doi: 10.3934/era.2021021
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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 Early access 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

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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

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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

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

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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

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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

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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

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