Double Bruhat Cells and Symplectic Groupoids

Generalized Bruhat Cells and Completeness of Hamiltonian Flows of Kogan-Zelevinsky Integrable Systems

“…More precisely, ρẇ denotes the right Poisson action Proof. We prove Lemma A.3 using a similar result for (B − wB − , π st ) in [35,Remark 9]. Consider the map where S is the map that switches the two factors, and Iẇ is given in (91).…”

“…In addition to carrying Kogan-Zelevinsky integrable systems, which we have now proved to have complete Hamiltonian flows with property Q, it is shown in [35] that G u,u also has the natural structure of a Poisson groupoid over the Bruhat cell BuB/B ⊂ G/B. In fact, each symplectic leaf of π st in G u,u is shown in [35] to be a symplectic groupoid over BuB/B. It is very natural, then, to formulate any compatibility between the Kogan-Zelevinsky integrable systems and the groupoid structure on G u,u , and to ask whether there is any groupoid-theoretical interpretation of the C l(u) -action on G u,u defined by the Kogan-Zelevinsky systems.…”

“…Proof. By [35,Lemma 10], both N − and C v = (N v) ∩ (vN − ) are coisotropic submanifolds of G with respect to the Poisson structure π st . It follows by the multiplicativity of π st that N − vN − = N − C v is also coisotropic with respect to π st .…”

“…Recall that (B − , π st ) and (B, −π st ) form a pair of dual Poisson Lie groups under the pairing , (b − ,b) of their Lie algebras. Define the Poisson structure π ′ on B − × (BwB/B) byπ ′ = π st × (ρ B − , λ + ) (−π 1 ) = (π st , 0) + (0, −π 1 ) − m i=1 (x L i , 0) ∧ (0, λ + (ξ i )).It is shown in[35, Remark 9] that J ′ẇ (π st ) = π ′ as Poisson structures on B − × (BwB/B).Consider (see[17, §1.5]) the involutive anti-automorphism Θ of G determined byΘ(t) = t, t ∈ T, Θ(u α (c)) = u −α (−c), α ∈ Γ, c ∈ C.It is easy to see from the definition that Θ(π st ) = π st . Consider now the sequence of Poisson isomorphisms(BwB, π st ) Θ −→ (B − w −1 B − , π st ) J ′ Θ(ẇ) −→ (B − × (Bw −1 B/B), π ′ ) Θ×Θ −→ (B × (B − \B − wB − ), π (2) ) S −→ ((B − \B − wB − ) × B, π (3) ) I −1 w ×Id B−→ ((BwB/B) × B, π (4) ),…”

Symplectic groupoids for cluster manifolds

“…The source fibers of a symplectic groupoid are symplectic orthogonal to the target fibers. Some important examples of symplectic groupoids include: the Kostant-Kirillov-Souriau Poisson structures [11], the Drinfeld doubles of Poisson Lie groups [32], the double Bruhat cells [31], the blow-up groupoids of log-symplectic manifolds [26], and the symplectic doubles of (the positive part of) the cluster X -varieties [16]. We note that symplectic groupoids are a special case of Poisson groupoids.…”