotesthe
j j j
classicalreflectioninthex -direction,whichleadtoasecondsetofso(m,C)-generators L⊥
j a,b
and dR⊥(e ). Similarly, we will find in the next section two separate discrete Spingroups,
a,b
one involving the operators R and the other involving the operators S e⊥.
j j j
3 Discrete Spingroup
In this section we definea discrete SpingroupSpin(m) and show that it is a doublecover of
the special orthogonal group SO(m). The structure of this proof reflects the proof that the
Spingroup in Euclidean Clifford analysis is a double cover of SO(m), see for example [3].
However, there are two ways in which both settings are different: first of all the discrete
Spingroup Spin(m), as defined below, will consist purely of vectors in the operators R ,
j
j = 1,...,m. Elements of this Spingroup will thus have to act on a discrete function
before one can consider the value in a point of the grid. Second, as the operators R
j
behave as generators of a Cliffordalgebra of signatuur (m,0), i.e. R2 = +1, a lot of steps
j
differ in minus-signs. As will be explained in section 3.1, there is a second (orthogonal)
Spingroup Spin⊥(m) defined within discrete Clifford analysis involving the operators S e⊥
j j
which generate a Clifford algebra of signature (0,m). We choose to omit the proof of that
section andjustrefer to [3] and instead give the proof involving theoperators R explicitly.
j
Denote with R1 the linear vectorspace R1 = ω = m ω R : ω ∈ R, j = 1,...,m .
m m j=1 j j j
Even though ω is in fact an operator, we wilnl also call it a vector, which is justifiabole
P
m
since its action of the groundstate 1 gives us actual vectors: ω[1] = ω e . Then unit
j=1 j j
vectors are operators ω ∈ R1 such that |ω|2 =: m ω2 = 1. Since {R ,R } = 2δ , this
m j=1 j P j k j,k
implies that
m m P m
ωω = ω2R2+ ω ω R R = ω2 = 1.
j j j k j k j
j=1 j=1k6=j j=1
X XX X
Forapointa = (a ,...,a )∈ Zmwecanconsiderthecorrespondingvectora = m a R ∈
1 m j=1 j j
R1 which we will also denote a.
m P
Throughout this paper we will consider the following (anti-)involutions on C :
2m,0
∗
• The reversion a 7→ a∗, which is defined on the basis elements e± = e± and it
j j
is linearly extended to the entire Clifford algebra as (ab)∗ = (cid:16)b∗a∗(cid:17). We will also
extend their action also to R1 by R∗ = R ; this is motivated by the fact that
m j j
R = e+R++e−R− with R± scalar operators, see [10].
j j j j j j
• Theconjugationa 7→ aisthecompositionofthecomplexconjugationwiththeaction,
defined on basis elements as e± = −e± and linearly extended to the whole Clifford
j j
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algebra as ab = ba. We will also consider its action on R1 where in particular
m
R = −R .
j j
• The main involution a 7→ a which is a = a∗. In particular it holds that R = −R .
j j
Definition 3. Consider two operators X and Y, then we define
e e f
1
hX,Yi= (XY +Y X).
2
For two vectors a and b ∈ R1 , we find the inner product ha,bi = m a b ∈ R. Two
m j=1 j j
vectors a and b are then called orthogonal if and only if ha,bi = 0. Each vector a 6= 0 of
R1 is invertible, with inverse element a−1 = a . P
m |a|2
Definition 4. The (discrete) Clifford group is the multiplicative group
n
Γ(m)= ω : n ∈ N, ω ∈R1 \{0} .
i i m
( )
i=1
Y
Let M = {1,...,m}. Any element a ∈ Γ(m) can be decomposed as
a = a R , a ∈R.
A A A
A⊆M
X
For A = {a ,...,a } ⊆ M with 1 6 a < a < ... < a 6 m, we denote R = R ...R
1 k 1 2 k A a1 ak
and
|a|2 = |a |2.
A
A
X
Lemma 1. For a,b ∈ Γ(m) it holds that
aa∗ = |a|2, aa = |a|2, |ab| = |a||b|.
Proof. By definition, an element a ∈ Γ(m) consists of products of non-zero vectors a =
e
ω ω ...ω , with ω ∈ R1 \{0}. For ω , we find that ω ω∗ = ω ω = |ω |2 and so we get
1 2 k i m i i i i i i
that
aa∗ = ω ω ...ω ω∗...ω∗ω∗ = |ω |2|ω |2 ... |ω |2 > 0.
1 2 k k 2 1 1 2 k
We thus see that aa∗ is scalar. If on the other hand, we decompose a = a R , then
A A A
the (only) scalar part in the product aa∗ is a2. We may conclude that
A A P
aa∗ = Pa2 = |a|2.
A
A
X
a∗
Analogously, we can show that a∗a = |a|2. We thus see that for a ∈ Γ(m): a−1 = .
|a|2
From this it also follows that (a−1)∗ = a = (a∗)−1.
|a|2
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If we now take a,b ∈ Γ(m), then |ab|2 = (ab)(ab)∗ = abb∗a∗ = |a|2|b|2 and consequently
|ab| = |a||b|. Finally, consider
−1 ω ω a∗ a
(a)−1 = (−1)kω ...ω = (−1)k k ... 1 = (−1)k = .
1 k |ω |2 |ω |2 |a|2 |a|2
k 1
(cid:16) (cid:17)
e
With every a ∈ Γ(m), we can introduce the corresponding linear transformation χ(a) :
R1 → R1 :
m m
χ(a)(x)= ax(a)−1.
Lemma 2. Let a ∈ Γ(m) and x ∈ R1 , then it holds that ax(a)−1 ∈ R1 and the map
m m
e
χ(a) :R1 → R1 : x 7→ ax(a)−1
m m e
is a bijective isometry, i.e. χ(a) ∈ O(m).
e
Proof. Take x, y ∈ R1 , then xy + yx = 2 x,y ∈ R. Multiplication on the right of
m
yx = 2hx,yi−xy with y shows that yxy = 2hx,yiy−|y|2x. This is a linear combination
(cid:10) (cid:11)
of two elements of R1 with real coefficients and as such also an element of R1 . Since the
m m
−y
inverse element of y ∈ R1 is given by (y)−1 = , we see immediately that
m |y|2
e e yxy 2hx,yi
χ(y)(x) = yxy−1 = − = − y+x ∈ R1 .
|y|2 |y|2 m
e
Now take a ∈ Γ(m), then a = y y ...y for some y ∈ R1 \{0} and a = (−1)ny ...y .
1 2 n i m 1 n
Thus (a)−1 = (−1)ny−1...y−1 and hence
n 1
e
χ(a)(x) = ax(a)−1 = (−1)ny ... y xy−1 ... y−1 ∈R1 .
e 1 n n 1 m
The map χ(a) is clearly injectivee: χ(a)(x)= χ(a)(y) if and only if axa−1 = aya−1 if and
only if x = y. It is surjective since for a given a ∈ Γ(m) and x ∈R1 we find that
m
e e
χ(a) a−1xa = x,
where a−1xa ∈ R1 because it is equal to(cid:0)χ(a−1)(cid:1)(x), a−1 ∈Γ(m).
m e
Finally, the map χ(a) is an isometry, meaning that |x|2 = |χ(a)(x)|2:
e
axa−1 2 = axa−1 axa−1 ∗ = axa−1 (a∗)−1 x∗a∗
−1
(cid:12) (cid:12) = (cid:0)ax a∗a(cid:1)(cid:0) xa∗(cid:1)= |a|−2ax2a∗ = |a|−2|x|2|a|2 = |x|2.
(cid:12) e (cid:12) e e e e
(cid:16) (cid:17)
g
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Definition 5. We define the unit sphere Sm−1 to be the subspace of R1 containing unit
m
vectors, i.e. ω ∈ R1 such that ω2 =|ω|2 =1.
m
Lemma 3. Take ω ∈ Sm−1, then χ(ω)(x)[1] is the orthogonal reflection with respect to the
hyperplane ω⊥[1].
Proof. From the previous lemma, we know that
1
χ(ω)(x) = − ωxω = −2hx,ωiω+x.
|ω|2
If we decompose x = m x R as x = λω +t with λ ∈ R and t ∈ ω⊥, then hx,ωi =
j=1 j j
λ|ω|2 = λ and hence
P
χ(ω)(x) = −2λω+λω+t = −λω+t.
Thus χ(ω)(x)[1] is exactly the orthogonal reflection of x[1] with respect to the hyperplane
ω⊥[1].
Definition 6. The Pin group is the multiplicative group
n
Pin(m):= x : n ∈ N, x ∈ Sm−1, ∀i= 1,...,n .
i i
( )
i=1
Y
The Spingroup is the multiplicative group
2n
Spin(m) := x : n ∈ N, x ∈ Sm−1, ∀i= 1,...,2n .
i i
( )
i=1
Y
Every element a in Pin(m) corresponds to an element −χ(a) that is the composition of n
orthogonal reflections and thus to an element of O(m). In fact, since a and −a correspond
with the same bijective isometrie χ(a), we call Pin(m) a double cover of O(m).
Every element a in Spin(m) corresponds with an element χ(a) that is the composition of
an even number of orthogonal reflections and thus, by Hamilton’s theorem, an element of
SO(m). We call Spin(m) a double cover of SO(m).
3.1 Orthogonal Spingroup
In a completely similar fashion, we can define the linear vectorspace R1,⊥ of vectors in the
m
operators S e⊥, j = 1,...,m:
j j
m
R1,⊥ = ω = ω S e⊥ : ω ∈ R, j = 1,...,m
m  j j j j 
 Xj=1 
 
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and the unit sphere Sm−1,⊥ = ω ∈ R1,⊥ : ω2 = −|ω|2 = −1 .
m
Duetotherelation S e⊥,S e⊥n = −2δ ,thevectorsω ∈ Ro1,⊥satisfyω2 = − m ω2 =
j j k k j,k m j=1 j
−|ω|2. For a non-zenro vector ω oof Rm1,⊥, the inverse is given by ω−1 = −|ωω|2. P
Let Γ⊥(m) be the associated Clifford group
n
Γ⊥(m) = ω : n ∈ N, ω ∈ R1 \{0} .
i i m
( )
i=1
Y
e
Again one can associate a bijective isometry χ(a) ∈O(m) with every element a ∈ Γ⊥(m):
a ∈ Γ⊥(m)7→ χ(a) : R1,⊥ → R1,⊥, χ(a)(x) = axa−1.
m m
In particular, if a is a unit vector ω ∈ Sm−1,⊥, i.e. ω2 = −|ω|2 = −1, then χ(ω) is the
e
orthogonal reflection with respect to the hyperplane ω⊥, and as such, the Spingroup
2n
Spin⊥(m) = ω : n ∈N, ω ∈ Sm−1,⊥, ∀i= 1,...,2n
i i
( )
i=1
Y
is a double cover of SO(m).
In the following section we will now introduce a Spin(m)-representation within the space
of discrete polynomials in the m variables ξ ,...,ξ .
1 m
4 Spin(m)-representation
Consider the following actions of s ∈ Spin(m) on discrete Clifford-algebra valued polyno-
mials f(ξ ,...,ξ ):
1 m
H1(s)f(ξ ,...,ξ ) = sf(s¯ξs)s¯,
1 m
H0(s)f(ξ ,...,ξ ) = f(s¯ξs),
1 m
L(s)f(ξ ,...,ξ ) = sf(s¯ξs).
1 m
We will show that the operators ∂ and ∆ are L(s)-, resp. H1(s)- and H0(s)-invariant.
Remark 1. Both H-actions are ∆-invariant and preserve the space H , for k ∈ N. How-
k
ever, the difference lies in which function space they preserve. As in Euclidean Clifford
analysis, the H1-action of the classical Spingroup preserves k-vector, we would expect the
H1-action of the discrete Spingroup to show a similar treat. This will be a topic for further
research.
Remark 2. Because every discrete function can be expressed by its Taylor series, i.e. in
terms of discrete polynomials, this action is readily extendable to all discrete functions.
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We now explicitly prove that the discrete Dirac operator is invariant underthe L(s)-action
of s∈ Spin(m). We will start with some auxiliary lemmas
Lemma 4. Let ω = m ω R ∈ R1 , then
j=1 j j m
P m m
h∂,ωi = ω ∂ R , hξ,ωi = ω ξ R .
j j j j j j
j=1 j=1
X X
Proof. Thisfollows fromthedefinitionandthecommutator relations {ξ ,R } = 2δ ξ R
j k j,k j j
and {∂ ,R } = 2δ ∂ R :
j k j,k j j
m m m
1
hξ,ωi = ω {ξ ,R } = ω ξ R ,
k j k j j j
2
j=1k=1 j=1
XX X
m m m
1
h∂,ωi = ω {∂ ,R } = ω ∂ R .
k j k j j j
2
j=1k=1 j=1
XX X
Note that both ξ and R commute with hξ,ωi, for all j = 1,...,m, and thus also ξ and ω
j j
commute with hξ,ωi.
Corollary 1. The discrete Dirac operator ∂ and vector variable ξ admit the following
decompositions:
m m
∂ = h∂,R iR , ξ = hξ,R iR .
j j j j
j=1 j=1
X X
Proof. This follows from ∂ = m ∂ , h∂,R i = ∂ R and R2 = 1. Similarly for the
j=1 j j j j j
second statement.
P
Lemma 5. Let ω ∈ Sm−1 be a unit vector, i.e. ω2 = 1. Define the operator η = ωξω =
ω(ξ +...+ξ )ω. Then η = m η where
1 m j=1 j
P
η = 2hξ,ωiω R −ξ = hη,R iR
j j j j j j
and {R ,η } = 2R η δ .
j s j j j,s
Proof. Consider η = ωξω. Since ξω = 2hξ,ωi−ωξ and since ω is a unit vector, we get
η = ωξω = 2hξ,ωiω−ξ.
10