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

The Z n 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 � Symmetries functions � Leverage the relations between these topics to make advances

Today’s menu � Model: The Dirac-Dunkl equation associated to Z n 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 Z n 2 Laplace-Dunkl operator on R n � Dunkl operators: families of differential-difference operators associated to finite reflection groups � Consider reflection group Z n 2 = Z 2 ×···× Z n 2 � Dunkl operators T 1 ,..., T n (on R n ) defined as T i = ∂ x i + µ i (1 − r i ) i = 1 ,..., n x i with µ 1 ,..., µ n > 0 and r i f ( x 1 ,..., x i ,..., x n ) = f ( x 1 ,..., − x i ,..., x n ) � Clearly T i T j = T j T i for all i , j = 1 ,..., n � Laplace-Dunkl operator � n T 2 ∆ = i i = 1

The Z n 2 Dirac-Dunkl operator on R n � Clifford algebra Cl n = 〈 e 1 ,..., e n 〉 { e i , e j } = e i e j + e j e i = − 2 δ ij i , j = 1 ,..., n � Dirac-Dunkl operator D and "position" operator x � n � n D = e i T i x = e i x i i = 1 i = 1 � One has D 2 = ∆ x 2 = −|| x || 2 where � n i = 1 x 2 i

Some notation � Let [ n ] = { 1 ,..., n } � Let A ⊂ [ n ] and define "intermediate" Dirac-Dunkl and position operators � � D A = e i T i x A = e i T i i ∈ A i ∈ A � Similarly � � || x A || 2 = T 2 x 2 ∆ A = i i i ∈ A i ∈ A � 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 E A = � i ∈ A x i ∂ x i Proposition (De Bie et al., Trans Amer Math Soc 2012) For A ⊂ [ n ] , the operators D A and x A generate the Lie superalgebra osp (1 | 2) with defining relations { x A , D A } = − 2( E A + γ A ) , [ D A , E A + γ A ] = D A , [ E A + γ A , x A ] = x A where [ x , y ] = xy − yx stands for the commutator and where � γ A = | A | 2 + µ i . i ∈ A

sCasimir and spherical Dirac-Dunkl operators � From osp (1 , 2) symmetry, consider the sCasimir operator � � S A = 1 [ x A , D A ] − 1 2 has the property { S A , D A } = 0 { S A , x A } = 0 � Spherical Dirac-Dunkl operators � n � Γ [ n ] = S [ n ] r i Γ A = S A = r i i = 1 i ∈ A � Have the property [ Γ A , D A ] = 0 [ Γ A , x A ] = 0 � Also commute with E [ n ] (hence well-defined action on S n − 1 )

Dunkl monogenics � Let P ( R n ) = R [ x 1 ,..., x n ] ring of polynomials � Then P ( R n ) = � ∞ k = 0 P k ( R n ) where P k ( R n ) is space of homogeneous polynomials of degree k � Let V be Cl n -module (fixed once and for all) � Space of monogenics M k ( R n ; V ) of degree k defined as � M k ( R n ; V ) = Ker D ( P k ( R n ) ⊗ V ) Proposition (Ørsted et al, Adv. Appl. Clifford Alg. ) One has the decomposition � k x j M k − j ( R n ; V ) P k ( R n ) ⊗ V = j = 0

The Dirac-Dunkl equation on S n − 1 Proposition Let Ψ k ∈ M k ( R n ; V ) be a monogenic of degree k, then Ψ k satisfies the Dirac–Dunkl equation Γ [ n ] Ψ k = ( − 1) k ( k + γ [ n ] − 1/2) Ψ k ( ⋆ ) where γ [ n ] = � n i = 1 µ i + n 2 Proof. Stems directly from osp (1 , 2) symmetry Γ [ n ] Ψ k = ( − 1) k Ψ k = ( − 1) k � � � � x [ n ] D [ n ] − D [ n ] x [ n ] − 1 − D [ n ] x [ n ] − 1 Ψ k 2 2 = ( − 1) k Ψ k = ( − 1) k + 1 � � � � − x [ n ] D [ n ] − D [ n ] x [ n ] − 1 { x [ n ] , D [ n ] } + 1 Ψ k 2 2 = ( − 1) k + 1 � � Ψ k = ( − 1) k ( k + γ [ n ] − 1/2) Ψ k , − 2( E [ n ] + γ [ n ] ) + 1 2

Goals � Determine the symmetries of ( ⋆ ) � Investigate the corresponding symmetry algebra � Determine wavefunctions Ψ k � Describe the action of symmetries on the wavefunctions

Symmetries of the Dirac-Dunkl equation on S n − 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 Γ A satisfies 1. [ Γ A , D ] = [ Γ A , x ] = 0 2. [ Γ A , Γ [ n ] ] = 0 Proof follows from the fact that Γ A commute with D A , x A and act only in x i for x i ∈ A � Hence Γ A for all A ⊂ [ n ] are symmetries � For A �= B , clearly Γ A , Γ B will not commute in general Remark: Γ � = − 1 Γ { i } = µ i 2

Some formulas For the sake of being explicit �� � � � M ij + | A |− 1 Γ A = + µ k r k r i 2 { i , j } ⊂ A k ∈ A i ∈ A M ij reads M ij = e i e j ( x i T j − x j T i ) � Γ A 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 Γ A with A ⊂ [ n ] of the Dirac–Dunkl and spherical Dirac–Dunkl operators satisfy the anticommutation relations { Γ A , Γ B } = Γ ( A ∪ B ) \ ( A ∩ B ) + 2 Γ A ∩ B Γ A ∪ B + 2 Γ A \ ( A ∩ B ) Γ B \ ( A ∩ B ) . Proof is by direct calculations, and multiple inductions on the cardinality of the involved sets. � Call this algebra A n � Clearly A n ⊂ A n + 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 � A n is higher rank generalization of Bannai-Ito algebra � Take n = 3, symmetry algebra of Γ [3] is generated by K 3 = Γ { 1 , 2 } K 1 = Γ { 2 , 3 } K 2 = Γ { 1 , 3 } � Commutation relations are { K 1 , K 2 } = K 3 + ω 3 { K 2 , K 3 } = K 1 + ω 1 { K 3 , K 1 } = K 2 + ω 2 where ω 1 , ω 2 , ω 3 are given by ω 1 = 2 µ 1 Γ [3] + 2 µ 2 µ 3 ω 2 = 2 µ 2 Γ [3] + 2 µ 1 µ 3 ω 3 = 2 µ 3 Γ [3] + 2 µ 1 µ 2 . � ω i are central elements, but on M k ( R 3 ; 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 A n has an important (maximal) Abelian subalgebra Y n Y n = 〈 Γ [2] , Γ [3] ,..., Γ [ n − 1] 〉 � Thus A n has rank ( n − 2) � Non-unique, indeed Z n is also (maximal) Abelian subalgebra Z n = 〈 Γ { 2 , 3 } , Γ { 2 , 3 , 4 } ,..., Γ { 2 , 3 ,..., n } 〉 � Need to understand the wavefunctions Ψ k and action of Γ A ’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 Ψ k which are solutions of the Dirac-Dunkl equation on S n − 1 Γ [ n ] Ψ k = ( − 1) k ( k + γ [ n ] − 1/2) Ψ k ( ⋆ ) µ j x j : P k ( R j − 1 ) ⊗ V → M k ( R j ; V ) � Key: there is an isomorphism CK µ j � This isomorphism CK x j is explicit Proposition (De Bie, G., Vinet Comm Math. Phys. (2016)) µ j The isomorphism between P k ( R j − 1 ) ⊗ V and M k ( R j ; V ) denoted by CK x j is explicitly defined by the operator µ j CK x j = � � I µ j − 1/2 ( x j D [ j − 1] ) + 1 � 2 e j x j D [ j − 1] � Γ ( µ j + 1/2) I µ j + 1/2 ( x j D [ j − 1] ) , with � I α ( x ) = ( 2 x ) α I α ( x ) and I α ( x ) the modified Bessel functions

� Constructive: one considers a power series in the operators x j and D [ j − 1] and solves for the coefficients to ensure that the result is in M k ( R n ; V ) � More explicitly ⌊ k 2 ⌋ � Γ ( µ j + 1/2) µ j 2 2 ℓ ℓ ! Γ ( ℓ + µ j + 1/2) x 2 ℓ j D 2 ℓ CK x j ( p ) = [ j − 1] p ℓ = 0 ⌊ k − 1 2 ⌋ e j x j D [ j ] � Γ ( µ j + 1/2) 2 2 ℓ ℓ ! Γ ( ℓ + µ j + 3/2) x 2 ℓ j D 2 ℓ [ j − 1] p . + 2 ℓ = 0 with p ∈ P k ( R n ) ⊗ V � How is this helpful ? � We can combine this theorem with the (Fischer) decomposition presented earlier � k x j M k − j ( R n ; V ) P k ( R n ) ⊗ V = j = 0 to construct a basis

Constructing the basis � Let { v s } for s = 1 ,..., dim V be a basis for representation space V of Cl n µ j � Combining CK x j , one can write � k � � ∼ � � = CK µ n = CK µ n x k − j M k ( R n ; V ) ∼ P k ( R n − 1 ) ⊗ V [ n − 1] M j ( R n − 1 ; V ) x n x n j = 0 � k �� � � = CK µ n x k − j [ n − 1] CK µ n − 1 P j ( R n − 2 ) ⊗ V ∼ x n x n − 1 j = 0 � k �� j � � � = CK µ n x k − j [ n − 1] CK µ n − 1 x j − ℓ ∼ [ n − 2] M ℓ ( R n − 2 ; V ) ∼ = ··· x n x n − 1 j = 0 ℓ = 0 � We can go down the tower until we reach the space of Clifford-valued homogeneous polynomials of a certain degree j 1 in one variable, which is spanned by x j 1 1 v s