lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: 0c9359752bfc2d4f88353db8b8b68c23e355fbdc
Parent: 95cb9df40af61351c4f7f8587a44e46fffca4dca
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 16 Nov 2022 22:23:39 +0000

Edited the example of the Lie algebra of an algebraic group

Diffstat

1 file changed, 12 insertions, 7 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	19	12	7

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -65,15 +65,20 @@
   T_1 G\). In particular, \(\mathfrak{g}\) is finite-dimensional.
 \end{example}
 
+% TODOO: Is it worth discussing the non-affine case? If you can't even get an
+% equivalence of categories why should we introduce so much complexity?
 \begin{example}
-  Let \(G\) be an algebraic \(K\)-group and \(K[G]\) denote the ring of regular
-  functions \(G \to K\). We call a derivation \(D : K[G] \to K[G]\) left
+  Let \(G\) be an algebraic \(K\)-group and \(K[G]\) denote the coordinate ring
+  of \(G\) -- i.e. \(K[G] = \mathcal{O}(G)\) is the ring of global sections of
+  the structure sheaf of \(G\). We call a derivation \(D : K[G] \to K[G]\) left
   invariant if \(D(g \cdot f) = g \cdot D f\) for all \(g \in G\) and \(f \in
-  K[G]\) -- where the action of \(G\) in \(K[G]\) given by \((g \cdot f)(h) =
-  f(g^{-1} h)\). The commutator of left invariant derivations is invariant too,
-  so the space \(\operatorname{Lie}(G)\) of invariant derivations in \(K[G]\)
-  has the structure of a Lie algebra over \(K\) with brackets given by the
-  commutator of derivations. Again, \(\operatorname{Lie}(G) \cong T_1 G\) is
+  K[G]\) -- where the action of \(g \in G\) in \(K[G]\) given by the morphism
+  of sheafs \(\ell_{g^{-1}}^\sharp : \mathcal{O} \to \mathcal{O}\). The
+  commutator of left invariant derivations is invariant too, so the space
+  \(\operatorname{Lie}(G) = \operatorname{Der}(G)^G\) of invariant derivations
+  in \(K[G]\) has the structure of a Lie algebra over \(K\) with brackets given
+  by the commutator of derivations. Again, \(\operatorname{Lie}(G)\) is
+  isomorphic to the Zariski tangent space \(T_1 G\), which is
   finite-dimensional.
 \end{example}