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: fX11; X12g = X11X12; fX11; X22g = 2X12X21; fX12; X22g = X12X22; fX11; X21g = X11X21; fX12; X21g = 0; fX21; X22g = 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 2 A we have: I fa; fb; cgg + fb; fc; agg + fc; fa; bgg = 0, I fab; cg = afb; cg + fa; cgb. 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 2 A we have: I fa; fb; cgg + fb; fc; agg + fc; fa; bgg = 0, I fab; cg = afb; cg + fa; cgb. Example: X11 X12 Let A := O M2(K) = K . We dene a Poisson X21 X22 structure on A by setting: fX11; X12g = X11X12; fX11; X22g = 2X12X21; fX12; X22g = X12X22; fX11; X21g = X11X21; fX12; X21g = 0; fX21; X22g = 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: fX ; ag = α(a)X + δ(a) for all a 2 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 2 A: α(fa; bg) = fα(a); bg + fa; α(b)g; δ(fa; bg) = fδ(a); bg + fa; δ(b)g + 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 fY ; ag = α(a)Y and fX ; ag = 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 2 K . Suppose that δ extends to an iterative, locally nilpotent higher (η; α)-skew Poisson derivation (Di ) on A. Compare fY ; ag = α(a)Y and fX ; ag = 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 2 K . Suppose that δ extends to an iterative, locally nilpotent higher (η; α)-skew Poisson derivation (Di ) on A. 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 Poisson Deleting Derivation Homomorphism Theorem (Launois-L.) × Let A[X ; α; δ]P be a Poisson-Ore extension and η 2 K . Suppose that δ extends to an iterative, locally nilpotent higher (η; α)-skew Poisson derivation (Di ) on A. 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 fY ; ag = α(a)Y and fX ; ag = 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 2 K. A higher (η; α)-skew Poisson derivation is a sequence of K-linear maps 1 from to such that for all and all (Di ) := (Di )i=0 A A a; b 2 A n ≥ 0: n P (1) D0 = idA and Dn(ab) = Di (a)Dn−i (b), i=0 (2) Dn(fa; bg) = n P h i fDi (a); Dn−i (b)g + 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 2 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 2 K. A higher (η; α)-skew Poisson derivation is a sequence of K-linear maps 1 from to such that for all and all (Di ) := (Di )i=0 A A a; b 2 A n ≥ 0: n P (1) D0 = idA and Dn(ab) = Di (a)Dn−i (b), i=0 (2) Dn(fa; bg) = n P h i fDi (a); Dn−i (b)g + 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 2 A there exists na ≥ 0 such that Di (a) = 0 for all i ≥ na (locally nilpotent). i If char 0, then D1 for all . K = Di = i! i A class P of Poisson algebras 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 2 K. (1 ≤ j < i ≤ n) I δi extends to a higher (ηi ; αi )-skew Poisson derivation 1 , where ×. (2 ) (Di;k )k=0 ηi 2 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 2 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 2 P. Then the Poisson deleting derivations algorithm returns a Poisson ane space A = K[T1;:::; Tn] with fTi ; Tj g = λij Ti Tj such that: −1 ∼ ±1 ±1 for a multiplicative set in , I AS = K[T1 ;:::; Tn ] S A Corollary Let A 2 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 2 P. Then the Poisson deleting derivations algorithm returns a Poisson ane space A = K[T1;:::; Tn] with fTi ; Tj g = λij Ti Tj such that: −1 ∼ ±1 ±1 for a multiplicative set in , I AS = K[T1 ;:::; Tn ] S A I there exists an embedding: P.Spec (A) ,! P.Spec (A): Consequences Let A 2 P. Then the Poisson deleting derivations algorithm returns a Poisson ane space A = K[T1;:::; Tn] with fTi ; Tj g = λij Ti Tj such that: −1 ∼ ±1 ±1 for a multiplicative set in , I AS = K[T1 ;:::; Tn ] S A I there exists an embedding: P.Spec (A) ,! P.Spec (A): Corollary Let A 2 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 2 W , we set: P.Spec P.Spec w (A) := P 2 (A) j P\fT1;:::; Tng = fTi j i 2 wg : Then the partition of P.Spec (A): P.Spec G P.Spec (A) = w (A) w2W induces a partition of P.Spec (A): P.Spec G −1P.Spec (A) = ' w (A) 0 w2WP where 0 −1P.Spec (the set of WP := fw 2 W j ' w (A) 6= ;g Cauchon diagrams). Canonical partition and Cauchon Diagrams Let A 2 P, and A = K[T1;:::; Tn] with fTi ; Tj g = λij Ti Tj . Let W := P(f1;:::; ng). Then the partition of P.Spec (A): P.Spec G P.Spec (A) = w (A) w2W induces a partition of P.Spec (A): P.Spec G −1P.Spec (A) = ' w (A) 0 w2WP where 0 −1P.Spec (the set of WP := fw 2 W j ' w (A) 6= ;g Cauchon diagrams). Canonical partition and Cauchon Diagrams Let A 2 P, and A = K[T1;:::; Tn] with fTi ; Tj g = λij Ti Tj . Let W := P(f1;:::; ng). For w 2 W , we set: P.Spec P.Spec w (A) := P 2 (A) j P\fT1;:::; Tng = fTi j i 2 wg : induces a partition of P.Spec (A): P.Spec G −1P.Spec (A) = ' w (A) 0 w2WP where 0 −1P.Spec (the set of WP := fw 2 W j ' w (A) 6= ;g Cauchon diagrams). Canonical partition and Cauchon Diagrams Let A 2 P, and A = K[T1;:::; Tn] with fTi ; Tj g = λij Ti Tj . Let W := P(f1;:::; ng). For w 2 W , we set: P.Spec P.Spec w (A) := P 2 (A) j P\fT1;:::; Tng = fTi j i 2 wg : Then the partition of P.Spec (A): P.Spec G P.Spec (A) = w (A) w2W Canonical partition and Cauchon Diagrams Let A 2 P, and A = K[T1;:::; Tn] with fTi ; Tj g = λij Ti Tj .