# American Institute of Mathematical Sciences

2015, 2015(special): 19-28. doi: 10.3934/proc.2015.0019

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

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

Received  September 2014 Revised  September 2015 Published  November 2015

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.
Citation: Luis Álvarez–cónsul, David Fernández. Noncommutative bi-symplectic $\mathbb{N}Q$-algebras of weight 1. Conference Publications, 2015, 2015 (special) : 19-28. doi: 10.3934/proc.2015.0019
##### References:
 [1] M. Alexandrov, M. Kontsevich, A. Schwarz and O. Zaboronsky, The geometry of the Master Equation and Topological Quantum Field Theory,, Internat. J. Modern Phys. A 12, 7 (1997), 1405. [2] I. A. Batalin and G. A. Vilkovisky, Gauge algebra and quantization,, Phys. Lett., 102B (1981). [3] A. S. Cattaneo and G. Felder, On the AKSZ formalism of the Poisson sigma model,, Phys. Lett. B, 102 (1981). [4] W. Crawley-Boevey, Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities,, Comment. Math. Helv., 74 (1999), 548. [5] W. Crawley-Boevey, P. Etingof and V. Ginzburg, Noncommutative geometry and quiver algebras,, Adv. Math., 209 (2007), 274. [6] D. Cuntz and D. Quillen, Algebra extensions and nonsingularity,, J. Amer. Math. Soc., 8 (1995), 251. [7] D. R. Farkas and G. Letzter, Ring theory from symplectic geometry,, J. Pure Appl. Algebra, 125 (1998), 155. [8] V. Ginzburg, Lectures on Noncommutative Geometry,, Preprint arXiv:math.AG/0612139., (). [9] M. Kontsevich and A. Rosenberg, Noncommutative smooth spaces., In: The Gelfand Mathematical Seminars, (2000), 1996. [10] H. Matsumura, Commutative ring theory., Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, 8 (1989). [11] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, In: Quantization, 315 (2002). [12] D. Roytenberg, AKSZ-BV Formalism and Courant Algebroids-induced Topological Field Theories,, Lett. Math. Phys. 79, 2 (2007). [13] A. Schwarz, Geometry of Batalin-Vilkovisky quantization,, Commun. Math. Phys., 155 (1993), 249. [14] P. Ševera, Some title containing the words "homotopy'' and "symplectic'', e.g. this one,, In: Travaux mathématiques, (2005), 121. [15] M. Van den Bergh, Double Poisson algebras,, Trans. Amer. Math. Soc., 360 (2008), 5711.

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##### References:
 [1] M. Alexandrov, M. Kontsevich, A. Schwarz and O. Zaboronsky, The geometry of the Master Equation and Topological Quantum Field Theory,, Internat. J. Modern Phys. A 12, 7 (1997), 1405. [2] I. A. Batalin and G. A. Vilkovisky, Gauge algebra and quantization,, Phys. Lett., 102B (1981). [3] A. S. Cattaneo and G. Felder, On the AKSZ formalism of the Poisson sigma model,, Phys. Lett. B, 102 (1981). [4] W. Crawley-Boevey, Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities,, Comment. Math. Helv., 74 (1999), 548. [5] W. Crawley-Boevey, P. Etingof and V. Ginzburg, Noncommutative geometry and quiver algebras,, Adv. Math., 209 (2007), 274. [6] D. Cuntz and D. Quillen, Algebra extensions and nonsingularity,, J. Amer. Math. Soc., 8 (1995), 251. [7] D. R. Farkas and G. Letzter, Ring theory from symplectic geometry,, J. Pure Appl. Algebra, 125 (1998), 155. [8] V. Ginzburg, Lectures on Noncommutative Geometry,, Preprint arXiv:math.AG/0612139., (). [9] M. Kontsevich and A. Rosenberg, Noncommutative smooth spaces., In: The Gelfand Mathematical Seminars, (2000), 1996. [10] H. Matsumura, Commutative ring theory., Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, 8 (1989). [11] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, In: Quantization, 315 (2002). [12] D. Roytenberg, AKSZ-BV Formalism and Courant Algebroids-induced Topological Field Theories,, Lett. Math. Phys. 79, 2 (2007). [13] A. Schwarz, Geometry of Batalin-Vilkovisky quantization,, Commun. Math. Phys., 155 (1993), 249. [14] P. Ševera, Some title containing the words "homotopy'' and "symplectic'', e.g. this one,, In: Travaux mathématiques, (2005), 121. [15] M. Van den Bergh, Double Poisson algebras,, Trans. Amer. Math. Soc., 360 (2008), 5711.
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