Noncommutative bi-symplectic $\mathbb{N}Q$-algebras of weight 1

Luis  Álvarez–cónsul; David  Fernández

doi:10.3934/proc.2015.0019

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2015, Volume 2015: 19-28. Doi: 10.3934/proc.2015.0019 open access

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Noncommutative bi-symplectic $\mathbb{N}Q$-algebras of weight 1

Luis Álvarez–cónsul^1, and
David Fernández^1,

1.
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Nicolás Cabrera 13–15, Cantoblanco, 28049 Madrid

Received: September 2014

Revised: September 2015

Published: October 2015

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Abstract

It is well known that symplectic $\mathbb{N}Q$-manifolds of weight 1 are in 1-1 correspondence with Poisson manifolds. In this article, we prove a version of this correspondence in the framework of noncommutative algebraic geometry based on double derivations, as introduced by W. Crawley-Boevey, P. Etingof and V. Ginzburg. More precisely, we define noncommutative bi-symplectic $\mathbb{N}Q$-algebras and prove that bi-symplectic $\mathbb{N}Q$-algebras of weight 1 are in 1-1 correspondence with double Poisson algebras, as previously defined by M. Van den Bergh.

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Mathematics Subject Classification: Primary: 14A22.

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