- Commit
- 346f273a88d07593d4ac0493c91590223976fb0f
- Parent
- 9608f8083486fc6fcb95a9242b96baa141e815ab
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Changed the notation for the algebra of derivations
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Changed the notation for the algebra of derivations
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}