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The Unapologetic Mathematician
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Mathematics for the interested outsider
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The Unapologetic Mathematician
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2025-12-24 15:16:48
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The Unapologetic Mathematician The Unapologetic Mathematician Mathematics for the interested outsider Home About this weblog The Submodule of Invariants If is a module of a Lie algebra , there is one submodule that turns out to be rather interesting: the submodule of vectors such that for all . We call these vectors “invariants” of . As an illustration of how interesting these are, consider the modules we looked at last time . What are the invariant linear maps from one module to another ? We consider the action of on a linear map : Or, in other words: That is, a linear map is invariant if and only if it intertwines the actions on and . That is, . Next, consider the bilinear forms on . Here we calculate That is, a bilinear form is invariant if and only if it is associative, in the sense that the Killing form is: September 21, 2012 Posted by John Armstrong | Algebra , Lie Algebras , R...
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