Double quasi-Poisson algebras are pre-Calabi-Yau

Abstract

In this article we prove that double quasi-Poisson algebras, which are noncommutative analogues of quasi-Poisson manifolds, naturally give rise to pre-Calabi-Yau algebras. This extends one of the main results in [11] (see also [10]), where a relationship between pre-Calabi-Yau algebras and double Poisson algebras was found. However, a major difference between the pre-Calabi-Yau algebra constructed in the mentioned articles and the one constructed in this work is that the higher multiplications indexed by even integers of the underlying A∞-algebra structure of the pre-Calabi-Yau algebra associated to a double quasi-Poisson algebra do not vanish, but are given by nice cyclic expressions multiplied by explicitly determined coefficients involving the Bernoulli numbers.

On the noncommutative Poisson geometry of certain wild character varieties

“…Finally, note that Theorem 4.5 together with the main result of [28] give a pre-Calabi-Yau algebra structure on the Boalch algebra B(∆), which we expect to be non-degenerate, giving rise to a (right) Calabi-Yau structure. It would be interesting to show how this structure descends to the fission algebra F q (∆).…”

Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems

“…They have been the object of several studies [6,46,47,48,49,61], and our aim is to explore morphisms between double Poisson algebras. Noting that Van den Bergh also introduced the analogous notion of double quasi-Poisson brackets [59,60], it is natural to extend our investigation to morphisms between the corresponding algebras, called double quasi-Poisson algebras, which currently attract attention [4,11,12,19,20,21,25,40].…”

“…{ {c, d} } 3→2 1,f us = 0 , { {c, d} } cd − d ⊗ ce 1 ), { {ψ(c), ψ(d)} } 2→1 2,f us = 0 , { {ψ(c), ψ(d)} } 3→1 2,f us = 1 ψ(c)ψ(d) − ψ(d) ⊗ ψ(c)e 1 . (C 25). We consider that c is a generator of third type of the form c = ae 32 e 21 for a ∈ êAe 3 .…”