doi: 10.3934/era.2021021

On inner Poisson structures of a quantum cluster algebra without coefficients

Department of Mathematics, Zhejiang University (Yuquan Campus), Hangzhou, Zhejiang 310027, China

* Corresponding author

Received  July 2020 Revised  January 2021 Published  March 2021

The main aim of this article is to characterize inner Poisson structure on a quantum cluster algebra without coefficients. Mainly, we prove that inner Poisson structure on a quantum cluster algebra without coefficients is always a standard Poisson structure. We introduce the concept of so-called locally inner Poisson structure on a quantum cluster algebra and then show it is equivalent to locally standard Poisson structure in the case without coefficients. Based on the result from [7] we obtain finally the equivalence between locally inner Poisson structure and compatible Poisson structure in this case.

Citation: Fang Li, Jie Pan. On inner Poisson structures of a quantum cluster algebra without coefficients. Electronic Research Archive, doi: 10.3934/era.2021021
References:
[1]

A. Berenstein and A. Zelevinsky, Quantum cluster algebras, Adv. Math., 195 (2005), 405-455.  doi: 10.1016/j.aim.2004.08.003.  Google Scholar

[2]

C. Geiß, B. Leclerc and J. Schröer, Cluster structures on quantum coordinate rings, Selecta Math, 19 (2013) 337–397. doi: 10.1007/s00029-012-0099-x.  Google Scholar

[3]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster Algebras and Poisson Geometry, Mathematical Surveys and Monographs Volume 167, American Mathematical Society Providence, Rhode Island, 2010. doi: 10.1090/surv/167.  Google Scholar

[4]

M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math., 79 (1964), 59-103.  doi: 10.2307/1970484.  Google Scholar

[5]

K. R. Goodearl and M. T. Yakimov, Quantum cluster algebra structures on quantum nilpotent algebras, Mem. Amer. Math. Soc., 247 (2017), no.1169, arXiv: 1309.7869. doi: 10.1090/memo/1169.  Google Scholar

[6]

R. Inoue and T. Nakanishi, Difference equations and cluster algebras I: Poisson bracket for integrable difference equations, in Infinite Analysis 2010 - Developments in Quantum Integrable Systems, RIMS Kokyuroku Bessatsu, Vol.B28, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, 63–88, arXiv: 1012.5574.  Google Scholar

[7]

F. Li and J. Pan, Poisson structure and second quantization of quantum cluster algebras, preprint, arXiv: 2003.12257v3. Google Scholar

[8]

Y. YaoY. Ye and P. Zhang, Quiver Poisson algebras, J. Algebra, 312 (2007), 570-589.  doi: 10.1016/j.jalgebra.2007.03.034.  Google Scholar

show all references

References:
[1]

A. Berenstein and A. Zelevinsky, Quantum cluster algebras, Adv. Math., 195 (2005), 405-455.  doi: 10.1016/j.aim.2004.08.003.  Google Scholar

[2]

C. Geiß, B. Leclerc and J. Schröer, Cluster structures on quantum coordinate rings, Selecta Math, 19 (2013) 337–397. doi: 10.1007/s00029-012-0099-x.  Google Scholar

[3]

M. Gekhtman, M. Shapiro and A. Vainshtein, Cluster Algebras and Poisson Geometry, Mathematical Surveys and Monographs Volume 167, American Mathematical Society Providence, Rhode Island, 2010. doi: 10.1090/surv/167.  Google Scholar

[4]

M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math., 79 (1964), 59-103.  doi: 10.2307/1970484.  Google Scholar

[5]

K. R. Goodearl and M. T. Yakimov, Quantum cluster algebra structures on quantum nilpotent algebras, Mem. Amer. Math. Soc., 247 (2017), no.1169, arXiv: 1309.7869. doi: 10.1090/memo/1169.  Google Scholar

[6]

R. Inoue and T. Nakanishi, Difference equations and cluster algebras I: Poisson bracket for integrable difference equations, in Infinite Analysis 2010 - Developments in Quantum Integrable Systems, RIMS Kokyuroku Bessatsu, Vol.B28, Res. Inst. Math. Sci. (RIMS), Kyoto, 2011, 63–88, arXiv: 1012.5574.  Google Scholar

[7]

F. Li and J. Pan, Poisson structure and second quantization of quantum cluster algebras, preprint, arXiv: 2003.12257v3. Google Scholar

[8]

Y. YaoY. Ye and P. Zhang, Quiver Poisson algebras, J. Algebra, 312 (2007), 570-589.  doi: 10.1016/j.jalgebra.2007.03.034.  Google Scholar

[1]

Yusi Fan, Chenrui Yao, Liangyun Chen. Structure of sympathetic Lie superalgebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021020

[2]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565

[3]

Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, 2021, 15 (3) : 519-537. doi: 10.3934/ipi.2021003

[4]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003

[5]

Youjun Deng, Hongyu Liu, Xianchao Wang, Dong Wei, Liyan Zhu. Simultaneous recovery of surface heat flux and thickness of a solid structure by ultrasonic measurements. Electronic Research Archive, , () : -. doi: 10.3934/era.2021027

[6]

Christoforidou Amalia, Christian-Oliver Ewald. A lattice method for option evaluation with regime-switching asset correlation structure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1729-1752. doi: 10.3934/jimo.2020042

[7]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3319-3341. doi: 10.3934/dcds.2020407

[8]

Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks & Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007

[9]

Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039

[10]

Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021038

[11]

Yulia O. Belyaeva, Björn Gebhard, Alexander L. Skubachevskii. A general way to confined stationary Vlasov-Poisson plasma configurations. Kinetic & Related Models, 2021, 14 (2) : 257-282. doi: 10.3934/krm.2021004

[12]

Arianna Giunti. Convergence rates for the homogenization of the Poisson problem in randomly perforated domains. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021009

[13]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

[14]

Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3069-3096. doi: 10.3934/dcdsb.2020220

[15]

Anastasiia Panchuk, Frank Westerhoff. Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021117

[16]

Patrick Beißner, Emanuela Rosazza Gianin. The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 23-52. doi: 10.3934/puqr.2021002

[17]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447

[18]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030

[19]

Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237

[20]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003

 Impact Factor: 0.263

Article outline

[Back to Top]