Poisson Deleting Derivations Algorithm: Poisson Birational Equivalence and Poisson Spectrum

Poisson deleting derivations algorithm: Poisson birational equivalence and Poisson spectrum

César Lecoutre

University of Glasgow

AMS-EMS-SMP International Meeting June 2015 Example: X11 X12 Let A := O M2(K) = K . We dene a Poisson X21 X22 structure on A by setting:

{X11, X12} = X11X12, {X11, X22} = 2X12X21, {X12, X22} = X12X22, {X11, X21} = X11X21, {X12, X21} = 0, {X21, X22} = X21X22. We generalise this Poisson structure to O Mm,p(K) .

Poisson Algebra

Denition A Poisson K-algebra A is a commutative algebra endowed with a Poisson bracket, that is a skew-symmetric K-bilinear map from A × A to A such that for all a, b, c ∈ A we have:

I {a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0,

I {ab, c} = a{b, c} + {a, c}b. Poisson Algebra

Denition A Poisson K-algebra A is a commutative algebra endowed with a Poisson bracket, that is a skew-symmetric K-bilinear map from A × A to A such that for all a, b, c ∈ A we have:

I {a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0,

I {ab, c} = a{b, c} + {a, c}b. Example: X11 X12 Let A := O M2(K) = K . We dene a Poisson X21 X22 structure on A by setting:

{X11, X12} = X11X12, {X11, X22} = 2X12X21, {X12, X22} = X12X22, {X11, X21} = X11X21, {X12, X21} = 0, {X21, X22} = X21X22. We generalise this Poisson structure to O Mm,p(K) . Poisson-Ore extension A ,→ A[X ]

Let A be a Poisson algebra and α, δ be linear maps on A. By setting:

{X , a} = α(a)X + δ(a) for all a ∈ A,

we dene a Poisson bracket on the polynomial ring A[X ] if and only if α, δ are derivations of A and satises for all a, b ∈ A:

α({a, b}) = {α(a), b} + {a, α(b)}, δ({a, b}) = {δ(a), b} + {a, δ(b)} + α(a)δ(b) − δ(a)α(b).

We denote this Poisson algebra by A[X ; α, δ]P and called it a Poisson-Ore extension. Then there exists a Poisson algebra isomorphism:

±1 =∼ ±1 F : A[Y ; α, 0]P −→ A[X ; α, δ]P , dened by

X 1 F (a) = D (a)X −i , and F (Y ) = X . ηi i i≥0

Compare {Y , a} = α(a)Y and {X , a} = α(a)X + δ(a).

Poisson Deleting Derivation Homomorphism

Theorem (Launois-L.) × Let A[X ; α, δ]P be a Poisson-Ore extension and η ∈ K . Suppose that δ extends to an iterative, locally nilpotent higher (η, α)-skew Poisson derivation (Di ) on A. Compare {Y , a} = α(a)Y and {X , a} = α(a)X + δ(a).

Poisson Deleting Derivation Homomorphism

±1 =∼ ±1 F : A[Y ; α, 0]P −→ A[X ; α, δ]P , dened by

X 1 F (a) = D (a)X −i , and F (Y ) = X . ηi i i≥0 Poisson Deleting Derivation Homomorphism

±1 =∼ ±1 F : A[Y ; α, 0]P −→ A[X ; α, δ]P , dened by

X 1 F (a) = D (a)X −i , and F (Y ) = X . ηi i i≥0

Compare {Y , a} = α(a)Y and {X , a} = α(a)X + δ(a). i If char 0, then D1 for all . K = Di = i! i

Higher Poisson Derivation

Let A be a Poisson K-algebra, α a Poisson derivation, and η ∈ K.

A higher (η, α)-skew Poisson derivation is a sequence of K-linear maps ∞ from to such that for all and all (Di ) := (Di )i=0 A A a, b ∈ A n ≥ 0: n P (1) D0 = idA and Dn(ab) = Di (a)Dn−i (b), i=0 (2) Dn({a, b}) = n P h i {Di (a), Dn−i (b)} + i αDn−i (a)Di (b) − Di (a)αDn−i (b) , i=0 (3) Dnα = αDn + nηDn, (4) i+j for all 0 (iterative), Di Dj = i Di+j i, j ≥ (5) For all a ∈ A there exists na ≥ 0 such that Di (a) = 0 for all i ≥ na (locally nilpotent). Higher Poisson Derivation

Let A be a Poisson K-algebra, α a Poisson derivation, and η ∈ K.

I A = K[X1][X2; α2, δ2]P ··· [Xn; αn, δn]P is an iterated Poisson-Ore extension.

I αi (Xj ) = λij Xj for some scalar λij ∈ K. (1 ≤ j < i ≤ n) I δi extends to a higher (ηi , αi )-skew Poisson derivation ∞ , where ×. (2 ) (Di,k )k=0 ηi ∈ K ≤ i ≤ n I αi Dj,k = Dj,k αi + kλij Dj,k for all k ≥ 0. (2 ≤ j < i ≤ n)

The class P contains (the coordinate rings of) Poisson matrix variety O Mm,p(K) , odd and even Poisson euclidean spaces and symplectic Poisson spaces. I there exists an embedding:

P.Spec (A) ,→ P.Spec (A).

Corollary Let A ∈ P. Then A satises the Poisson analogue of the quantum Gel'fand-Kirillov problem. Moreover, if char K = 0, then A satises the Poisson Dixmier-Moeglin equivalence.

Consequences

Let A ∈ P. Then the Poisson deleting derivations algorithm returns a Poisson ane space A = K[T1,..., Tn] with {Ti , Tj } = λij Ti Tj such that: −1 ∼ ±1 ±1 for a multiplicative set in , I AS = K[T1 ,..., Tn ] S A Corollary Let A ∈ P. Then A satises the Poisson analogue of the quantum Gel'fand-Kirillov problem. Moreover, if char K = 0, then A satises the Poisson Dixmier-Moeglin equivalence.

Consequences

P.Spec (A) ,→ P.Spec (A). Consequences

P.Spec (A) ,→ P.Spec (A).

Corollary Let A ∈ P. Then A satises the Poisson analogue of the quantum Gel'fand-Kirillov problem. Moreover, if char K = 0, then A satises the Poisson Dixmier-Moeglin equivalence. For w ∈ W , we set:

P.Spec P.Spec w (A) := P ∈ (A) | P∩{T1,..., Tn} = {Ti | i ∈ w} .

Then the partition of P.Spec (A):

P.Spec G P.Spec (A) = w (A) w∈W

induces a partition of P.Spec (A):

P.Spec G −1P.Spec (A) = ϕ w (A) 0 w∈WP

where 0 −1P.Spec (the set of WP := {w ∈ W | ϕ w (A) 6= ∅} Cauchon diagrams).

Canonical partition and Cauchon Diagrams

Let A ∈ P, and A = K[T1,..., Tn] with {Ti , Tj } = λij Ti Tj . Let W := P({1,..., n}). Then the partition of P.Spec (A):

P.Spec G P.Spec (A) = w (A) w∈W

induces a partition of P.Spec (A):

P.Spec G −1P.Spec (A) = ϕ w (A) 0 w∈WP

where 0 −1P.Spec (the set of WP := {w ∈ W | ϕ w (A) 6= ∅} Cauchon diagrams).

Canonical partition and Cauchon Diagrams

Let A ∈ P, and A = K[T1,..., Tn] with {Ti , Tj } = λij Ti Tj . Let W := P({1,..., n}). For w ∈ W , we set: P.Spec P.Spec w (A) := P ∈ (A) | P∩{T1,..., Tn} = {Ti | i ∈ w} . induces a partition of P.Spec (A):

P.Spec G −1P.Spec (A) = ϕ w (A) 0 w∈WP

where 0 −1P.Spec (the set of WP := {w ∈ W | ϕ w (A) 6= ∅} Cauchon diagrams).

Canonical partition and Cauchon Diagrams

Let A ∈ P, and A = K[T1,..., Tn] with {Ti , Tj } = λij Ti Tj . Let W := P({1,..., n}). For w ∈ W , we set: P.Spec P.Spec w (A) := P ∈ (A) | P∩{T1,..., Tn} = {Ti | i ∈ w} .

Then the partition of P.Spec (A):

P.Spec G P.Spec (A) = w (A) w∈W Canonical partition and Cauchon Diagrams

Let A ∈ P, and A = K[T1,..., Tn] with {Ti , Tj } = λij Ti Tj . Let W := P({1,..., n}). For w ∈ W , we set: P.Spec P.Spec w (A) := P ∈ (A) | P∩{T1,..., Tn} = {Ti | i ∈ w} .

Then the partition of P.Spec (A):

P.Spec G P.Spec (A) = w (A) w∈W

induces a partition of P.Spec (A):

P.Spec G −1P.Spec (A) = ϕ w (A) 0 w∈WP

where 0 −1P.Spec (the set of WP := {w ∈ W | ϕ w (A) 6= ∅} Cauchon diagrams). Theorem When char 2, the set of Cauchon diagrams 0 for is in K 6= WP A bijection with the set of all m × p rectangular arrays whose boxes are colored in black or white with the property that if a box is black, then all the boxes above it are black or all the boxes on its left are black.

Combinatorial description of Cauchon diagram for Poisson matrices A = O(Mm,p(K)) In the case we have: W := P({1,..., m} × {1,..., p}), and we obtain the following description of the set of Cauchon diagrams 0 of . WP ⊆ W A Combinatorial description of Cauchon diagram for Poisson matrices A = O(Mm,p(K)) In the case we have: W := P({1,..., m} × {1,..., p}), and we obtain the following description of the set of Cauchon diagrams 0 of . WP ⊆ W A Theorem When char 2, the set of Cauchon diagrams 0 for is in K 6= WP A bijection with the set of all m × p rectangular arrays whose boxes are colored in black or white with the property that if a box is black, then all the boxes above it are black or all the boxes on its left are black.