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

Commit
346f273a88d07593d4ac0493c91590223976fb0f
Parent
9608f8083486fc6fcb95a9242b96baa141e815ab
Author
Pablo <pablo-escobar@riseup.net>
Date

Changed the notation for the algebra of derivations

Diffstat

1 file changed, 12 insertions, 12 deletions

Status File Name N° Changes Insertions Deletions
Modified sections/introduction.tex 24 12 12
diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -34,23 +34,23 @@
 \end{example}
 
 \begin{example}
-  Let \(A\) be an associative \(K\)-algebra and \(\mathcal{D}_A\) be the space
-  of all derivations on \(A\) -- i.e. all linear maps \(D : A \to A\)
+  Let \(A\) be an associative \(K\)-algebra and \(\operatorname{Der}(A)\) be
+  the space of all derivations on \(A\) -- i.e. all linear maps \(D : A \to A\)
   satisfying the Leibniz rule \(D(a \cdot b) = a \cdot D b + (D a) \cdot b\).
-  The commutator \([D_1, D_2]\) of two derivations \(D_1, D_2 \in
-  \mathcal{D}_A\) in the ring \(\operatorname{End}(A)\) of \(K\)-linear
-  endomorphisms of \(A\) is, once again, a derivation. Hence \(\mathcal{D}_A\)
-  is a Lie algebra.
+  The commutator \([D, D']\) of two derivations \(D, D' \in
+  \operatorname{Der}(A)\) in the ring \(\operatorname{End}(A)\) of \(K\)-linear
+  endomorphisms of \(A\) is a derivation. Hence \(\operatorname{Der}(A)\) is a
+  Lie algebra.
 \end{example}
 
 \begin{example}
   Given a smooth manifold \(M\), the space \(\mathfrak{X}(M)\) of all smooth
-  vector fields is canonically identifyed with the
-  \(\mathcal{D}_{C^\infty(M)}\) -- where a field \(X \in \mathfrak{X}(M)\) is
-  identified with the map \(C^\infty(M) \to C^\infty(M)\) which takes a
-  function \(f \in C^\infty(M)\) to its derivative in the direction of \(X\).
-  This gives \(\mathfrak{X}(M)\) the structure of a Lie algebra over
-  \(\mathbb{R}\).
+  vector fields is canonically identifyed with
+  \(\operatorname{Der}(C^\infty(M))\) -- where a field \(X \in
+  \mathfrak{X}(M)\) is identified with the map \(C^\infty(M) \to C^\infty(M)\)
+  which takes a function \(f \in C^\infty(M)\) to its derivative in the
+  direction of \(X\). This gives \(\mathfrak{X}(M)\) the structure of a Lie
+  algebra over \(\mathbb{R}\).
 \end{example}
 
 \begin{example}