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Took care of some TODOs.

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cionx committed Apr 28, 2019
1 parent e8e353e commit 2c28f940f6d7177ab502a340b0a1bc093d561431
@@ -51,7 +51,7 @@
%%% FONTS
\usepackage[silent]{fontspec}
\usepackage[warnings-off={mathtools-colon,mathtools-overbracket}]{unicode-math} % has to be loaded after mathtools and amssymb
\setmainfont[RawFeature={+calt}, Numbers=Lining]{EB Garamond}
\setmainfont[Numbers=Lining]{EB Garamond}
\setmathfont{TeX Gyre Termes Math}
\setmathfont[range=\mathup/{num,latin,Latin},Numbers=Lining]{EBGaramond12-Regular}
\setmathfont[range=\mathit/{latin,Latin}]{EBGaramond12-Italic}
@@ -173,6 +173,7 @@
\newcommand{\liealgebra}[1]{#1\nobreakdash-Lie~algebra}
\newcommand{\liealgebras}[1]{#1\nobreakdash-Lie~algebras}
\newcommand{\linear}[1]{#1\nobreakdash-linear}
\newcommand{\matrices}[1]{#1\nobreakdash-matrices}
\newcommand{\many}[1]{#1\nobreakdash-many}
\newcommand{\module}[1]{#1\nobreakdash-module}
\newcommand{\modules}[1]{#1\nobreakdash-modules}
@@ -272,6 +273,7 @@
}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\sign}{sgn}
\DeclareMathOperator{\tr}{tr}

% relations and operators
@@ -131,7 +131,7 @@ \subsection{Concrete Version of the Poincar\'{e}--Birkhoff--Witt Theorem}


\begin{example}
Once the PBW~theorem is proven we can give a counterexample for \cref{generators of associated graded}:
Once the PBW~theorem is proven we can give a counterexample to part~\ref*{generators of associated graded part} of \cref{generators of associated graded}:
Let~$\glie = \sllie_2(\kf)$ and let~$A \defined \Univ(\glie)$ with the standard filtration induced from the standard grading of the tensor algebra~$\Tensor(\glie)$.
We already know that~$\Univ(\glie)$ is generated as a~{\algebra{$\kf$}} by the elements~$e$,~$h$~and~$f$.
It follows from the relation~$[e,f] = h$ that~$A$ is already generated by the two elements~$e$~and~$f$.
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