2021/11/04 by Jianlong Lu, Lu, Jianlong · 1 citation
Physics and Astronomy · #Black Holes and Theoretical Physics #FOS: Physical sciences #High Energy Physics - Phenomenology (hep-ph) #Neutrino Physics Research #Particle physics theoretical and experimental studies
paper · pdf · doi:10.48550/arxiv.2111.02729
openalex publication_date 2021/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We respectfully reply to Wang et al.'s reply to our comment to their [JHEP 09 (2021) 053]. Some more subtleties and details are discussed for hopefully better clarification and understanding, especially about the conditions in Hilbert's finiteness theorem. We also disprove Wang et al.'s incorrect assertion in their reply [arXiv:2110.13865 [hep-ph]] that there is an isomorphism between \rm U(n,ℂ) and \rm GL(n,ℂ). We give two proofs for our argument. One proof is in the context of Lie group for arbitrary positive integer n, and the other proof is in the context of abstract group for n=3 which can also be generalized to n≥ 3. In order to better illustrate the intrinsic differences between a compact Lie group and its complexification, we choose \rm SU(n,ℂ) and \rm SL(n,ℂ) as a pair of examples and give two proofs for the generally non-isomorphic relation between them. One proof is in the context of Lie group for integers n≥ 2, and the other proof is in the context of abstract group for n=3 which can again be generalized to n≥ 3.