2 Dirac-Dunkl operator and a higher rank Bannai-Ito algebra Vincent - - PowerPoint PPT Presentation

The Zn

2 Dirac-Dunkl operator and a higher

rank Bannai-Ito algebra

Vincent X. Genest

Department of Mathematics Massachusetts Institute of Technology

Collaborators:

• H. De Bie (Ghent)
• L. Vinet (Montreal)

General context

Exactly Solvable Models

• Algebraic

Structures

• Special

functions

• Symmetries
• Leverage the relations between these topics to make advances

Today’s menu

Model: The Dirac-Dunkl equation associated to Zn

2

Algebraic structures:

Lie superalgebra osp(1,2) The Bannai-Ito algebra at rank n

Symmetries: Nested constants of motion arising as sCasimir operators Special functions:

Clifford-valued multivariate Jacobi-type orthogonal polynomials

stemming from Cauchy-Kovaleskaia extension

Multivariate Bannai-Ito polynomials

The Zn

2 Laplace-Dunkl operator on Rn

Dunkl operators: families of differential-difference operators associated

to finite reflection groups

Consider reflection group Zn

2 = Z2 ×···×Zn 2

Dunkl operators T1,...,Tn (on Rn) defined as

Ti = ∂xi + µi xi (1−ri) i = 1,...,n with µ1,...,µn > 0 and ri f(x1,...,xi,...,xn) = f(x1,...,−xi,...,xn)

Clearly TiTj = TjTi for all i,j = 1,...,n Laplace-Dunkl operator

∆ =

n

• i=1

T2

i

The Zn

2 Dirac-Dunkl operator on Rn

Clifford algebra Cln = 〈e1,...,en〉

{ei,ej} = eiej +ejei = −2δij i,j = 1,...,n

Dirac-Dunkl operator D and "position" operator x

D =

n

• i=1

eiTi x =

n

• i=1

eixi

One has

D2 = ∆ x2 = −||x||2 where n

i=1 x2 i

Some notation

Let [n] = {1,...,n} Let A ⊂ [n] and define "intermediate" Dirac-Dunkl and position

• perators

DA =

• i∈A

eiTi xA =

• i∈A

eiTi

Similarly

∆A =

• i∈A

T2

i

||xA||2 =

• i∈A

x2

i

Observe that D[n] = D and x[n] = x Likewise, for ℓ ≤ n, we shall use

D[ℓ] = D{1,...,ℓ} x[ℓ] = x{1,...,ℓ}

The osp(1,2) dynamical symmetry

Introduce the Euler operator EA =

i∈A xi ∂xi

Proposition (De Bie et al.,Trans Amer Math Soc 2012)

For A ⊂ [n], the operators DA and xA generate the Lie superalgebra

• sp(1|2) with defining relations

{xA,DA} = −2(EA +γA), [DA,EA +γA] = DA, [EA +γA,xA] = xA where [x,y] = xy−yx stands for the commutator and where γA = |A| 2 +

• i∈A

µi.

sCasimir and spherical Dirac-Dunkl operators

From osp(1,2) symmetry, consider the sCasimir operator

SA = 1 2

• [xA,DA]−1
• has the property

{SA,DA} = 0 {SA,xA} = 0

Spherical Dirac-Dunkl operators

Γ[n] = S[n]

n

• i=1

ri ΓA = SA =

• i∈A

ri

Have the property

[ΓA,DA] = 0 [ΓA,xA] = 0

Also commute with E[n] (hence well-defined action on Sn−1)

Dunkl monogenics

Let P (Rn) = R[x1,...,xn] ring of polynomials Then P (Rn) = ∞

k=0 Pk(Rn) where Pk(Rn) is space of homogeneous

polynomials of degree k

Let V be Cln-module (fixed once and for all) Space of monogenics Mk(Rn;V) of degree k defined as

Mk(Rn;V) = Ker D

• (Pk(Rn)⊗V)

Proposition (Ørsted et al, Adv. Appl. Clifford Alg.)

One has the decomposition Pk(Rn)⊗V =

k

• j=0

xj Mk−j(Rn;V)

The Dirac-Dunkl equation on Sn−1

Proposition

Let Ψk ∈ Mk(Rn;V) be a monogenic of degree k, then Ψk satisfies the Dirac–Dunkl equation Γ[n]Ψk = (−1)k(k+γ[n] −1/2)Ψk (⋆) where γ[n] = n

i=1 µi + n 2

Proof.

Stems directly from osp(1,2) symmetry Γ[n]Ψk = (−1)k 2

• x[n]D[n] −D[n]x[n] −1
• Ψk = (−1)k

2

• −D[n]x[n] −1
• Ψk

= (−1)k 2

• −x[n]D[n] −D[n]x[n] −1
• Ψk = (−1)k+1

2

• {x[n],D[n]}+1
• Ψk

= (−1)k+1 2

• −2(E[n] +γ[n])+1
• Ψk = (−1)k(k+γ[n] −1/2)Ψk,

Goals

Determine the symmetries of (⋆) Investigate the corresponding symmetry algebra Determine wavefunctions Ψk Describe the action of symmetries on the wavefunctions

Symmetries of the Dirac-Dunkl equation on Sn−1

We construct joint symmetries of Dirac-Dunkl (D) and spherical

Dirac-Dunkl (Γ[n]) operators

Use the nested osp(1,2) sCasimirs for all A ⊂ [n]

Lemma

For A ⊂ [n], the operator ΓA satisfies

• 1. [ΓA,D] = [ΓA,x] = 0
• 2. [ΓA,Γ[n]] = 0

Proof follows from the fact that ΓA commute with DA, xA and act only in xi for xi ∈ A

Hence ΓA for all A ⊂ [n] are symmetries For A = B, clearly ΓA, ΓB will not commute in general

Remark: Γ = −1 2 Γ{i} = µi

Some formulas

For the sake of being explicit ΓA =

• {i,j}⊂A

Mij + |A|−1 2 +

• k∈A

µkrk

i∈A

ri Mij reads Mij = eiej(xiTj −xjTi)

ΓA generate symmetry algebra of the Dirac-Dunkl equation (⋆) Algebra is determined by the commutation relations of Γ’s

Symmetry algebra

Proposition (De Bie, Genest, Vinet (2015))

The symmetries ΓA with A ⊂ [n] of the Dirac–Dunkl and spherical Dirac–Dunkl operators satisfy the anticommutation relations {ΓA,ΓB} = Γ(A∪B)\(A∩B) +2ΓA∩BΓA∪B +2ΓA\(A∩B)ΓB\(A∩B). Proof is by direct calculations, and multiple inductions on the cardinality of the involved sets.

Call this algebra An Clearly An ⊂ An+1 Can be show that all Γ{i,j} are sufficient generating set

Unsual algebra ! Let’s look at the n = 3 case.

The n = 3 case: The Bannai-Ito algebra

An is higher rank generalization of Bannai-Ito algebra Take n = 3, symmetry algebra of Γ[3] is generated by

K3 = Γ{1,2} K1 = Γ{2,3} K2 = Γ{1,3}

Commutation relations are

{K1,K2} = K3 +ω3 {K2,K3} = K1 +ω1 {K3,K1} = K2 +ω2 where ω1, ω2, ω3 are given by ω1 = 2µ1Γ[3] +2µ2µ3 ω2 = 2µ2Γ[3] +2µ1µ3 ω3 = 2µ3Γ[3] +2µ1µ2.

ωi are central elements, but on Mk(R3;V) they become numbers Structure associated to bispectrality of BI polynomials n = 3 case detailed in [De Bie, Genest, Vinet Comm. Math (2016)]

Abelian subalgebras

Algebra An has an important (maximal) Abelian subalgebra Yn

Yn = 〈Γ[2],Γ[3],...,Γ[n−1]〉

Thus An has rank (n−2) Non-unique, indeed Zn is also (maximal) Abelian subalgebra

Zn = 〈Γ{2,3},Γ{2,3,4},...,Γ{2,3,...,n}〉

Need to understand the wavefunctions Ψk and action of ΓA’s We’ll need a generalization of a construction known in Clifford analysis

as the Cauchy-Kovalevskaia extension

Wavefuntions and the CK isomorphism

We construct a basis of orthogonal wavefunctions Ψk which are

solutions of the Dirac-Dunkl equation on Sn−1 Γ[n]Ψk = (−1)k(k+γ[n] −1/2) Ψk (⋆)

Key: there is an isomorphism CK

µj xj : Pk(Rj−1)⊗V → Mk(Rj;V)

This isomorphism CK

µj xj is explicit

Proposition (De Bie, G., Vinet Comm Math. Phys. (2016))

The isomorphism between Pk(Rj−1)⊗V and Mk(Rj;V) denoted by CK

µj xj

is explicitly defined by the operator CK

µj xj =

Γ(µj +1/2)

• Iµj−1/2(xj D[j−1])+ 1

2ej xj D[j−1] Iµj+1/2(xj D[j−1])

• ,

with Iα(x) = ( 2

x )αIα(x) and Iα(x) the modified Bessel functions

Constructive: one considers a power series in the operators xj and D[j−1]

and solves for the coefficients to ensure that the result is in Mk(Rn;V)

More explicitly

CK

µj xj (p) = ⌊ k

2 ⌋

• ℓ=0

Γ(µj +1/2) 22ℓℓ!Γ(ℓ+µj +1/2) x2ℓ

j D2ℓ [j−1] p

+ ejxjD[j] 2

⌊ k−1

2 ⌋

• ℓ=0

Γ(µj +1/2) 22ℓℓ!Γ(ℓ+µj +3/2)x2ℓ

j D2ℓ [j−1] p.

with p ∈ Pk(Rn)⊗V

How is this helpful ? We can combine this theorem with the (Fischer) decomposition

presented earlier Pk(Rn)⊗V =

k

• j=0

xj Mk−j(Rn;V) to construct a basis

Constructing the basis

Let {vs} for s = 1,...,dim V be a basis for representation space V of Cln Combining CK

µj xj , one can write

Mk(Rn;V) ∼ = CKµn

xn

• Pk(Rn−1)⊗V

∼ = CKµn

xn

k

• j=0

xk−j

[n−1]Mj(Rn−1;V)

• ∼

= CKµn

xn

k

• j=0

xk−j

[n−1]CKµn−1 xn−1

• Pj(Rn−2)⊗V
• ∼

= CKµn

xn

k

• j=0

xk−j

[n−1]CKµn−1 xn−1

• j
• ℓ=0

xj−ℓ

[n−2]Mℓ(Rn−2;V)

• ∼

= ···

We can go down the tower until we reach the space of Clifford-valued

homogeneous polynomials of a certain degree j1 in one variable, which is spanned by xj1

1 vs

Wavefunctions

Proposition (De Bie, G., Vinet (2015))

Let j be defined as j = (j1,j2,...,jn−2,jn−1 = k−n−2

i=1 ji) where j1,...,jn−2

are non-negative integers such that n−2

i=1 ji ≤ k. Consider the set of

functions Ψs

j(x1,...,xn) defined by

Ψs

j(x1,...,xn) =

CKµn

xn

• xjn−1

[n−1]CKµn−1 xn−1

• xjn−2

[n−2]

• ···CKµ3

x3 [xj2 [2]CKµ2 x2 [xj1 1 ]

• vs,

with s ∈ I. Then the functions Ψs

j form a basis for the space Mk(Rn;V) of

k-homogeneous Dunkl monogenics.

Since CK

µj xj is known, these wavefunctions can be made explicit

Long calculations involved, but relatively straightforward

Wavefunctions

Some notation

|jℓ| = j1 +j2 +···+jℓ ||x[j]||2 = x2

1 +···+x2 j

Proposition (De Bie, G., Vinet (2015))

The basis functions Ψs

j have the expression

Ψs

j(x1,...,xn) =

− − − →

• 3
• ℓ=n
• Qjℓ−1(xℓ,x[ℓ−1])mj1(x2,x1)vs,

Qjℓ−1(xℓ,x[ℓ−1]) = β!x[ℓ]2β (µℓ +1/2)β ×                              P(|jℓ−2|+γ[ℓ−1]−1,µℓ−1/2)

β

• x2

ℓ−x[ℓ−1]2

x[ℓ]2

• jℓ−1 = 2β

−

eℓxℓx[ℓ−1] x[ℓ]2

P(|jℓ−2|+γ[ℓ−1],µℓ+1/2)

β−1

x2

[ℓ]−x[ℓ−1]2

x[ℓ]2

• x[ℓ−1] P(|jℓ−2|+γ[ℓ−1],µℓ−1/2)

β

• x2

ℓ−x[ℓ−1]2

x[ℓ]2

• jℓ−1 = 2β+1

−eℓxℓ β+|jℓ−2|+γ[ℓ−1]

β+µℓ+1/2

• P(|jℓ−2|+γ[ℓ−1]−1,µℓ+1/2)

β

• x2

ℓ−x[ℓ−1]2

x[ℓ]2

mj1(x2,x1) = (−1)ββ!(x2

1 +x2 2)β

(µ2 +1/2)β ×                            P(µ1−1/2,µ2−1/2)

β

• x2

2−x2 1

x2

1+x2 2

• j1 = 2β

− e2e1x2x1

x2

1+x2 2

P(µ1+1/2,µ2+1/2)

β−1

• x2

2−x2 1

x2

1+x2 2

• ,

x1P(µ1+1/2,µ2−1/2)

β

• x2

2−x2 1

x2

1+x2 2

• j1 = 2β+1

+ β+µ1+1/2

β+µ2+1/2

• e2e1x2 P(µ1−1/2,µ2+1/2)

β

• x2

2−x2 1

x2

1+x2 2

• P(α,β)

n

(x) are the Jacobi polynomials

(a)n = Γ(a+n)

Γ(a)

stands for the Pochhammer symbol

Algebra Yn and orthogonality

Wavefunctions Ψs

j(x1,...,xn) are joint eigenfunctions of the

commutative subalgebra Yn

Proposition (De Bie, G., Vinet (2015))

The wavefunctions Ψs

j ∈ Mk(Rn;V) satisfy

Γ[ℓ]Ψs

j(x1,...,xn) = λℓ(j)Ψs j(x1,...,xn),

where the eigenvalues are given by λℓ(j) = (−1)|jℓ−1|(|jℓ−1|+γ[ℓ] −1/2).

Up to a normalization factor, one has

• Sn−1

|x1|2µ1dx1 ···|xn|2µndxn

• Ψs′

j′ (x1,...,xn)

∗ ·Ψs

j(x1,...,xn) = δjj′δss′,

where ∗ is complex conjugation and · stands for an appropriately defined inner product on V; one has also ei∗ = −ei

Algebra Zn and Bannai-Ito polynomials

From the common eigenfunctions Ψs

j(x1,...,xn) of Yn, one can easily

construct a set of joint eigenfunctions for the Abelian subalgebra Zn

Let π = (12···n) be the cyclic permutation acting on x1,...,xn, e1,...,en

and µ1,...,µn, then common eigenfunctions Φs

j′(x1,...,xn) of Zn

Φs

j′(x1,...,xn) = πΨs j′(x1,...,xn)

π = (123···n),

Φs

j′(x1,...,xn) also form a basis for Mk(Rn;V)

Since both basis are orthonormal and finite dimensional, there exist a

unitary transformation between the two bases

Conjecture

The matrix elements of the intertwining operator between the basis functions Ψs

j(x1,...,xn) and Φs j′(x1,...,xn) are expressed in terms of

• rthogonal polynomials which could be taken to define the multivariate

extension of the Bannai–Ito polynomials.

Representations of An

It is clear that functions Ψs

j(x1,...,xn) will support representations of

An

To specify these representations, it is needed to find the action of the

• perators ΓA on these functions

This can be done via the introduction of raising/lowering operators

K±

ℓ =

(Γ{ℓ+1,ℓ+2} ±Γ[ℓ+2]\{ℓ+1})(Γ[ℓ+1] ∓1/2)−(Γ{ℓ+2} ±Γ[ℓ+2])(Γ[ℓ] ±Γ{ℓ+1}). for ℓ ∈ [n−2] and observing that [K±

ℓ ,Γ[j]] = 0

j = ℓ+1 Conversely when j = ℓ+1 one has {K±

ℓ ,Γ[ℓ+1]} = ±K± ℓ

Final remarks

There exists an equivalent scalar model that does not involve the

Clifford algebra

Much work to be done to understand An further (representations, etc.) The conjecture should be checked ! Opens the door for higher rank versions of the Racah algebra, which

would describe the bispectrality of the multivariate Racah polynomials

One could also investigate other root systems