Modified double Poisson brackets

Abstract

Abstract. We propose a non-skew-symmetric generalization of the original definition of double Poisson Bracket by M. Van den Bergh. It allows one to explicitly construct more general class of H 0 -Poisson structures on finitely generated associative algebras. We show that modified double Poisson brackets inherit certain major properties of the double Poisson brackets.

Double Lie algebras of a nonzero weight

“…We show that this double Lie algebra M has the only one nonzero proper ideal which occurs to be isomorphic to M. Finally, we prove that every λ-double Lie algebra structure on a vector space V generates a unique modified double Poisson algebra structure on As V . This general result confirms the conjecture of S. Arthamonov (2017) [4], which says that the double bracket defined on the three-dimensional vector space V = Span{a 1 , a 2 , a 3 } as follows,…”

“…In the middle of 2010s, S. Arthamonov introduced a notion of modified double Poisson algebra [3,4] with weakened versions of anti-commutativity and Jacobi identity. This notion allowed S. Arthamonov to study the Kontsevich system and give more examples of H 0 -Poisson structures arisen from double brackets.…”

Hamiltonian operators and related differential-algebraic Balinsky-Novikov, Riemann and Leibniz type structures on nonassociative noncommutative algebras

“…Proceed now to a slightly generalized construction of Hamiltonian operators on a phase space, generated by associative noncommutative algebra A-valued matrices, which was first studied in [25,35,76,78] in case of the noncommutative operator algebras and continued later in [62,51,52,53,62,66,67,71] in case of general associative noncommutative algebras. This natural and simple generalization appeared to be very useful [5,6,94,96,62,66,67] for describing a wide class of new Lax type integrable nonlinear Hamiltonian systems on associative noncommutative algebras, interesting for diverse applications in modern quantum physics.…”

“…In a supplement the classical Poisson manifold approach, closely related to our construction of Hamiltonian operators, generated by nonassociative and noncommutative algebras, is briefly revisited. In particular, its natural and simple generalization appeared to be useful [5,6,94,96,62,66,67] for describing a wide class of Lax type integrable nonlinear Hamiltonian systems on associative noncommutative algebras, initiated first in [25,35,76,78] in case of the noncommutative operator algebras and continued later in [62,51,52,53,62,66,67,71] in case of general associative noncommutative algebras.…”

Nonabelian elliptic Poisson structures on projective spaces

“…In particular, he described noncommutative versions of differential forms and symplectic Poisson brackets. These ideas were deepen and further developed in [2,3,4,5,6,8,14] where, in particular, a more general framework applicable to a wide class of associative algebras was suggested, a lot of examples were constructed and various applications from quivers representation theory to integrable systems were outlined.…”