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

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, Spain, Spain

Received September 2014 Revised September 2015 Published November 2015

Abstract
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Under a Creative Commons license

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.

Keywords: Quivers, Bi-symplectic structures, $\mathbb{N}$Q-algebras, Double Poisson algebras, Noncommutative algebraic geometry..

Mathematics Subject Classification: Primary: 14A2.

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) 1405-1429.
[2]	I. A. Batalin and G. A. Vilkovisky, Gauge algebra and quantization, Phys. Lett., 102B (1981), 27.
[3]	A. S. Cattaneo and G. Felder, On the AKSZ formalism of the Poisson sigma model, Phys. Lett. B, 102 (1981), 27.
[4]	W. Crawley-Boevey, Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities, Comment. Math. Helv., 74 (1999), 548-574.
[5]	W. Crawley-Boevey, P. Etingof and V. Ginzburg, Noncommutative geometry and quiver algebras, Adv. Math., 209 (2007), 274-336.
[6]	D. Cuntz and D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc., 8 (1995), 251-289.
[7]	D. R. Farkas and G. Letzter, Ring theory from symplectic geometry, J. Pure Appl. Algebra, 125 (1998), 155-190.
[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, 1996-1999, 85-108, Bikhäuser, Boston, 2000.
[10]	H. Matsumura, Commutative ring theory. Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge-New York, 1989.
[11]	D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, In: Quantization, Poisson brackets and Beyond. Th. Voronov (ed.), Contemp. Math., 315, Amer. Math. Soc., Providence, RI, 2002.
[12]	D. Roytenberg, AKSZ-BV Formalism and Courant Algebroids-induced Topological Field Theories, Lett. Math. Phys. 79, 2 (2007) 143159.
[13]	A. Schwarz, Geometry of Batalin-Vilkovisky quantization, Commun. Math. Phys., 155 (1993), 249-260.
[14]	P. Ševera, Some title containing the words "homotopy'' and "symplectic'', e.g. this one, In: Travaux mathématiques, Fasc. XVI, pp. 121-137. Univ. Luxemb., Luxembourg, 2005.
[15]	M. Van den Bergh, Double Poisson algebras, Trans. Amer. Math. Soc., 360 (2008), 5711-5769.

show all references

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) 1405-1429.
[2]	I. A. Batalin and G. A. Vilkovisky, Gauge algebra and quantization, Phys. Lett., 102B (1981), 27.
[3]	A. S. Cattaneo and G. Felder, On the AKSZ formalism of the Poisson sigma model, Phys. Lett. B, 102 (1981), 27.
[4]	W. Crawley-Boevey, Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities, Comment. Math. Helv., 74 (1999), 548-574.
[5]	W. Crawley-Boevey, P. Etingof and V. Ginzburg, Noncommutative geometry and quiver algebras, Adv. Math., 209 (2007), 274-336.
[6]	D. Cuntz and D. Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc., 8 (1995), 251-289.
[7]	D. R. Farkas and G. Letzter, Ring theory from symplectic geometry, J. Pure Appl. Algebra, 125 (1998), 155-190.
[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, 1996-1999, 85-108, Bikhäuser, Boston, 2000.
[10]	H. Matsumura, Commutative ring theory. Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge-New York, 1989.
[11]	D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, In: Quantization, Poisson brackets and Beyond. Th. Voronov (ed.), Contemp. Math., 315, Amer. Math. Soc., Providence, RI, 2002.
[12]	D. Roytenberg, AKSZ-BV Formalism and Courant Algebroids-induced Topological Field Theories, Lett. Math. Phys. 79, 2 (2007) 143159.
[13]	A. Schwarz, Geometry of Batalin-Vilkovisky quantization, Commun. Math. Phys., 155 (1993), 249-260.
[14]	P. Ševera, Some title containing the words "homotopy'' and "symplectic'', e.g. this one, In: Travaux mathématiques, Fasc. XVI, pp. 121-137. Univ. Luxemb., Luxembourg, 2005.
[15]	M. Van den Bergh, Double Poisson algebras, Trans. Amer. Math. Soc., 360 (2008), 5711-5769.

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