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(1)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Projectively equivariant quantization over

R

p|q

Fabian Radoux

(2)

Projectively equivariant quantization over Rp|q Fabian Radoux

Introduction

Classical observables: functions on a symplectic manifold (M, ω).

Quantum observables: operators on a Hilbert space H. Dirac problem: find a bijection Q : C∞(M) → L(H) such that

Q(1M) = IdH, Q(¯f ) = Q(f )∗,

(3)

Projectively equivariant quantization over Rp|q Fabian Radoux

Introduction

Classical observables: functions on a symplectic manifold (M, ω).

Quantum observables: operators on a Hilbert space H.

Dirac problem: find a bijection Q : C∞(M) → L(H) such that

Q(1M) = IdH, Q(¯f ) = Q(f )∗,

(4)

Projectively equivariant quantization over Rp|q Fabian Radoux

Introduction

Classical observables: functions on a symplectic manifold (M, ω).

Quantum observables: operators on a Hilbert space H. Dirac problem: find a bijection Q : C∞(M) → L(H) such that

Q(1M) = IdH, Q(¯f ) = Q(f )∗,

(5)

Projectively equivariant quantization over Rp|q Fabian Radoux

Introduction

Classical observables: functions on a symplectic manifold (M, ω).

Quantum observables: operators on a Hilbert space H. Dirac problem: find a bijection Q : C∞(M) → L(H) such that

Q(1M) = IdH,

Q(¯f ) = Q(f )∗,

(6)

Projectively equivariant quantization over Rp|q Fabian Radoux

Introduction

Classical observables: functions on a symplectic manifold (M, ω).

Quantum observables: operators on a Hilbert space H. Dirac problem: find a bijection Q : C∞(M) → L(H) such that

Q(1M) = IdH, Q(¯f ) = Q(f )∗,

(7)

Projectively equivariant quantization over Rp|q Fabian Radoux

Introduction

Classical observables: functions on a symplectic manifold (M, ω).

Quantum observables: operators on a Hilbert space H. Dirac problem: find a bijection Q : C∞(M) → L(H) such that

Q(1M) = IdH, Q(¯f ) = Q(f )∗, Q({f , g }) = i

(8)

Projectively equivariant quantization over Rp|q Fabian Radoux Hamiltonian H, observable F , dF dt = {H, F } 7−→ dQ(F ) dt = i ~[Q(H), Q(F )] Prequantization: if H = L2(M), if ω = d α, Q(f ) = ~ iXf + f − hXf, αi.

(9)

Projectively equivariant quantization over Rp|q Fabian Radoux Hamiltonian H, observable F , dF dt = {H, F } 7−→ dQ(F ) dt = i ~[Q(H), Q(F )] Prequantization: if H = L2(M), if ω = d α, Q(f ) = ~ iXf + f − hXf, αi.

(10)

Projectively equivariant quantization over Rp|q Fabian Radoux Hamiltonian H, observable F , dF dt = {H, F } 7−→ dQ(F ) dt = i ~[Q(H), Q(F )] Prequantization: if H = L2(M), if ω = d α, Q(f ) = ~ iXf + f − hXf, αi.

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Problem of the prequantization: if M = T∗M, L2(T∗M) is too big−→reduction of H.

Polarization of (M, ω): distribution P ⊂ T M. P allows to reduce H, H is replaced by HP.

The observable f is quantizable if Q(f ) preserves HP. The set of quantizable observables: A.

(12)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Problem of the prequantization: if M = T∗M, L2(T∗M) is too big−→reduction of H.

Polarization of (M, ω): distribution P ⊂ T M.

P allows to reduce H, H is replaced by HP.

The observable f is quantizable if Q(f ) preserves HP. The set of quantizable observables: A.

(13)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Problem of the prequantization: if M = T∗M, L2(T∗M) is too big−→reduction of H.

Polarization of (M, ω): distribution P ⊂ T M. P allows to reduce H, H is replaced by HP.

The observable f is quantizable if Q(f ) preserves HP. The set of quantizable observables: A.

(14)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Problem of the prequantization: if M = T∗M, L2(T∗M) is too big−→reduction of H.

Polarization of (M, ω): distribution P ⊂ T M. P allows to reduce H, H is replaced by HP.

The observable f is quantizable if Q(f ) preserves HP.

The set of quantizable observables: A.

(15)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Problem of the prequantization: if M = T∗M, L2(T∗M) is too big−→reduction of H.

Polarization of (M, ω): distribution P ⊂ T M. P allows to reduce H, H is replaced by HP.

The observable f is quantizable if Q(f ) preserves HP. The set of quantizable observables: A.

(16)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Problem of the prequantization: if M = T∗M, L2(T∗M) is too big−→reduction of H.

Polarization of (M, ω): distribution P ⊂ T M. P allows to reduce H, H is replaced by HP.

The observable f is quantizable if Q(f ) preserves HP. The set of quantizable observables: A.

(17)

Projectively equivariant quantization over

Rp|q Fabian Radoux

If M = T∗M, if P is the vertical polarization, then

HP ∼= L2(M), A ∼= S≤1(M),

QG(Xi(x )pi + A(x )) = ~

iX

i(x )∂

i+ A(x ).

Is it possible to extend the geometric quantization to S(M)?

Is this prolongation unique?

(18)

Projectively equivariant quantization over

Rp|q Fabian Radoux

If M = T∗M, if P is the vertical polarization, then

HP ∼= L2(M), A ∼= S≤1(M),

QG(Xi(x )pi + A(x )) = ~

iX

i(x )∂

i+ A(x ).

Is it possible to extend the geometric quantization to S(M)?

Is this prolongation unique?

(19)

Projectively equivariant quantization over

Rp|q Fabian Radoux

If M = T∗M, if P is the vertical polarization, then

HP ∼= L2(M), A ∼= S≤1(M),

QG(Xi(x )pi + A(x )) = ~

iX

i(x )∂

i+ A(x ).

Is it possible to extend the geometric quantization to S(M)?

Is this prolongation unique?

(20)

Projectively equivariant quantization over

Rp|q Fabian Radoux

If M = T∗M, if P is the vertical polarization, then

HP ∼= L2(M), A ∼= S≤1(M),

QG(Xi(x )pi + A(x )) = ~

iX

i(x )∂

i+ A(x ).

Is it possible to extend the geometric quantization to S(M)?

Is this prolongation unique?

(21)

Projectively equivariant quantization over

Rp|q Fabian Radoux

If M = T∗M, if P is the vertical polarization, then

HP ∼= L2(M), A ∼= S≤1(M),

QG(Xi(x )pi + A(x )) = ~

iX

i(x )∂

i+ A(x ).

Is it possible to extend the geometric quantization to S(M)?

Is this prolongation unique?

(22)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Equivariant quantization: action of a Lie group G on M: Φ : G × M → M.

Equivariant quantization Q: linear bijection Q: S(M) → D(M) s.t. σ(Q(S)) = S and s.t. Q(Φ∗gS) = Φ∗gQ(S ) ∀g ∈ G .

Q(Lh∗S ) = Lh∗Q(S ) ∀h ∈ g.

Idea: to take G sufficiently small to have a quantization and sufficiently big to have the uniqueness.

Projective case (P. Lecomte, V. Ovsienko):

PGL(m + 1, R) acts on RPm.

X ∈ sl(m + 1, R) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ sl(m + 1, R).

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Equivariant quantization: action of a Lie group G on M: Φ : G × M → M.

Equivariant quantization Q: linear bijection Q: S(M) → D(M) s.t. σ(Q(S)) = S and s.t. Q(Φ∗gS) = Φ∗gQ(S ) ∀g ∈ G .

Q(Lh∗S ) = Lh∗Q(S ) ∀h ∈ g.

Idea: to take G sufficiently small to have a quantization and sufficiently big to have the uniqueness.

Projective case (P. Lecomte, V. Ovsienko):

PGL(m + 1, R) acts on RPm.

X ∈ sl(m + 1, R) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ sl(m + 1, R).

(24)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Equivariant quantization: action of a Lie group G on M: Φ : G × M → M.

Equivariant quantization Q: linear bijection Q: S(M) → D(M) s.t. σ(Q(S)) = S and s.t. Q(Φ∗gS) = Φ∗gQ(S ) ∀g ∈ G .

Q(Lh∗S ) = Lh∗Q(S ) ∀h ∈ g.

Idea: to take G sufficiently small to have a quantization and sufficiently big to have the uniqueness.

Projective case (P. Lecomte, V. Ovsienko):

PGL(m + 1, R) acts on RPm.

X ∈ sl(m + 1, R) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ sl(m + 1, R).

(25)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Equivariant quantization: action of a Lie group G on M: Φ : G × M → M.

Equivariant quantization Q: linear bijection Q: S(M) → D(M) s.t. σ(Q(S)) = S and s.t. Q(Φ∗gS) = Φ∗gQ(S ) ∀g ∈ G .

Q(Lh∗S ) = Lh∗Q(S ) ∀h ∈ g.

Idea: to take G sufficiently small to have a quantization and sufficiently big to have the uniqueness.

Projective case (P. Lecomte, V. Ovsienko):

PGL(m + 1, R) acts on RPm.

X ∈ sl(m + 1, R) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ sl(m + 1, R).

(26)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Equivariant quantization: action of a Lie group G on M: Φ : G × M → M.

Equivariant quantization Q: linear bijection Q: S(M) → D(M) s.t. σ(Q(S)) = S and s.t. Q(Φ∗gS) = Φ∗gQ(S ) ∀g ∈ G .

Q(Lh∗S ) = Lh∗Q(S ) ∀h ∈ g.

Idea: to take G sufficiently small to have a quantization and sufficiently big to have the uniqueness.

Projective case (P. Lecomte, V. Ovsienko):

PGL(m + 1, R) acts on RPm.

X ∈ sl(m + 1, R) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ sl(m + 1, R).

(27)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Equivariant quantization: action of a Lie group G on M: Φ : G × M → M.

Equivariant quantization Q: linear bijection Q: S(M) → D(M) s.t. σ(Q(S)) = S and s.t. Q(Φ∗gS) = Φ∗gQ(S ) ∀g ∈ G .

Q(Lh∗S ) = Lh∗Q(S ) ∀h ∈ g.

Idea: to take G sufficiently small to have a quantization and sufficiently big to have the uniqueness.

Projective case (P. Lecomte, V. Ovsienko):

PGL(m + 1, R) acts on RPm.

X ∈ sl(m + 1, R) 7→ X∗ vector field on Rm.

(28)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Equivariant quantization: action of a Lie group G on M: Φ : G × M → M.

Equivariant quantization Q: linear bijection Q: S(M) → D(M) s.t. σ(Q(S)) = S and s.t. Q(Φ∗gS) = Φ∗gQ(S ) ∀g ∈ G .

Q(Lh∗S ) = Lh∗Q(S ) ∀h ∈ g.

Idea: to take G sufficiently small to have a quantization and sufficiently big to have the uniqueness.

Projective case (P. Lecomte, V. Ovsienko):

PGL(m + 1, R) acts on RPm.

X ∈ sl(m + 1, R) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ sl(m + 1, R).

(29)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Conformal case (C. Duval, P. Lecomte, V. Ovsienko):

SO(p + 1, q + 1) acts on Sp× Sq.

X ∈ so(p + 1, q + 1) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ so(p + 1, q + 1). Casimir operator method:

l: Semi-simple Lie algebra endowed with a non degenerate Killing form K ,

K (X , Y ) := tr (ad(X) ◦ ad(Y)) (V , β): representation of l.

(ui : i 6 n): basis of l; (ui0 : i 6 n): Killing-dual basis

(K (ui, uj0) = δi ,j).

Casimir operator corresponding to (V , β):

n

X

i =1

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Conformal case (C. Duval, P. Lecomte, V. Ovsienko):

SO(p + 1, q + 1) acts on Sp× Sq.

X ∈ so(p + 1, q + 1) 7→ X∗ vector field on Rm.

∃Q : LXQ(S ) = Q(LXS) ∀X ∈ so(p + 1, q + 1). Casimir operator method:

l: Semi-simple Lie algebra endowed with a non degenerate Killing form K ,

K (X , Y ) := tr (ad(X) ◦ ad(Y)) (V , β): representation of l.

(ui : i 6 n): basis of l; (ui0 : i 6 n): Killing-dual basis

(K (ui, uj0) = δi ,j).

Casimir operator corresponding to (V , β):

n

X

i =1

(31)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Conformal case (C. Duval, P. Lecomte, V. Ovsienko):

SO(p + 1, q + 1) acts on Sp× Sq.

X ∈ so(p + 1, q + 1) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ so(p + 1, q + 1).

Casimir operator method:

l: Semi-simple Lie algebra endowed with a non degenerate Killing form K ,

K (X , Y ) := tr (ad(X) ◦ ad(Y)) (V , β): representation of l.

(ui : i 6 n): basis of l; (ui0 : i 6 n): Killing-dual basis

(K (ui, uj0) = δi ,j).

Casimir operator corresponding to (V , β):

n

X

i =1

(32)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Conformal case (C. Duval, P. Lecomte, V. Ovsienko):

SO(p + 1, q + 1) acts on Sp× Sq.

X ∈ so(p + 1, q + 1) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ so(p + 1, q + 1).

Casimir operator method:

l: Semi-simple Lie algebra endowed with a non degenerate Killing form K ,

K (X , Y ) := tr (ad(X) ◦ ad(Y))

(V , β): representation of l.

(ui : i 6 n): basis of l; (ui0 : i 6 n): Killing-dual basis

(K (ui, uj0) = δi ,j).

Casimir operator corresponding to (V , β):

n

X

i =1

(33)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Conformal case (C. Duval, P. Lecomte, V. Ovsienko):

SO(p + 1, q + 1) acts on Sp× Sq.

X ∈ so(p + 1, q + 1) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ so(p + 1, q + 1).

Casimir operator method:

l: Semi-simple Lie algebra endowed with a non degenerate Killing form K ,

K (X , Y ) := tr (ad(X) ◦ ad(Y)) (V , β): representation of l.

(ui : i 6 n): basis of l; (ui0 : i 6 n): Killing-dual basis

(K (ui, uj0) = δi ,j).

Casimir operator corresponding to (V , β):

n

X

i =1

(34)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Conformal case (C. Duval, P. Lecomte, V. Ovsienko):

SO(p + 1, q + 1) acts on Sp× Sq.

X ∈ so(p + 1, q + 1) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ so(p + 1, q + 1).

Casimir operator method:

l: Semi-simple Lie algebra endowed with a non degenerate Killing form K ,

K (X , Y ) := tr (ad(X) ◦ ad(Y)) (V , β): representation of l.

(ui : i 6 n): basis of l; (ui0 : i 6 n): Killing-dual basis

(K (ui, uj0) = δi ,j).

Casimir operator corresponding to (V , β):

n

X

i =1

(35)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Conformal case (C. Duval, P. Lecomte, V. Ovsienko):

SO(p + 1, q + 1) acts on Sp× Sq.

X ∈ so(p + 1, q + 1) 7→ X∗ vector field on Rm. ∃Q : LXQ(S ) = Q(LXS) ∀X ∈ so(p + 1, q + 1).

Casimir operator method:

l: Semi-simple Lie algebra endowed with a non degenerate Killing form K ,

K (X , Y ) := tr (ad(X) ◦ ad(Y)) (V , β): representation of l.

(ui : i 6 n): basis of l; (ui0 : i 6 n): Killing-dual basis

(K (ui, uj0) = δi ,j).

Casimir operator corresponding to (V , β):

n

X

i =1

(36)

Projectively equivariant quantization over

Rp|q

Fabian Radoux (S(Rm), L) and (D(Rm), L) are representations of g.

C and C: Casimir operators of g on S(M) and D(M). If C (S ) = αS and L ◦ Q = Q ◦ L, then

C(Q(S)) = αQ(S).

In non-critical situations: if C (S ) = αS , then ∃! Q(S ) s.t. C(Q(S)) = αQ(S ), σ(Q(S )) = S .

In these conditions: L(Q(S)) = Q(L(S )) because: σ(L(Q(S ))) = σ(Q(L(S )) = L(S );

(37)

Projectively equivariant quantization over

Rp|q

Fabian Radoux (S(Rm), L) and (D(Rm), L) are representations of g.

C and C: Casimir operators of g on S(M) and D(M).

If C (S ) = αS and L ◦ Q = Q ◦ L, then C(Q(S)) = αQ(S).

In non-critical situations: if C (S ) = αS , then ∃! Q(S ) s.t. C(Q(S)) = αQ(S ), σ(Q(S )) = S .

In these conditions: L(Q(S)) = Q(L(S )) because: σ(L(Q(S ))) = σ(Q(L(S )) = L(S );

(38)

Projectively equivariant quantization over

Rp|q

Fabian Radoux (S(Rm), L) and (D(Rm), L) are representations of g.

C and C: Casimir operators of g on S(M) and D(M). If C (S ) = αS and L ◦ Q = Q ◦ L, then

C(Q(S)) = αQ(S).

In non-critical situations: if C (S ) = αS , then ∃! Q(S ) s.t. C(Q(S)) = αQ(S ), σ(Q(S )) = S .

In these conditions: L(Q(S)) = Q(L(S )) because: σ(L(Q(S ))) = σ(Q(L(S )) = L(S );

(39)

Projectively equivariant quantization over

Rp|q

Fabian Radoux (S(Rm), L) and (D(Rm), L) are representations of g.

C and C: Casimir operators of g on S(M) and D(M). If C (S ) = αS and L ◦ Q = Q ◦ L, then

C(Q(S)) = αQ(S).

In non-critical situations: if C (S ) = αS , then ∃! Q(S ) s.t. C(Q(S)) = αQ(S ), σ(Q(S )) = S .

In these conditions: L(Q(S)) = Q(L(S )) because: σ(L(Q(S ))) = σ(Q(L(S )) = L(S );

(40)

Projectively equivariant quantization over

Rp|q

Fabian Radoux (S(Rm), L) and (D(Rm), L) are representations of g.

C and C: Casimir operators of g on S(M) and D(M). If C (S ) = αS and L ◦ Q = Q ◦ L, then

C(Q(S)) = αQ(S).

In non-critical situations: if C (S ) = αS , then ∃! Q(S ) s.t. C(Q(S)) = αQ(S ), σ(Q(S )) = S .

In these conditions: L(Q(S)) = Q(L(S )) because:

σ(L(Q(S ))) = σ(Q(L(S )) = L(S );

(41)

Projectively equivariant quantization over

Rp|q

Fabian Radoux (S(Rm), L) and (D(Rm), L) are representations of g.

C and C: Casimir operators of g on S(M) and D(M). If C (S ) = αS and L ◦ Q = Q ◦ L, then

C(Q(S)) = αQ(S).

In non-critical situations: if C (S ) = αS , then ∃! Q(S ) s.t. C(Q(S)) = αQ(S ), σ(Q(S )) = S .

In these conditions: L(Q(S)) = Q(L(S )) because: σ(L(Q(S ))) = σ(Q(L(S )) = L(S );

(42)

Projectively equivariant quantization over

Rp|q

Fabian Radoux (S(Rm), L) and (D(Rm), L) are representations of g.

C and C: Casimir operators of g on S(M) and D(M). If C (S ) = αS and L ◦ Q = Q ◦ L, then

C(Q(S)) = αQ(S).

In non-critical situations: if C (S ) = αS , then ∃! Q(S ) s.t. C(Q(S)) = αQ(S ), σ(Q(S )) = S .

In these conditions: L(Q(S)) = Q(L(S )) because: σ(L(Q(S ))) = σ(Q(L(S )) = L(S );

(43)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Projectively equivariant quantization on R

p|q

(P.

Mathonet, R.)

f ∈ C∞p|q:

f (x1, . . . , xp) =P

I ⊆{1,...,q}fI(x1, . . . , xp)θI,

θiθj = −θjθi

Vector field X : superderivation of C∞p|q: X .(fg ) = (X .f )g + (−1)˜f ˜Xf (X .g )

λ-density over Rp|q: smooth function f ∈ C∞p|q s.t. LλXf = X (f ) + λ div(X )f , where div(X ) = p+q X i =1 (−1)y˜ifXi ∂yiXi. (V , ρ): representation of gl(p|q).

(44)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Projectively equivariant quantization on R

p|q

(P.

Mathonet, R.)

f ∈ C∞p|q:

f (x1, . . . , xp) =P

I ⊆{1,...,q}fI(x1, . . . , xp)θI,

θiθj = −θjθi

Vector field X : superderivation of C∞p|q:

X .(fg ) = (X .f )g + (−1)˜f ˜Xf (X .g )

λ-density over Rp|q: smooth function f ∈ C∞p|q s.t. LλXf = X (f ) + λ div(X )f , where div(X ) = p+q X i =1 (−1)y˜ifXi ∂yiXi. (V , ρ): representation of gl(p|q).

(45)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Projectively equivariant quantization on R

p|q

(P.

Mathonet, R.)

f ∈ C∞p|q:

f (x1, . . . , xp) =P

I ⊆{1,...,q}fI(x1, . . . , xp)θI,

θiθj = −θjθi

Vector field X : superderivation of C∞p|q:

X .(fg ) = (X .f )g + (−1)˜f ˜Xf (X .g )

λ-density over Rp|q: smooth function f ∈ C∞p|q s.t. LλXf = X (f ) + λ div(X )f , where div(X ) = p+q X i =1 (−1)y˜ifXi ∂yiXi. (V , ρ): representation of gl(p|q).

(46)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Projectively equivariant quantization on R

p|q

(P.

Mathonet, R.)

f ∈ C∞p|q:

f (x1, . . . , xp) =P

I ⊆{1,...,q}fI(x1, . . . , xp)θI,

θiθj = −θjθi

Vector field X : superderivation of C∞p|q:

X .(fg ) = (X .f )g + (−1)˜f ˜Xf (X .g )

λ-density over Rp|q: smooth function f ∈ C∞p|q s.t. LλXf = X (f ) + λ div(X )f , where div(X ) = p+q X i =1 (−1)y˜ifXi ∂yiXi. (V , ρ): representation of gl(p|q).

(47)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Projectively equivariant quantization on R

p|q

(P.

Mathonet, R.)

f ∈ C∞p|q:

f (x1, . . . , xp) =P

I ⊆{1,...,q}fI(x1, . . . , xp)θI,

θiθj = −θjθi

Vector field X : superderivation of C∞p|q:

X .(fg ) = (X .f )g + (−1)˜f ˜Xf (X .g )

λ-density over Rp|q: smooth function f ∈ C∞p|q s.t. LλXf = X (f ) + λ div(X )f , where div(X ) = p+q X i =1 (−1)y˜ifXi ∂yiXi. (V , ρ): representation of gl(p|q).

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Projectively equivariant quantization over Rp|q Fabian Radoux LX(f ⊗ v ) = X (f ) ⊗ v + (−1)X ˜ef PijfJ j i ⊗ ρ(eji)v , where Jij = (−1)y˜iX +1e (∂ yiXj).

V for densities: Bδ: 1|0-dimensional vector space

spanned by one element u.

Representation ρ of gl(p|q) on Bδ: ρ(A)u = −δstr(A)u.

Sk δ = C ∞p|q⊗ (Bδ⊗ Sk Rp|q). Differential operator D ∈ Dkλ,µ: D(f ) = X |α|6k fα∂xα11· · · ∂ αp xp∂ αp+1 θ1 · · · ∂ αp+q θq f . LXD = LµX ◦ D − (−1)X ˜˜DD ◦ Lλ X.

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Projectively equivariant quantization over Rp|q Fabian Radoux LX(f ⊗ v ) = X (f ) ⊗ v + (−1)X ˜ef PijfJ j i ⊗ ρ(eji)v , where Jij = (−1)y˜iX +1e (∂ yiXj).

V for densities: Bδ: 1|0-dimensional vector space

spanned by one element u.

Representation ρ of gl(p|q) on Bδ: ρ(A)u = −δstr(A)u.

Sk δ = C ∞p|q⊗ (Bδ⊗ Sk Rp|q). Differential operator D ∈ Dkλ,µ: D(f ) = X |α|6k fα∂xα11· · · ∂ αp xp∂ αp+1 θ1 · · · ∂ αp+q θq f . LXD = LµX ◦ D − (−1)X ˜˜DD ◦ Lλ X.

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Projectively equivariant quantization over Rp|q Fabian Radoux LX(f ⊗ v ) = X (f ) ⊗ v + (−1)X ˜ef PijfJ j i ⊗ ρ(eji)v , where Jij = (−1)y˜iX +1e (∂ yiXj).

V for densities: Bδ: 1|0-dimensional vector space

spanned by one element u.

Representation ρ of gl(p|q) on Bδ: ρ(A)u = −δstr(A)u.

Sk δ = C ∞p|q⊗ (Bδ⊗ Sk Rp|q). Differential operator D ∈ Dkλ,µ: D(f ) = X |α|6k fα∂xα11· · · ∂ αp xp∂ αp+1 θ1 · · · ∂ αp+q θq f . LXD = LµX ◦ D − (−1)X ˜˜DD ◦ Lλ X.

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Projectively equivariant quantization over Rp|q Fabian Radoux LX(f ⊗ v ) = X (f ) ⊗ v + (−1)X ˜ef PijfJ j i ⊗ ρ(eji)v , where Jij = (−1)y˜iX +1e (∂ yiXj).

V for densities: Bδ: 1|0-dimensional vector space

spanned by one element u.

Representation ρ of gl(p|q) on Bδ: ρ(A)u = −δstr(A)u.

Sk δ = C ∞p|q⊗ (Bδ⊗ Sk Rp|q). Differential operator D ∈ Dkλ,µ: D(f ) = X |α|6k fα∂xα11· · · ∂ αp xp∂ αp+1 θ1 · · · ∂ αp+q θq f . LXD = LµX ◦ D − (−1)X ˜˜DD ◦ Lλ X.

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Projectively equivariant quantization over Rp|q Fabian Radoux LX(f ⊗ v ) = X (f ) ⊗ v + (−1)X ˜ef PijfJ j i ⊗ ρ(eji)v , where Jij = (−1)y˜iX +1e (∂ yiXj).

V for densities: Bδ: 1|0-dimensional vector space

spanned by one element u.

Representation ρ of gl(p|q) on Bδ: ρ(A)u = −δstr(A)u.

Sk δ = C ∞p|q⊗ (Bδ⊗ Sk Rp|q). Differential operator D ∈ Dkλ,µ: D(f ) = X |α|6k fα∂α1 x1 · · · ∂ αp xp∂ αp+1 θ1 · · · ∂ αp+q θq f . LXD = LµX ◦ D − (−1)X ˜˜DD ◦ Lλ X.

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Projectively equivariant quantization over Rp|q Fabian Radoux LX(f ⊗ v ) = X (f ) ⊗ v + (−1)X ˜ef PijfJ j i ⊗ ρ(eji)v , where Jij = (−1)y˜iX +1e (∂ yiXj).

V for densities: Bδ: 1|0-dimensional vector space

spanned by one element u.

Representation ρ of gl(p|q) on Bδ: ρ(A)u = −δstr(A)u.

Sk δ = C ∞p|q⊗ (Bδ⊗ Sk Rp|q). Differential operator D ∈ Dkλ,µ: D(f ) = X |α|6k fα∂α1 x1 · · · ∂ αp xp∂ αp+1 θ1 · · · ∂ αp+q θq f . LXD = LµX ◦ D − (−1)X ˜˜DD ◦ Lλ X.

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Projectively equivariant quantization over

Rp|q

Fabian Radoux Space of symbols: graded space associated with Dλ,µ.

Space of symbols isomorphic to Sδ=L∞ l =0Sδl, δ = µ − λ. Isomorphism: σk : Dk → Sk : D 7→ X |α|=k fα⊗ e1α1 ∨ · · · ∨ e αp+q p+q .

Quantization on Rp|q: linear bijection Q : Sδ → Dλ,µ

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Projectively equivariant quantization over

Rp|q

Fabian Radoux Space of symbols: graded space associated with Dλ,µ.

Space of symbols isomorphic to Sδ=L∞ l =0Sδl, δ = µ − λ. Isomorphism: σk : Dk → Sk : D 7→ X |α|=k fα⊗ e1α1 ∨ · · · ∨ e αp+q p+q .

Quantization on Rp|q: linear bijection Q : Sδ → Dλ,µ

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Projectively equivariant quantization over

Rp|q

Fabian Radoux Space of symbols: graded space associated with Dλ,µ.

Space of symbols isomorphic to Sδ=L∞ l =0Sδl, δ = µ − λ. Isomorphism: σk : Dk → Sk : D 7→ X |α|=k fα⊗ e1α1∨ · · · ∨ e αp+q p+q .

Quantization on Rp|q: linear bijection Q : Sδ → Dλ,µ

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Projectively equivariant quantization over

Rp|q

Fabian Radoux Space of symbols: graded space associated with Dλ,µ.

Space of symbols isomorphic to Sδ=L∞ l =0Sδl, δ = µ − λ. Isomorphism: σk : Dk → Sk : D 7→ X |α|=k fα⊗ e1α1∨ · · · ∨ e αp+q p+q .

Quantization on Rp|q: linear bijection Q : Sδ → Dλ,µ

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Projective superalgebra of vector fields

pgl(p + 1|q) = gl(p + 1|q)/RId ←→ subalgebra of vector fields over Rp|q.

Ω subset of Rp+1 equal to {(x0, . . . , xp) : x0> 0}.

H(Ω): space of restrictions of homogeneous functions over Rp+1|q to Ω.

Bijective correspondence i : C∞p|q → H(Ω).

Homomorphism hp+1,q: gl(p + 1|q) → Vect(Rp+1|q).

If A ∈ gl(p + 1|q), hp+1,q(A) preserves H(Ω).

π(hp+1,q(A)) ∈ Vect(Rp|q):

π(hp+1,q(A))(f ) = i−1◦ hp+1,q(A) ◦ i (f ), where

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Projective superalgebra of vector fields

pgl(p + 1|q) = gl(p + 1|q)/RId ←→ subalgebra of vector fields over Rp|q.

Ω subset of Rp+1 equal to {(x0, . . . , xp) : x0> 0}.

H(Ω): space of restrictions of homogeneous functions over Rp+1|q to Ω.

Bijective correspondence i : C∞p|q → H(Ω).

Homomorphism hp+1,q: gl(p + 1|q) → Vect(Rp+1|q).

If A ∈ gl(p + 1|q), hp+1,q(A) preserves H(Ω).

π(hp+1,q(A)) ∈ Vect(Rp|q):

π(hp+1,q(A))(f ) = i−1◦ hp+1,q(A) ◦ i (f ), where

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Projective superalgebra of vector fields

pgl(p + 1|q) = gl(p + 1|q)/RId ←→ subalgebra of vector fields over Rp|q.

Ω subset of Rp+1 equal to {(x0, . . . , xp) : x0> 0}. H(Ω): space of restrictions of homogeneous functions over Rp+1|q to Ω.

Bijective correspondence i : C∞p|q → H(Ω).

Homomorphism hp+1,q: gl(p + 1|q) → Vect(Rp+1|q).

If A ∈ gl(p + 1|q), hp+1,q(A) preserves H(Ω).

π(hp+1,q(A)) ∈ Vect(Rp|q):

π(hp+1,q(A))(f ) = i−1◦ hp+1,q(A) ◦ i (f ), where

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Projective superalgebra of vector fields

pgl(p + 1|q) = gl(p + 1|q)/RId ←→ subalgebra of vector fields over Rp|q.

Ω subset of Rp+1 equal to {(x0, . . . , xp) : x0> 0}. H(Ω): space of restrictions of homogeneous functions over Rp+1|q to Ω.

Bijective correspondence i : C∞p|q → H(Ω).

Homomorphism hp+1,q: gl(p + 1|q) → Vect(Rp+1|q).

If A ∈ gl(p + 1|q), hp+1,q(A) preserves H(Ω).

π(hp+1,q(A)) ∈ Vect(Rp|q):

π(hp+1,q(A))(f ) = i−1◦ hp+1,q(A) ◦ i (f ), where

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Projective superalgebra of vector fields

pgl(p + 1|q) = gl(p + 1|q)/RId ←→ subalgebra of vector fields over Rp|q.

Ω subset of Rp+1 equal to {(x0, . . . , xp) : x0> 0}. H(Ω): space of restrictions of homogeneous functions over Rp+1|q to Ω.

Bijective correspondence i : C∞p|q → H(Ω).

Homomorphism hp+1,q: gl(p + 1|q) → Vect(Rp+1|q).

If A ∈ gl(p + 1|q), hp+1,q(A) preserves H(Ω).

π(hp+1,q(A)) ∈ Vect(Rp|q):

π(hp+1,q(A))(f ) = i−1◦ hp+1,q(A) ◦ i (f ), where

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Projective superalgebra of vector fields

pgl(p + 1|q) = gl(p + 1|q)/RId ←→ subalgebra of vector fields over Rp|q.

Ω subset of Rp+1 equal to {(x0, . . . , xp) : x0> 0}. H(Ω): space of restrictions of homogeneous functions over Rp+1|q to Ω.

Bijective correspondence i : C∞p|q → H(Ω).

Homomorphism hp+1,q: gl(p + 1|q) → Vect(Rp+1|q).

If A ∈ gl(p + 1|q), hp+1,q(A) preserves H(Ω).

π(hp+1,q(A)) ∈ Vect(Rp|q):

π(hp+1,q(A))(f ) = i−1◦ hp+1,q(A) ◦ i (f ), where

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Projective superalgebra of vector fields

pgl(p + 1|q) = gl(p + 1|q)/RId ←→ subalgebra of vector fields over Rp|q.

Ω subset of Rp+1 equal to {(x0, . . . , xp) : x0> 0}. H(Ω): space of restrictions of homogeneous functions over Rp+1|q to Ω.

Bijective correspondence i : C∞p|q → H(Ω).

Homomorphism hp+1,q: gl(p + 1|q) → Vect(Rp+1|q).

If A ∈ gl(p + 1|q), hp+1,q(A) preserves H(Ω).

π(hp+1,q(A)) ∈ Vect(Rp|q):

π(hp+1,q(A))(f ) = i−1◦ hp+1,q(A) ◦ i (f ), where

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Projectively equivariant quantization over

Rp|q Fabian Radoux

π ◦ hp+1,q(Id) = 0, thus π ◦ hp+1,q induces a

homomorphism from pgl(p + 1|q) to Vect(Rp|q).

pgl(p + 1|q) ←→ g−1⊕ g0⊕ g1, g−1= Rp|q,

g0 = gl(p|q) and g1 = (Rp|q)∗.

g−1: constant vector fields, g0: linear vector fields, g1:

quadratic vector fields.

Projectively equivariant quantization: quantization Q s.t. LXh◦ Q = Q ◦ LXh for every h ∈ pgl(p + 1|q).

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Projectively equivariant quantization over

Rp|q Fabian Radoux

π ◦ hp+1,q(Id) = 0, thus π ◦ hp+1,q induces a

homomorphism from pgl(p + 1|q) to Vect(Rp|q).

pgl(p + 1|q) ←→ g−1⊕ g0⊕ g1, g−1= Rp|q,

g0 = gl(p|q) and g1 = (Rp|q)∗.

g−1: constant vector fields, g0: linear vector fields, g1:

quadratic vector fields.

Projectively equivariant quantization: quantization Q s.t. LXh◦ Q = Q ◦ LXh for every h ∈ pgl(p + 1|q).

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Projectively equivariant quantization over

Rp|q Fabian Radoux

π ◦ hp+1,q(Id) = 0, thus π ◦ hp+1,q induces a

homomorphism from pgl(p + 1|q) to Vect(Rp|q).

pgl(p + 1|q) ←→ g−1⊕ g0⊕ g1, g−1= Rp|q,

g0 = gl(p|q) and g1 = (Rp|q)∗.

g−1: constant vector fields, g0: linear vector fields, g1:

quadratic vector fields.

Projectively equivariant quantization: quantization Q s.t. LXh◦ Q = Q ◦ LXh for every h ∈ pgl(p + 1|q).

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Projectively equivariant quantization over

Rp|q Fabian Radoux

π ◦ hp+1,q(Id) = 0, thus π ◦ hp+1,q induces a

homomorphism from pgl(p + 1|q) to Vect(Rp|q).

pgl(p + 1|q) ←→ g−1⊕ g0⊕ g1, g−1= Rp|q,

g0 = gl(p|q) and g1 = (Rp|q)∗.

g−1: constant vector fields, g0: linear vector fields, g1:

quadratic vector fields.

Projectively equivariant quantization: quantization Q s.t. LXh◦ Q = Q ◦ LXh for every h ∈ pgl(p + 1|q).

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Construction of the quantization

QAff: inverse of the total symbol map σ

LXS = QAff−1◦ LX ◦ QAff(S ), for every S ∈ Sδ and

X ∈ Vect(Rp|q).

Projectively equivariant quantization←→ pgl(p + 1|q)−module isomorphism from the representation (Sδ, L) to the representation (Sδ, L).

Map γ:

γ : g → gl(Sδ, Sδ) : h 7→ γ(h) = LXh− LXh.

γ vanishes on g−1⊕ g0, γ(h) : Sδk → S k−1 δ .

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Construction of the quantization

QAff: inverse of the total symbol map σ

LXS = QAff−1◦ LX ◦ QAff(S ), for every S ∈ Sδ and

X ∈ Vect(Rp|q).

Projectively equivariant quantization←→ pgl(p + 1|q)−module isomorphism from the representation (Sδ, L) to the representation (Sδ, L).

Map γ:

γ : g → gl(Sδ, Sδ) : h 7→ γ(h) = LXh− LXh.

γ vanishes on g−1⊕ g0, γ(h) : Sδk → S k−1 δ .

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Construction of the quantization

QAff: inverse of the total symbol map σ

LXS = QAff−1◦ LX ◦ QAff(S ), for every S ∈ Sδ and

X ∈ Vect(Rp|q).

Projectively equivariant quantization←→ pgl(p + 1|q)−module isomorphism from the representation (Sδ, L) to the representation (Sδ, L).

Map γ:

γ : g → gl(Sδ, Sδ) : h 7→ γ(h) = LXh− LXh.

γ vanishes on g−1⊕ g0, γ(h) : Sδk → S k−1 δ .

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Construction of the quantization

QAff: inverse of the total symbol map σ

LXS = QAff−1◦ LX ◦ QAff(S ), for every S ∈ Sδ and

X ∈ Vect(Rp|q).

Projectively equivariant quantization←→ pgl(p + 1|q)−module isomorphism from the representation (Sδ, L) to the representation (Sδ, L).

Map γ:

γ : g → gl(Sδ, Sδ) : h 7→ γ(h) = LXh− LXh.

γ vanishes on g−1⊕ g0, γ(h) : Sδk → S k−1 δ .

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Projectively equivariant quantization over

Rp|q Fabian Radoux

Construction of the quantization

QAff: inverse of the total symbol map σ

LXS = QAff−1◦ LX ◦ QAff(S ), for every S ∈ Sδ and

X ∈ Vect(Rp|q).

Projectively equivariant quantization←→ pgl(p + 1|q)−module isomorphism from the representation (Sδ, L) to the representation (Sδ, L).

Map γ:

γ : g → gl(Sδ, Sδ) : h 7→ γ(h) = LXh− LXh.

γ vanishes on g−1⊕ g0, γ(h) : Sδk → S k−1 δ .

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Projectively equivariant quantization over Rp|q Fabian Radoux Casimir operators:

l: Lie superalgebra endowed with a nondegenerate even supersymmetric bilinear form K .

(V , β): representation of l. (ui : i 6 n): homogeneous basis of l; (u0i : i 6 n): K -dual basis (K (ui, uj0) = δi ,j). Casimir operator of (V , β): n X i =1 (−1)u˜iβ(u i)β(u0i) = n X i =1 β(ui0)β(ui).

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Projectively equivariant quantization over Rp|q Fabian Radoux Casimir operators:

l: Lie superalgebra endowed with a nondegenerate even supersymmetric bilinear form K .

(V , β): representation of l. (ui : i 6 n): homogeneous basis of l; (u0i : i 6 n): K -dual basis (K (ui, uj0) = δi ,j). Casimir operator of (V , β): n X i =1 (−1)u˜iβ(u i)β(u0i) = n X i =1 β(ui0)β(ui).

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Projectively equivariant quantization over Rp|q Fabian Radoux Casimir operators:

l: Lie superalgebra endowed with a nondegenerate even supersymmetric bilinear form K .

(V , β): representation of l. (ui : i 6 n): homogeneous basis of l; (u0i : i 6 n): K -dual basis (K (ui, uj0) = δi ,j). Casimir operator of (V , β): n X i =1 (−1)u˜iβ(u i)β(u0i) = n X i =1 β(ui0)β(ui).

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Projectively equivariant quantization over Rp|q Fabian Radoux Casimir operators:

l: Lie superalgebra endowed with a nondegenerate even supersymmetric bilinear form K .

(V , β): representation of l. (ui : i 6 n): homogeneous basis of l; (u0i : i 6 n): K -dual basis (K (ui, uj0) = δi ,j). Casimir operator of (V , β): n X i =1 (−1)u˜iβ(u i)β(u0i) = n X i =1 β(ui0)β(ui).

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Projectively equivariant quantization over Rp|q Fabian Radoux Casimir operators:

l: Lie superalgebra endowed with a nondegenerate even supersymmetric bilinear form K .

(V , β): representation of l. (ui : i 6 n): homogeneous basis of l; (u0i : i 6 n): K -dual basis (K (ui, uj0) = δi ,j). Casimir operator of (V , β): n X i =1 (−1)u˜iβ(u i)β(u0i) = n X i =1 β(ui0)β(ui).

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Projectively equivariant quantization over Rp|q Fabian Radoux If q 6= p + 1, pgl(p + 1|q) ∼= sl(p + 1|q). Killing form of sl(p + 1|q):

K (A, B) = str(ad (A)ad (B)) = 2(p + 1 − q) str(AB).

The Casimir operator C of pgl(p + 1|q) ∼= sl(p + 1|q) on (Sδk, L) is equal to α(k, δ)Id, where

α(k, δ) = p − q 2 δ 22k + p − q 2 δ + k(k + (p − q)) (p − q + 1) .

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Projectively equivariant quantization over Rp|q Fabian Radoux If q 6= p + 1, pgl(p + 1|q) ∼= sl(p + 1|q). Killing form of sl(p + 1|q):

K (A, B) = str(ad (A)ad (B)) = 2(p + 1 − q) str(AB).

The Casimir operator C of pgl(p + 1|q) ∼= sl(p + 1|q) on (Sδk, L) is equal to α(k, δ)Id, where

α(k, δ) = p − q 2 δ 22k + p − q 2 δ + k(k + (p − q)) (p − q + 1) .

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Projectively equivariant quantization over Rp|q Fabian Radoux If q 6= p + 1, pgl(p + 1|q) ∼= sl(p + 1|q). Killing form of sl(p + 1|q):

K (A, B) = str(ad (A)ad (B)) = 2(p + 1 − q) str(AB).

The Casimir operator C of pgl(p + 1|q) ∼= sl(p + 1|q) on (Sδk, L) is equal to α(k, δ)Id, where

α(k, δ) = p − q 2 δ 22k + p − q 2 δ + k(k + (p − q)) (p − q + 1) .

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Projectively equivariant quantization over

Rp|q Fabian Radoux

The Casimir operator C of pgl(p + 1|q) ∼= sl(p + 1|q) on (Sδk, L) is equal to C + N, where N is defined in this way:

N : Sδk → Sδk−1 : S 7→ 2X

i

γ(i)LXeiS ,

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Projectively equivariant quantization over

Rp|q

Fabian Radoux A value of δ is critical if there exist k, l ∈ N with l < k

such that α(k, δ) = α(l , δ).

The set of critical values: C= ∪∞k=1Ck, Ck =  2k − l + p − q p − q + 1 : l = 1, . . . , k  . Remarks:

1 “Same” values as in the classical case: m is replaced by

p − q.

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Projectively equivariant quantization over

Rp|q

Fabian Radoux A value of δ is critical if there exist k, l ∈ N with l < k

such that α(k, δ) = α(l , δ). The set of critical values:

C= ∪∞k=1Ck, Ck =  2k − l + p − q p − q + 1 : l = 1, . . . , k  . Remarks:

1 “Same” values as in the classical case: m is replaced by

p − q.

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Projectively equivariant quantization over

Rp|q

Fabian Radoux A value of δ is critical if there exist k, l ∈ N with l < k

such that α(k, δ) = α(l , δ). The set of critical values:

C= ∪∞k=1Ck, Ck =  2k − l + p − q p − q + 1 : l = 1, . . . , k  . Remarks:

1 “Same” values as in the classical case: m is replaced by

p − q.

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Projectively equivariant quantization over

Rp|q

Fabian Radoux A value of δ is critical if there exist k, l ∈ N with l < k

such that α(k, δ) = α(l , δ). The set of critical values:

C= ∪∞k=1Ck, Ck =  2k − l + p − q p − q + 1 : l = 1, . . . , k  . Remarks:

1 “Same” values as in the classical case: m is replaced by

p − q.

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Projectively equivariant quantization over

Rp|q Fabian Radoux

If δ is not critical, then there exists a unique projectively equivariant quantization.

Proof:

1 For every S ∈ Sk

δ, ∃! ˆS s.t. C(ˆS ) = α(k, δ)ˆS and s.t. ˆ

S = S + Sk−1+ · · · + S0, where Sl ∈ Sδl for all l 6 k − 1.

2 Q(S ) := ˆS .

3 If S ∈ Sk

δ, Q(LXhS ) = LXh(Q(S )) because they are

eigenvectors of C of eigenvalue α(k, δ) and because their term of degree k is exactly LXhS .

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Projectively equivariant quantization over

Rp|q Fabian Radoux

If δ is not critical, then there exists a unique projectively equivariant quantization.

Proof:

1 For every S ∈ Sk

δ, ∃! ˆS s.t. C(ˆS ) = α(k, δ)ˆS and s.t. ˆ

S = S + Sk−1+ · · · + S0, where Sl ∈ Sδl for all l 6 k − 1.

2 Q(S ) := ˆS .

3 If S ∈ Sk

δ, Q(LXhS ) = LXh(Q(S )) because they are

eigenvectors of C of eigenvalue α(k, δ) and because their term of degree k is exactly LXhS .

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Projectively equivariant quantization over

Rp|q Fabian Radoux

If δ is not critical, then there exists a unique projectively equivariant quantization.

Proof:

1 For every S ∈ Sk

δ, ∃! ˆS s.t. C(ˆS ) = α(k, δ)ˆS and s.t. ˆ

S = S + Sk−1+ · · · + S0, where Sl ∈ Sδl for all l 6 k − 1.

2 Q(S ) := ˆS .

3 If S ∈ Sk

δ, Q(LXhS ) = LXh(Q(S )) because they are

eigenvectors of C of eigenvalue α(k, δ) and because their term of degree k is exactly LXhS .

(90)

Projectively equivariant quantization over

Rp|q Fabian Radoux

If δ is not critical, then there exists a unique projectively equivariant quantization.

Proof:

1 For every S ∈ Sk

δ, ∃! ˆS s.t. C(ˆS ) = α(k, δ)ˆS and s.t. ˆ

S = S + Sk−1+ · · · + S0, where Sl ∈ Sδl for all l 6 k − 1.

2 Q(S ) := ˆS . 3 If S ∈ Sk

δ, Q(LXhS ) = LXh(Q(S )) because they are

eigenvectors of C of eigenvalue α(k, δ) and because their term of degree k is exactly LXhS .

(91)

Projectively equivariant quantization over Rp|q Fabian Radoux Divergence operator: div : Sδk → Sδk−1 : S 7→ p+q X j =1 (−1)y˜ji (εj)∂yjS . Theorem

If δ is not critical, then the map Q : Sδ → Dλ,µ defined by

Q(S )(f ) =

k

X

r =0

Ck,rQAff(divrS )(f ), for all S ∈ Sδk

is the unique sl(p + 1|q)-equivariant quantization if Ck,r = Qr j =1((p − q + 1)λ + k − j ) r !Qr j =1(p − q + 2k − j − (p − q + 1)δ) for all r > 1.

(92)

Projectively equivariant quantization over Rp|q Fabian Radoux Divergence operator: div : Sδk → Sδk−1 : S 7→ p+q X j =1 (−1)y˜ji (εj)∂yjS . Theorem

If δ is not critical, then the map Q : Sδ → Dλ,µ defined by

Q(S )(f ) =

k

X

r =0

Ck,rQAff(divrS )(f ), for all S ∈ Sδk

is the unique sl(p + 1|q)-equivariant quantization if Ck,r = Qr j =1((p − q + 1)λ + k − j ) r !Qr j =1(p − q + 2k − j − (p − q + 1)δ) for all r > 1.

(93)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Case q = p + 1: pgl(p + 1|p + 1) not endowed with a non degenerate bilinear symmetric invariant form.

pgl(p + 1|p + 1) is not simple, codimension one ideal: psl(p + 1|p + 1);

pgl(p + 1|p + 1) = psl(p + 1|p + 1) ⊕ RId. Analyse of the existence of a

psl(p + 1|p + 1)-equivariant quantization.

Killing form of psl(p + 1|p + 1) vanishes, but K defined by

K ([A], [B]) = strAB

(94)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Case q = p + 1: pgl(p + 1|p + 1) not endowed with a non degenerate bilinear symmetric invariant form. pgl(p + 1|p + 1) is not simple, codimension one ideal: psl(p + 1|p + 1);

pgl(p + 1|p + 1) = psl(p + 1|p + 1) ⊕ RId.

Analyse of the existence of a

psl(p + 1|p + 1)-equivariant quantization.

Killing form of psl(p + 1|p + 1) vanishes, but K defined by

K ([A], [B]) = strAB

(95)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Case q = p + 1: pgl(p + 1|p + 1) not endowed with a non degenerate bilinear symmetric invariant form. pgl(p + 1|p + 1) is not simple, codimension one ideal: psl(p + 1|p + 1);

pgl(p + 1|p + 1) = psl(p + 1|p + 1) ⊕ RId. Analyse of the existence of a

psl(p + 1|p + 1)-equivariant quantization.

Killing form of psl(p + 1|p + 1) vanishes, but K defined by

K ([A], [B]) = strAB

(96)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Case q = p + 1: pgl(p + 1|p + 1) not endowed with a non degenerate bilinear symmetric invariant form. pgl(p + 1|p + 1) is not simple, codimension one ideal: psl(p + 1|p + 1);

pgl(p + 1|p + 1) = psl(p + 1|p + 1) ⊕ RId. Analyse of the existence of a

psl(p + 1|p + 1)-equivariant quantization.

Killing form of psl(p + 1|p + 1) vanishes, but K defined by

K ([A], [B]) = strAB

(97)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Casimir operator C of psl(p + 1|p + 1) acting on Sk δ is

equal to 2k(k − 1)Id.

α1,δ = α0,δ: critical situation.

If k 6= 1, Q is given by the same formula as in the case q 6= p + 1.

If k = 1,

Q1 : S 7→ Q(S ) = QAff(S + t div(S ))

defines a psl(p + 1|p + 1)-equivariant quantization for every t ∈ R (vector fields in psl(p + 1|p + 1) are divergence-free).

The psl(p + 1, p + 1)-equivariant quantizations are pgl(p + 1, p + 1)-equivariant (equivariance with respect to the Euler vector field corresponding to Id).

(98)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Casimir operator C of psl(p + 1|p + 1) acting on Sk δ is

equal to 2k(k − 1)Id.

α1,δ = α0,δ: critical situation.

If k 6= 1, Q is given by the same formula as in the case q 6= p + 1.

If k = 1,

Q1 : S 7→ Q(S ) = QAff(S + t div(S ))

defines a psl(p + 1|p + 1)-equivariant quantization for every t ∈ R (vector fields in psl(p + 1|p + 1) are divergence-free).

The psl(p + 1, p + 1)-equivariant quantizations are pgl(p + 1, p + 1)-equivariant (equivariance with respect to the Euler vector field corresponding to Id).

(99)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Casimir operator C of psl(p + 1|p + 1) acting on Sk δ is

equal to 2k(k − 1)Id.

α1,δ = α0,δ: critical situation.

If k 6= 1, Q is given by the same formula as in the case q 6= p + 1.

If k = 1,

Q1 : S 7→ Q(S ) = QAff(S + t div(S ))

defines a psl(p + 1|p + 1)-equivariant quantization for every t ∈ R (vector fields in psl(p + 1|p + 1) are divergence-free).

The psl(p + 1, p + 1)-equivariant quantizations are pgl(p + 1, p + 1)-equivariant (equivariance with respect to the Euler vector field corresponding to Id).

(100)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Casimir operator C of psl(p + 1|p + 1) acting on Sk δ is

equal to 2k(k − 1)Id.

α1,δ = α0,δ: critical situation.

If k 6= 1, Q is given by the same formula as in the case q 6= p + 1.

If k = 1,

Q1: S 7→ Q(S ) = QAff(S + t div(S ))

defines a psl(p + 1|p + 1)-equivariant quantization for every t ∈ R (vector fields in psl(p + 1|p + 1) are divergence-free).

The psl(p + 1, p + 1)-equivariant quantizations are pgl(p + 1, p + 1)-equivariant (equivariance with respect to the Euler vector field corresponding to Id).

(101)

Projectively equivariant quantization over

Rp|q Fabian Radoux

Casimir operator C of psl(p + 1|p + 1) acting on Sk δ is

equal to 2k(k − 1)Id.

α1,δ = α0,δ: critical situation.

If k 6= 1, Q is given by the same formula as in the case q 6= p + 1.

If k = 1,

Q1: S 7→ Q(S ) = QAff(S + t div(S ))

defines a psl(p + 1|p + 1)-equivariant quantization for every t ∈ R (vector fields in psl(p + 1|p + 1) are divergence-free).

The psl(p + 1, p + 1)-equivariant quantizations are pgl(p + 1, p + 1)-equivariant (equivariance with respect to the Euler vector field corresponding to Id).

(102)

Projectively equivariant quantization over

Rp|q Fabian Radoux

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