ON SOME GRAPH PROPERTIES OF HEISENBERG LIE ALGEBRAS
DOI:
https://doi.org/10.54607/hcmue.js.22.10.4490(2025)Abstract
Heisenberg Lie Algebras play an important role in both mathematics and physics, in particular in quantum mechanics. These algebras provide a framework for expressing the uncertainty principle and commutation relations between physical quantities in quantum systems. Many aspects, properties, and applications of Heisenberg Lie algebras have been widely studied. In this paper, we focus on the discrete representation of Heisenberg Lie algebras. Specifically, we study the commuting graphs associated with all Heisenberg Lie algebras, describe their structural forms, and present key properties such as connectedness and diameter.Downloads
References
Anderson, D. F., & Badawi, A. (2008). The total graph of a commutative ring. Journal of Algebra, 320(7), 2706-2719.
Anderson, D. F. (2011). Zero-divisor graphs in commutative rings. Commutative Algebra, Noetherian and Non-Noetherian Perspectives/Springer-Verlag. https://doi.org/10.1006/jabr.1998.7840
Akbari, S., Ghandehari, M., Hadian, M., & Mohammadian, A. (2004). On commuting graphs of semisimple rings. Linear algebra and its applications, 390, 345-355. https://doi.org/10.1016/j.laa.2004.05.001
Akbari, S., Bidkhori, H., & Mohammadian, A. (2008). Commuting graphs of matrix algebras. Communications in Algebra, 36(11), 4020-4031. https://doi.org/10.1080/00927870802174538
Akbari, S., & Raja, P. (2006). Commuting graphs of some subsets in simple rings. Linear algebra and its applications, 416(2-3), 1038-1047. https://doi.org/10.1016/j.laa.2006.01.006
Cao, M. N., Mai, H. B., & Bui, X. H. (2023). On diameters of commuting graphs of matrix algebras over division rings. Journal of Algebra and Its Applications, 24(06), 2550148. https://doi.org/10.1142/S0219498825501488
Chartrand, G. (1996). Graphs and Digraphs. Chapman and Hall.
Dorbidi, H. R. (2023). The diameter of commuting graph of matrix ring of division rings. Journal of Algebra and Its Applications, 24(05), 2550121.
Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43(3), 172-198.
Humphreys, J. E. (1972). Introduction to Lie Algebras and Representation Theory. Springer-Verlag.
Miguel, C. (2016). On the diameter of the commuting graph of the matrix ring over a centrally finite division ring. Linear Algebra and its Applications, 509, 276-285. https://doi.org/10.1016/j.laa.2016.08.001
Šnobl, L., & Winternitz, P. (2014). Classification and identification of Lie algebras (Vol. 33). American Mathematical Soc.
Wang, D., & Xia, C. (2017). Diameters of the Commuting Graphs of Simple Lie Algebras. J. Lie Theory, 27, 139-154.
