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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeAug 7th 2016
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThere is a stub [[adjoint representation]] which in my opinion should be the same entry as [[adjoint action]], hence should be merged. Words representation and action are in general equivalent; to each action $G\times M\to M$ one assigns a representation $G\to End M$ and viceversa (up to nicetess of inner hom spaces etc.). True, the specialists in Lie theory like to prefer calling representation when they have a **linear representation** but their own textbooks start with nonlinear case. Thus action of a Lie group on a Lie group is nonlinear hance usually the action terminology used while on a Lie algebra more often the representation is used, but it is not a rule, and the distinction does not survive in generalizations (like the Hopf algebra); any sensible entry, as the main entry "adjoint action" should relate th nonlinear case and its linearization hence should not be in separate entries. I wanted to write some references for adjoint action for quantum groups but gave up as I do not know into which of the entries and expect a decision on fiture fate of the two entries first.

   There is a stub adjoint representation which in my opinion should be the same entry as adjoint action, hence should be merged. Words representation and action are in general equivalent; to each action $G\times M\to M$ one assigns a representation $G\to End M$ and viceversa (up to nicetess of inner hom spaces etc.). True, the specialists in Lie theory like to prefer calling representation when they have a linear representation but their own textbooks start with nonlinear case. Thus action of a Lie group on a Lie group is nonlinear hance usually the action terminology used while on a Lie algebra more often the representation is used, but it is not a rule, and the distinction does not survive in generalizations (like the Hopf algebra); any sensible entry, as the main entry “adjoint action” should relate th nonlinear case and its linearization hence should not be in separate entries. I wanted to write some references for adjoint action for quantum groups but gave up as I do not know into which of the entries and expect a decision on fiture fate of the two entries first.