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On inner Poisson structures of a quantum cluster algebra without coefficients

  • 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[J]. Electronic Research Archive, 2021, 29(5): 2959-2972. doi: 10.3934/era.2021021

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  • 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.



    Creating this inaugural special issue on Engineering Applications of Artificial Intelligence (AI) is important due to the rapid technology advancement and the aim to reduce the manpower by incorporating Artificial Intelligence in various Industry 4.0 applications. As my research reflects the multi-disciplinarily of systems (consisting of mechanical, electrical, electronics, acoustical and marine engineering) from initial concepts to the modelling and AI simulation, creating graphical-user interface and their actual implementations and testing on sites. The special issue provides a good platform to share applied research results from different researchers around the world.

    For example, the phase partition-based ensemble learning framework upon least squares supports vector regression (LSSVR) was used for soft sensor modeling to improve the prediction accuracy in chemical and biological processes. As a result, the robotic grasping based on improved Gaussian mixture model was also proposed using the virtual robot experimentation platform. The face image recognition algorithm based on two-dimensional (2D) Gabor wavelet transform and Local Binary Pattern (LBP) was presented. It provides a better classification performance in different scales and directions affected by illumination, gesture, expression, and other factor's variation. With more consciousness in cyber-security, the paper that used the Kalman filter-based attack detection model was proposed. The block withholding delay attack and the countermeasure were also proposed in a similar occasion. The well-known convolutional neural network (CNN) based approach was applied to detect the obstacle for the unmanned surface vehicle. Subsequently, an effective classifier based on the CNN and regularized extreme learning machine (ELM) was adopted to reduce the classification time in the training and testing.

    In summary, this issue concluded with different engineering applications of AI. It is imperative that we continue to progress in our search for better engineering systems design and simulation using AI. The progress reported in this special issue suggests that achieving these aims is an attainable one. I hope that we can stay in contact and make this world a better place for a "deep" collaborative research.



    [1] Quantum cluster algebras. Adv. Math. (2005) 195: 405-455.
    [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
    [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
    [4] On the deformation of rings and algebras. Ann. of Math. (1964) 79: 59-103.
    [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
    [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.
    [7] F. Li and J. Pan, Poisson structure and second quantization of quantum cluster algebras, preprint, arXiv: 2003.12257v3.
    [8] Quiver Poisson algebras. J. Algebra (2007) 312: 570-589.
  • This article has been cited by:

    1. Xu Hao, Deyu Zhou, Ruiheng Zhong, Shunxi Li, Xianming Meng, Bo Liu, Electrification pathways for light-duty logistics vehicles based on perceived cost of ownership in Northern China, 2024, 3, 2831-932X, 10.20517/cf.2024.24
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