Poisson Clusters and Unique Factorization Ken Goodearl University - PowerPoint PPT Presentation

Poisson Clusters and Unique Factorization Ken Goodearl University of California at Santa Barbara [joint work with Milen Yakimov] 0

Quick cluster algebra sketch (geometric type; coeffs ∈ field) K ⊂ F = K ( y 1 , . . . , y N ) = rational function field clusters = transcendence bases for F / K initial cluster = ( y 1 , . . . , y N ) [1 , N ] ⊇ ex = set of exchangeable indices ( others are frozen ) M N × ex( Z ) ∋ B = exchange matrix ( with some conditions ) 1

Quick cluster algebra sketch (geometric type; coeffs ∈ field) K ⊂ F = K ( y 1 , . . . , y N ) = rational function field clusters = transcendence bases for F / K initial cluster = ( y 1 , . . . , y N ) [1 , N ] ⊇ ex = set of exchangeable indices ( others are frozen ) M N × ex( Z ) ∋ B = exchange matrix ( with some conditions ) mutation in direction k ∈ ex : cluster ( y 1 , . . . , y N ) ∼ � cluster ( y 1 , . . . , y k − 1 , y ′ k , y k +1 , . . . , y N ) B ∼ � B ′ and ( by formulas involving B ) 1

Quick cluster algebra sketch (geometric type; coeffs ∈ field) K ⊂ F = K ( y 1 , . . . , y N ) = rational function field clusters = transcendence bases for F / K initial cluster = ( y 1 , . . . , y N ) [1 , N ] ⊇ ex = set of exchangeable indices ( others are frozen ) M N × ex( Z ) ∋ B = exchange matrix ( with some conditions ) mutation in direction k ∈ ex : cluster ( y 1 , . . . , y N ) ∼ � cluster ( y 1 , . . . , y k − 1 , y ′ k , y k +1 , . . . , y N ) B ∼ � B ′ and ( by formulas involving B ) Iterate mutations in all ex directions cluster algebra := K -subalgebra of F generated by � all clusters from iterated mutations, together with y − 1 for k in some set inv ⊆ [1 , N ] \ ex k 1

upper cluster algebra := of K [ z ± 1 � | i ∈ ex ⊔ inv ] [ z i | i / ∈ ex ⊔ inv ] i for original cluster and one-step mutations in all ex directions 2

upper cluster algebra := of K [ z ± 1 � | i ∈ ex ⊔ inv ] [ z i | i / ∈ ex ⊔ inv ] i for original cluster and one-step mutations in all ex directions Laurent Phenomenon [Fomin-Zelevinsky] cluster algebra ⊆ upper cluster algebra ⊆ K [ y ± 1 1 , . . . , y ± 1 N ] 2

upper cluster algebra := of K [ z ± 1 � | i ∈ ex ⊔ inv ] [ z i | i / ∈ ex ⊔ inv ] i for original cluster and one-step mutations in all ex directions Laurent Phenomenon [Fomin-Zelevinsky] cluster algebra ⊆ upper cluster algebra ⊆ K [ y ± 1 1 , . . . , y ± 1 N ] Some known cluster algebras : homogeneous coordinate rings of • Grassmannians Gr ( m , n ) [Scott] • partial flag varieties in semisimple algebraic groups type ADE [Geiß-Leclerc-Schr¨ oer] Some known upper cluster algebras : coordinate rings of • double Bruhat cells in semisimple algebraic groups / C [Berenstein-Fomin-Zelevinsky] 2

Assume char( K ) = 0 from now on [ K = base field] Poisson algebra = a commutative algebra R with Lie bracket {− , −} : R × R − → R such that all { r , −} are derivations ( ↑ a Poisson bracket ) 3

Assume char( K ) = 0 from now on [ K = base field] Poisson algebra = a commutative algebra R with Lie bracket {− , −} : R × R − → R such that all { r , −} are derivations ( ↑ a Poisson bracket ) E.G. O ( M m , n ( K )) with the standard Sklyanin bracket : { X ij , X il } = X ij X il ( j < l ) { X ij , X kj } = X ij X kj ( i < k ) � 0 ( i < k , j > l ) { X ij , X kl } = 2 X il X kj ( i < k , j < l ) 3

Assume char( K ) = 0 from now on [ K = base field] Poisson algebra = a commutative algebra R with Lie bracket {− , −} : R × R − → R such that all { r , −} are derivations ( ↑ a Poisson bracket ) E.G. O ( M m , n ( K )) with the standard Sklyanin bracket : { X ij , X il } = X ij X il ( j < l ) { X ij , X kj } = X ij X kj ( i < k ) � 0 ( i < k , j > l ) { X ij , X kl } = 2 X il X kj ( i < k , j < l ) and coordinate rings of Poisson subvarieties of M m , n ( K ), such as GL n ( K ), double Bruhat cells of GL n ( K ) 3

Consider a cluster algebra A ⊆ F = K ( y 1 , . . . , y N ) Assume F is a Poisson algebra / K 4

Consider a cluster algebra A ⊆ F = K ( y 1 , . . . , y N ) Assume F is a Poisson algebra / K • a cluster ( z 1 , . . . , z N ) is log-canonical if { z i , z j } ∈ Kz i z j ∀ i , j • the cluster structure on A is Poisson-compatible iff all clusters are log-canonical 4

Poisson polynomial algebra ( Poisson version of skew poly ring ) R = K [ x 1 ][ x 2 ; σ 2 , δ 2 ] p · · · [ x N ; σ N , δ N ] p : a polynomial ring K [ x 1 , . . . , x N ] with Poisson bracket ∋ { x k , r } = σ k ( r ) x k + δ k ( r ) for all r ∈ K [ x 1 , . . . , x k − 1 ] ( σ k = a Poisson derivation; suitable identities for δ k ) 5

Poisson polynomial algebra ( Poisson version of skew poly ring ) R = K [ x 1 ][ x 2 ; σ 2 , δ 2 ] p · · · [ x N ; σ N , δ N ] p : a polynomial ring K [ x 1 , . . . , x N ] with Poisson bracket ∋ { x k , r } = σ k ( r ) x k + δ k ( r ) for all r ∈ K [ x 1 , . . . , x k − 1 ] ( σ k = a Poisson derivation; suitable identities for δ k ) R ( ↑ ) is a Poisson-nilpotent algebra iff ∃ K -torus H = ( K × ) r ∋ • H acts rationally on R by Poisson automorphisms • All x k are H -eigenvectors • All δ k are locally nilpotent • Each σ k given by action of h k ∈ Lie H , with h k · x k � = 0 5

Poisson polynomial algebra ( Poisson version of skew poly ring ) R = K [ x 1 ][ x 2 ; σ 2 , δ 2 ] p · · · [ x N ; σ N , δ N ] p : a polynomial ring K [ x 1 , . . . , x N ] with Poisson bracket ∋ { x k , r } = σ k ( r ) x k + δ k ( r ) for all r ∈ K [ x 1 , . . . , x k − 1 ] ( σ k = a Poisson derivation; suitable identities for δ k ) R ( ↑ ) is a Poisson-nilpotent algebra iff ∃ K -torus H = ( K × ) r ∋ • H acts rationally on R by Poisson automorphisms • All x k are H -eigenvectors • All δ k are locally nilpotent • Each σ k given by action of h k ∈ Lie H , with h k · x k � = 0 E.G. R = O ( M m , n ( K )) with Sklyanin bracket, H = ( K × ) m + n , ( α 1 , . . . , α m , β 1 , . . . , β n ) · X ij = α i β j X ij 5

In a Poisson algebra R : • Poisson ideal I ⊳ R : { R , I } ⊆ I • Poisson-normal element c ∈ R : { c , R } ⊆ cR • Poisson-prime element : Poisson-normal, prime element 6

In a Poisson algebra R : • Poisson ideal I ⊳ R : { R , I } ⊆ I • Poisson-normal element c ∈ R : { c , R } ⊆ cR • Poisson-prime element : Poisson-normal, prime element Thm. 1 [Yakimov-K.G.] Every Poisson-nilpotent algebra is an H -Poisson-UFD : Each nonzero H -stable, prime, Poisson ideal of R contains a Poisson-prime H -eigenvector. 6

In a Poisson algebra R : • Poisson ideal I ⊳ R : { R , I } ⊆ I • Poisson-normal element c ∈ R : { c , R } ⊆ cR • Poisson-prime element : Poisson-normal, prime element Thm. 1 [Yakimov-K.G.] Every Poisson-nilpotent algebra is an H -Poisson-UFD : Each nonzero H -stable, prime, Poisson ideal of R contains a Poisson-prime H -eigenvector. Consequence : All Poisson-normal H -eigenvectors in R are products of units and Poisson-prime H -eigenvectors, unique up to ordering and associates. 6

Initial clusters : Thm 2. [Yakimov-K.G.] Let R = K [ x 1 , . . . , x N ] be a Poisson-nilpotent algebra. ∃ Poisson-prime H -eigenvectors y k ∈ K [ x 1 , . . . , x k ] ∀ k ∋ • All Poisson-prime H -eigenvectors in K [ x 1 , . . . , x k ] are among the scalar multiples of y 1 , . . . , y k . � � • ( y 1 , . . . , y N ) is log-canonical { y k , y l } ∈ Ky k y l . K [ y 1 , . . . , y N ] ⊆ R ⊆ K [ y ± 1 1 , . . . , y ± 1 • N ]. 7

A Poisson-nilpotent algebra R = K [ x 1 , . . . , x N ] is symmetric if : • δ k ( x j ) ∈ K [ x j +1 , . . . , x k − 1 ] ∀ k > j • R = K [ x N , x N − 1 , . . . , x 1 ] is Poisson-nilpotent with • The same torus H • (a compatibility condition on scalars) 8

A Poisson-nilpotent algebra R = K [ x 1 , . . . , x N ] is symmetric if : • δ k ( x j ) ∈ K [ x j +1 , . . . , x k − 1 ] ∀ k > j • R = K [ x N , x N − 1 , . . . , x 1 ] is Poisson-nilpotent with • The same torus H • (a compatibility condition on scalars) Ξ N := { τ ∈ S N | τ ([1 , k ]) = an interval , ∀ k ∈ [2 , N ] } 8

A Poisson-nilpotent algebra R = K [ x 1 , . . . , x N ] is symmetric if : • δ k ( x j ) ∈ K [ x j +1 , . . . , x k − 1 ] ∀ k > j • R = K [ x N , x N − 1 , . . . , x 1 ] is Poisson-nilpotent with • The same torus H • (a compatibility condition on scalars) Ξ N := { τ ∈ S N | τ ([1 , k ]) = an interval , ∀ k ∈ [2 , N ] } If R is a symmetric Poisson-nilpotent algebra, then ∀ τ ∈ Ξ N : • R = K [ x τ (1) , x τ (2) , . . . , x τ ( N ) ] is Poisson-nilpotent. • The corresponding y -elements from Theorem 2 form a log-canonical cluster ( y τ, 1 , y τ, 2 , . . . , y τ, N ). 8

Thm 3. [Yakimov-K.G.] Let R = K [ x 1 , . . . , x N ] be a symmetric Poisson-nilpotent algebra ( with mild conditions on scalars ). Set ex := { k ∈ [1 , N ] | y k is not Poisson-prime in R } . • R is a Poisson-compatible cluster algebra. • R = the corresponding upper cluster algebra. • R is generated by the cluster variables y τ, k for τ ∈ Ξ N and k ∈ [1 , N ]. Also true for R [ y − 1 • | k ∈ inv ], any inv ⊆ [1 , N ] \ ex . k 9