Differentiable semigroups are Lie groups
We present here a modern, detailed proof to the following theorem which was introduced by Garrett Birkhoff [1] in 1938. If S is a local semigroup with neighborhood of 1 homeomorphic to a Banach space and with multiplication strongly differentiable at 1, then S is a local Lie Group. Although this the...
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| Format: | Article |
| Language: | English |
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Wiley
1995-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171295000652 |
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| _version_ | 1832563144013119488 |
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| author | John P. Holmes Mitch Anderson |
| author_facet | John P. Holmes Mitch Anderson |
| author_sort | John P. Holmes |
| collection | DOAJ |
| description | We present here a modern, detailed proof to the following theorem which was introduced
by Garrett Birkhoff [1] in 1938. If S is a local semigroup with neighborhood of 1 homeomorphic to a
Banach space and with multiplication strongly differentiable at 1, then S is a local Lie Group. Although
this theorem is more than 50 years old and remains the strongest result relating to Hilbert's fifth problem
in the infinite dimensional setting, it is frequently overlooked in favor of weaker results. Therefore, it
is the goal of the authors here to clarify its importance and to demonstrate a proofwhich is more accessible
to contemporary readers than the one offered by Birkhoff. |
| format | Article |
| id | doaj-art-431ee3a9321c4cfbb4bc9c1e10b43ab3 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1995-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-431ee3a9321c4cfbb4bc9c1e10b43ab32025-02-03T01:20:52ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251995-01-0118350953010.1155/S0161171295000652Differentiable semigroups are Lie groupsJohn P. Holmes0Mitch Anderson1Department of Mathematics (FAT), Auburn University, 36849-5310, Auburn, USADepartment of Mathematics, University of Hawaii at Hilo, Hilo 96720-4091, HI, USAWe present here a modern, detailed proof to the following theorem which was introduced by Garrett Birkhoff [1] in 1938. If S is a local semigroup with neighborhood of 1 homeomorphic to a Banach space and with multiplication strongly differentiable at 1, then S is a local Lie Group. Although this theorem is more than 50 years old and remains the strongest result relating to Hilbert's fifth problem in the infinite dimensional setting, it is frequently overlooked in favor of weaker results. Therefore, it is the goal of the authors here to clarify its importance and to demonstrate a proofwhich is more accessible to contemporary readers than the one offered by Birkhoff.http://dx.doi.org/10.1155/S0161171295000652Lie groupsLie algebrasstrong differentiabilitycanonical parametersCampbell-Baker-Hausdorf formula. |
| spellingShingle | John P. Holmes Mitch Anderson Differentiable semigroups are Lie groups International Journal of Mathematics and Mathematical Sciences Lie groups Lie algebras strong differentiability canonical parameters Campbell-Baker-Hausdorf formula. |
| title | Differentiable semigroups are Lie groups |
| title_full | Differentiable semigroups are Lie groups |
| title_fullStr | Differentiable semigroups are Lie groups |
| title_full_unstemmed | Differentiable semigroups are Lie groups |
| title_short | Differentiable semigroups are Lie groups |
| title_sort | differentiable semigroups are lie groups |
| topic | Lie groups Lie algebras strong differentiability canonical parameters Campbell-Baker-Hausdorf formula. |
| url | http://dx.doi.org/10.1155/S0161171295000652 |
| work_keys_str_mv | AT johnpholmes differentiablesemigroupsareliegroups AT mitchanderson differentiablesemigroupsareliegroups |