The Fueter Theorem and Dirac symmetries

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1 The Fueter Theorem and Dirac symmetries David Eelbode Departement of Mathematics and Computer Science University of Antwerp (partially joint work with V. Souček and P. Van Lancker)

2 General overview of lecture Define differential operators on R m (multivariate analysis) Generate solutions of a special form (the Fueter theorem) Proof by means of (Lie) symmetries of the operator Explore the underlying algebraic framework Find connections with special functions

3 Quaternionic analysis in H: notations Regularity in quaternionic analysis D q f = 0 with f : R 4 H : (t; x, y, z) f (t; x, y, z), where f = f 0 + if 1 + jf 2 + kf 3 (and i 2 = j 2 = k 2 = 1) The Fueter-operator on H-valued functions: D q := t + i x + j y + k z Quaternions can be written in polar form as q := t + (ix + jy + kz) = t + ρω (ω S 2 R 3 ) where ρ 2 = x 2 + y 2 + z 2 (Euclidean spatial squared norm)

4 Recent interest in quaternionic analysis Applications in theoretical physics (Frenkel - Libine) Schrödinger model for minimal representations of O(3, 3) Quaternionic analysis, representation theory and physics Functional calculus and spectral analysis (Alpay - Colombo - Sabadini - Struppa -...) Quaternionic Fourier transforms (Bujack - De Bie - Hitzer - Sangwine - Scheuermann) Relation with the conformal algebra: R 1,3 = M 2 2 (H)

5 Fueter s original result (1935) Construction of regular functions from holomorphic ones: F : ker z ker D q Input: a holomorphic function on Ω C f (z) = u(x, y) + iv(x, y) ker( x + i y ) Output: a q-regular function (solution for D q ) F (t; x, y, z) = F[f ] := 4 ( u(t, ρ) + ωv(t, ρ) ) the map F combines substitution + Laplace operator

6 Clifford analysis in a nutshell Inspired by the (massless) Dirac equation on R 1,3 : i γ µ µ ψ(t; x) = 0 matrices γ µ generate the Clifford algebra R 1,3 Clifford algebras exist in greater generality: (p, q) Z + Z + R p,q R p,q C = C p+q := C m Natural question: study the corresponding Dirac operator m R p,q = Alg(e 1,..., e m ) x := e j xj j=1

7 Defining relations for C m (similar for R p,q ): e a e b + e b e a = {e a, e b } = 2δ ab m anti-commuting complex (or hyperbolic) units Clifford analysis thus refines harmonic analysis: 2 x = (e 1 x e m xm ) 2 = m Study of functions with values in (subset of) C m typical example: a real subalgebra such as R 0,m best choice (algebraically speaking): a spinor space

8 Clifford analysis generalises complex analysis algebra isomorphism R 0,1 = C Dirac operator on R 2 factorises 2 analogue of Cauchy formula available (for all m) Clifford analysis generalises quaternionic analysis algebra isomorphism R 0,2 = H Dirac operator on R 4 factorises 4 Obvious question 1: how to generalise Fueter s theorem? (i.e. special Dirac solutions from holomorphic functions) Obvious question 2: do they have a special meaning?

9 Answers can be found in... Work done by Sommen-Tao-Kou F : ker z ker x first in odd dimensions m ( obvious generalisation) later in even dimensions m (Fourier multipliers) further generalisations ( shifted versions ) Essentially also substitution + Laplace operator F[f ](x 1,..., x m ) = m 1 [ 2 m u(ρ, xm ) + ωe m v(ρ, x m ) ] with j x je j = ρω + x m e m R m 1 R (branched splitting)

10 Plan for the lecture Give an alternative proof using symmetries for x We are interested in an algebraic approach: easier to see the link with special functions allows a generalisation to other invariant operators possible connection with other quadratic algebras possible connection with slice regularity

11 Symmetries for a differential operator Consider a differential operator D : C (R m, V) C (R m, V) One says: δ 1 End C (R m, V) is a generalised symmetry if there also exists an operator δ 2 End C (R m, V) such that Dδ 1 = δ 2 D δ 1 End ker D Special case: δ 1 = δ 2 a classical symmetry for D First order (generalised) symmetries span a Lie algebra [Miller]

12 Generalised symmetries for the Dirac operator (obvious) translation symmetry operators: [ xj, x ] = 0 rotational symmetry operators: [dl(e ij ), x ] = 0 ( dl(e ij )f (x) := x i xj x j xi 1 ) 2 e ij f (x), where e ij generates a rotation ( anti-symmetric matrix ) generalised Euler symmetry (E x = r r = j x j xj ): x (2E x + m 1) = (2E x + m + 1) x So far: symmetries belonging to ( so(m) R ) R m (you may recognise a parabolic Lie algebra/euclidean motions)

13 More symmetries available! Definition The (monogenic) Klein inversion operator is defined by means of I : ker x ker x : f (x) I[f ](x) := x ( ) x x m f x 2 easily verified that the following operator equality holds: I x I = x 2 x x I = I x 2 x = ( I x 2 I ) I x gives rise to another class of generalised symmetries: x (I xj I) = 1 x 2 (I x j I) x 2 x (1 j m)

14 The full conformal picture We have found the following Lie algebra of symmetries: Alg ( I xj I ) ( Alg ( dl(e ij ) ) ) R(2E x + m 1) Alg( xj ) Defines a realisation for the algebra so(1, m + 1) e.g. [ xj, I xk I] = δ jk (2E x + m 1) dl(e jk ) explains where the conformal weight (m 1) comes from in abstract form: R m ( so(m) R ) R m (1-graded)

15 Intermezzo: the Lie algebra sl(2) Classical matrix definition: sl(2) := {M C 2 2 : trace(m) = 0} Well-known vector space basis {X, Y, H} given by {( ) ( ) ( )} ,, Defines a Lie algebra for [A, B] := AB BA with [X, Y ] = H [H, X ] = +2X [H, Y ] = 2Y

16 Verma modules Take a highest weight vector v h in ker X Repeated action of Y sl(2) creates a vector space := Y j [v h ] V λ j=0 (provided we have that Y j [v h ] 0 for all j Z + ) Each of these vectors is an eigenvector for H sl(2) Result: a Verma module V λ H [ Y j [v h ] ] = (λ 2j)Y j [v h ] with λ C (with restrictions)

17 Specific subalgebra plays a crucial role Lemma The Lie algebra sl(2) can be realised as sl(2) = Alg( xj, I xj I, 1 m 2E x ) = Alg(X, Y, H) Arbitrary polynomial in (x 1,..., x m 1 ) generates a Verma module: V 1 m 2k := ( 1) j (I xm I) j P k (x 1,..., x m 1 ) j=0 weight spaces expressed in terms of Gegenbauer polynomials these Verma modules will serve as images of the Fueter map

18 Slightly deformed version of previous slide Definition The (poly-monogenic) inversion operator is defined by means of I α : f (x) I α [f ](x) := x ( ) x x m+α f x 2 (α C) typical case of interest: α = 2l with l Z + mapping properties differ from Klein inversion (α = 0) I 2l : ker x 2l+1 ker x 2l+1 indeed: f (x) x 2l I[f ](x) (extra factor x 2l to be killed)

19 Still a copy of sl(2) available Lemma The Lie algebra sl(2) can be realised as sl(2) = Alg( xj, I α xj I α, 1 m α 2E x ) Explicit formula for the creation operator : I α xm I α = x 2 xm + x m (2E x + α + m 1) + x e m In order to get a Fueter theorem we need two ingredients: a substitution map z = x + iy? the action of a Laplace power

20 The substitution map Earlier results based on the following: u + iv u + (ωe m )v = scalar + (ωe m )scalar We know that in the Clifford algebra one has that This suggests the following: (vector) (vector) = scalar + bivector z = x + iy (ē m x) = e m m j=1 m 1 e j x j = x m + e j e m x j j=1

21 Invoking the conformal symmetries The following can now easily be proved: ( x 2 xm + x m (2E x + 1) + x e m ) (ēm x) k = (k + 1)(ē m x) k+1 This leads to (choose the appropriate α C): (I 2 m xm I 2 m ) k [1] = k!(ē m x) k we have that I 2 m = I 2l l = m 2 2 (for m even) the operator I 3 m then preserves solutions for x 1+2l so does the creation operator I 2 m xm I 2 m = m 1 x

22 Fueter s theorem We have now shown (for odd dimensions m): ( ) x m 1 (ē m x) k = ( 1) m 2 1 x m 2 1 x (ē m x) k = 0 As this holds for all k, the result for holomorphic f (z) follows: f (z) ker z m 1 2 m f (ē m x) ker x Explicit expressions for Fueter images available: F[z k+m 1 ] (I 0 xm I 0 ) k [1] π m ( x k C m 2 1 with t = cos θ = xm x [ 1, +1] fixed by the choice of e m k ) (t)

23 The monogenic projection operator π m Complex case: 2 = 4 z z, so 2 f (z) = 0 f (z) = f 0 (z) + zf 1 (z) π 2 f0 (z) with z f 0 (z) = z f 1 (z) = 0 We know that 2 x = m, so x f (x) = 0 f (x) = f 0 (x) + xf 1 (x) π m f0 (x) with x f 0 (x) = x f 1 (x) = 0

24 The role of the Gegenbauer polynomials In a sense, Fueter s theorem does the following: ( c k z k π m C k,m x k + D ) k,m x k+m 2 C m 2 1 k (t) k Z k=0 Gegenbauers arise as a consequence of branching rules (both for harmonics and monogenics on R m ) k P k (R m, C m ) ker x = P j (R m 1, C m ) ker x j=0 Special role for j = 0: these are the Gegegenbauers

25 Different interpretation for Fueter s theorem Complex powers z k are mapped to special components for the restriction of homogeneous solutions for x on R m to R m 1 The result is a slice monogenic function (Colombo et al.) Obvious question: can one restrict from R m to R m p (p > 1) leads to bi-axial Fueter theorems (Jacobi polynomials) may lead to refinements of the slice theorems

26 Thank you for your attention!