# 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) 1405-1429. [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.
 [1] Doston Jumaniyozov, Ivan Kaygorodov, Abror Khudoyberdiyev. The algebraic classification of nilpotent commutative algebras. Electronic Research Archive, 2021, 29 (6) : 3909-3993. doi: 10.3934/era.2021068 [2] Henrique Bursztyn, Alejandro Cabrera, Matias del Hoyo. Poisson double structures. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021029 [3] Jin-Yun Guo, Cong Xiao, Xiaojian Lu. On $n$-slice algebras and related algebras. Electronic Research Archive, 2021, 29 (4) : 2687-2718. doi: 10.3934/era.2021009 [4] Grégory Berhuy. Algebraic space-time codes based on division algebras with a unitary involution. Advances in Mathematics of Communications, 2014, 8 (2) : 167-189. doi: 10.3934/amc.2014.8.167 [5] Andrew James Bruce, Janusz Grabowski. Symplectic ${\mathbb Z}_2^n$-manifolds. Journal of Geometric Mechanics, 2021, 13 (3) : 285-311. doi: 10.3934/jgm.2021020 [6] A. A. Kirillov. Family algebras. Electronic Research Announcements, 2000, 6: 7-20. [7] Steffen Konig and Changchang Xi. Cellular algebras and quasi-hereditary algebras: a comparison. Electronic Research Announcements, 1999, 5: 71-75. [8] Andrew N. W. Hone, Matteo Petrera. Three-dimensional discrete systems of Hirota-Kimura type and deformed Lie-Poisson algebras. Journal of Geometric Mechanics, 2009, 1 (1) : 55-85. doi: 10.3934/jgm.2009.1.55 [9] Stephen Doty and Anthony Giaquinto. Generators and relations for Schur algebras. Electronic Research Announcements, 2001, 7: 54-62. [10] Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39 [11] Valentin Ovsienko, MichaeL Shapiro. Cluster algebras with Grassmann variables. Electronic Research Announcements, 2019, 26: 1-15. doi: 10.3934/era.2019.26.001 [12] L. S. Grinblat. Theorems on sets not belonging to algebras. Electronic Research Announcements, 2004, 10: 51-57. [13] Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov. On matrix wreath products of algebras. Electronic Research Announcements, 2017, 24: 78-86. doi: 10.3934/era.2017.24.009 [14] Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, 2021, 29 (3) : 2457-2473. doi: 10.3934/era.2020124 [15] Santiago Cañez. Double groupoids and the symplectic category. Journal of Geometric Mechanics, 2018, 10 (2) : 217-250. doi: 10.3934/jgm.2018009 [16] Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021 [17] Randall Dougherty and Thomas Jech. Left-distributive embedding algebras. Electronic Research Announcements, 1997, 3: 28-37. [18] G. Mashevitzky, B. Plotkin and E. Plotkin. Automorphisms of categories of free algebras of varieties. Electronic Research Announcements, 2002, 8: 1-10. [19] María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 637-652. doi: 10.3934/dcds.2015.35.637 [20] A. Giambruno and M. Zaicev. Minimal varieties of algebras of exponential growth. Electronic Research Announcements, 2000, 6: 40-44.

Impact Factor: