- Commit
- 36b566c9870c636181c2723f27521cc9e48135ac
- Parent
- e2ec70dd6089a6518c1c393efb9b33b1bf4843ec
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Minor improvement in the notation of a proof
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Minor improvement in the notation of a proof
1 file changed, 16 insertions, 16 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Modified | sections/introduction.tex | 32 | 16 | 16 |
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -554,37 +554,37 @@ containing \(\mathfrak{g}\) as a Lie subalgebra. In practice this means\dots \begin{proof} Let \(f : \mathfrak{g} \to A\) be a homomorphism of Lie algebras. By the - universal property of free algebras, there is a homomorphism of algebras \(g - : T \mathfrak{g} \to A\) such that + universal property of free algebras, there is a homomorphism of algebras + \(\tilde f : T \mathfrak{g} \to A\) such that \begin{center} \begin{tikzcd} - T \mathfrak{g} \arrow[dotted]{dr}{g} & \\ + T \mathfrak{g} \arrow[dotted]{dr}{\tilde f} & \\ \mathfrak{g} \uar \rar[swap]{f} & A \end{tikzcd} \end{center} Since \(f\) is a homomorphism of Lie algebras, \[ - g([X, Y]) + \tilde f([X, Y]) = f([X, Y]) = [f(X), f(Y)] - = [g(X), g(Y)] - = g(X \otimes Y - Y \otimes X) + = [\tilde f(X), \tilde f(Y)] + = \tilde f(X \otimes Y - Y \otimes X) \] for all \(X, Y \in \mathfrak{g}\). Hence \(I = ([X, Y] - (X \otimes Y - Y - \otimes X) : X, Y \in \mathfrak{g}) \subset \ker g\) and therefore \(g\) - factors through the quotient \(\mathcal{U}(\mathfrak{g}) = \mfrac{T - \mathfrak{g}}{I}\). + \otimes X) : X, Y \in \mathfrak{g}) \subset \ker \tilde f\) and therefore + \(\tilde f\) factors through the quotient \(\mathcal{U}(\mathfrak{g}) = + \mfrac{T \mathfrak{g}}{I}\). \begin{center} \begin{tikzcd} - T \mathfrak{g} \rar{g} \dar & A \\ - \mathcal{U}(\mathfrak{g}) \arrow[swap, dotted]{ur}{\bar{g}} & + T \mathfrak{g} \rar{\tilde f} \dar & A \\ + \mathcal{U}(\mathfrak{g}) \arrow[swap, dotted]{ur}{\bar{\tilde f}} & \end{tikzcd} \end{center} - Combining the two previous diagrams, we can see that \(\bar{g}\) is indeed an - extension of \(f\). The uniqueness of the extension then follows from the - uniqueness of \(g\) and \(\bar{g}\). + Combining the two previous diagrams, we can see that \(\bar{\tilde f}\) is + indeed an extension of \(f\). The uniqueness of the extension then follows + from the uniqueness of \(\tilde f\) and \(\bar{\tilde f}\). \end{proof} We should point out this construction is functorial. Indeed, if @@ -603,8 +603,8 @@ algebras \(\mathcal{U}(f) : \mathcal{U}(\mathfrak{g}) \to \end{center} It is important to note, however, that \(\mathcal{U} : K\text{-}\mathbf{LieAlg} -\to K\text{-}\mathbf{Alg}\) is not the ``inverse'' of \(\operatorname{Lie} : -K\text{-}\mathbf{Alg} \to K\text{-}\mathbf{LieAlg}\). For instance, if +\to K\text{-}\mathbf{Alg}\) is not the ``inverse'' of our functor +\(K\text{-}\mathbf{Alg} \to K\text{-}\mathbf{LieAlg}\). For instance, if \(\mathfrak{g} = K\) is the \(1\)-dimensional Abelian Lie algebra then \(\mathcal{U}(\mathfrak{g}) \cong K[x]\), which is infinite-dimensional. Nevertheless, proposition~\ref{thm:universal-env-uni-prop} may be restated