Generators belong strictly to the Lie algebra rather than the Lie group itself
Is it true that generators belong strictly to the Lie algebra rather than the Lie group itself?
Yes. In differential geometry and Lie theory, generators are elements of the tangent space at the identity—the Lie algebra—which map into the Lie group via the exponential map rather than belonging to the group itself.
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The claim states that generators belong strictly to the Lie algebra rather than the Lie group itself. Standard mathematical definitions and multiple papers in the literature (e.g., [0], [1], [2], [7]) confirm that generators form the Lie algebra (the tangent space at the identity), which generates the Lie group through the exponential map. Therefore, the claim is well-supported by standard mathematical principles and the provided literature.
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Durov, N., Meljanac, S., Samsarov, A., & vSkoda, Z. (2006). A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra. doi.org. https://doi.org/10.1016/J.JALGEBRA.2006.08.025
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Celledoni, E. & Iserles, A. (2000). Approximating the exponential from a Lie algebra to a Lie group. doi.org. https://doi.org/10.1090/s0025-5718-00-01223-0
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Shumaylov, Z., Zaika, P., Rowbottom, J., Sherry, F., Weber, M., & Schönlieb, C. (2024). Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups. doi.org. https://doi.org/10.48550/arXiv.2410.02698
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Neretin, Y. A. (2024). A Lie group corresponding to the free Lie algebra and its universality. doi.org. https://doi.org/10.4171/ggd/976
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Abdigapparovich, N. O. (2018). Lie Algebra of Infinitesimal Generators of the Symmetry Group of the Heat Equation. doi.org. https://doi.org/10.4236/JAMP.2018.62035
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The claim states that generators belong strictly to the Lie algebra rather than the Lie group itself. Standard mathematical definitions and multiple papers in the literature (e.g., [0], [1], [2], [7]) confirm that generators form the Lie algebra (the tangent space at the identity), which generates the Lie group through the exponential map. Therefore, the claim is well-supported by standard mathematical principles and the provided literature.
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