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Added the basics on filtered and graded algebras.
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cionx committed Apr 13, 2019
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72 changes: 40 additions & 32 deletions generalstyle.sty
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Expand Up @@ -83,10 +83,12 @@
\chead{}
\ohead{\rightmark}

% more space for page numbers in the toc, default is 1.55
\renewcommand{\@pnumwidth}{2em}

% dont number subsubsections
\setcounter{secnumdepth}{1}


% colours for hyperref
\hypersetup{
colorlinks,
Expand All @@ -105,11 +107,13 @@
\theoremstyle{definition}
\newtheorem{claim}{Claim}[everything] % numbered version, numbering is only local
\newtheorem*{claim*}{Claim} % unnumbered version
\newtheorem{construction}[everything]{Construction}
\newtheorem{convention}[everything]{Convention}
\newtheorem{corollary}[everything]{Corollary}
\newtheorem{definition}[everything]{Definition}
\newtheorem{example}[everything]{Example}
\newtheorem{examples}[everything]{Examples}
\newtheorem{fluff}[everything]{}
\newtheorem{lemma}[everything]{Lemma}
\newtheorem{proposition}[everything]{Proposition}
\newtheorem{recall}[everything]{Recall}
Expand All @@ -130,8 +134,7 @@
\setlist[description, 1]{leftmargin=\labelwidth, font=\normalfont\itshape}
\setlist[description, 2]{leftmargin=\labelwidth}
\setlist[itemize, 1]{leftmargin=*}
\setlist[itemize, 2]{leftmargin=*, label={\textopenbullet}}
\setlist[itemize, 3]{leftmargin=*}
\setlist[itemize, 2]{leftmargin=*,label={\textopenbullet}}

% different kinds of lists
\newlist{equivalenceslist}{enumerate}{2}
Expand All @@ -143,40 +146,43 @@
\newcommand{\defemph}{\emph}

% text shortcuts, mostly involving nobreakdash
\newcommand{\adnilpotent}{$\ad$\nobreakdash-nil\-po\-tent}
\newcommand{\adsemisimple}{$\ad$\nobreakdash-semi\-sim\-ple}
\newcommand{\algebra}[1]{#1\nobreakdash-al\-ge\-bra}
\newcommand{\algebras}[1]{#1\nobreakdash-al\-ge\-bras}
\newcommand{\bilinear}[1]{#1\nobreakdash-bi\-lin\-ear}
\newcommand{\dimensional}[1]{#1\nobreakdash-di\-men\-sion\-al}
\newcommand{\adnilpotent}{$\ad$\nobreakdash-nilpotent}
\newcommand{\adsemisimple}{$\ad$\nobreakdash-semisimple}
\newcommand{\algebra}[1]{#1\nobreakdash-algebra}
\newcommand{\algebras}[1]{#1\nobreakdash-algebras}
\newcommand{\bilinear}[1]{#1\nobreakdash-bilinear}
\newcommand{\derivation}[1]{#1\nobreakdash-derivation}
\newcommand{\dimensional}[1]{#1\nobreakdash-dimensional}
\newcommand{\fold}[1]{#1\nobreakdash-fold}
\newcommand{\graded}[1]{#1\nobreakdash-graded}
\newcommand{\grading}[1]{#1\nobreakdash-grading}
\newcommand{\gradings}[1]{#1\nobreakdash-gradings}
\newcommand{\howmanyth}[1]{#1\nobreakdash-th}
\newcommand{\invariance}[1]{#1\nobreakdash-in\-vari\-ance}
\newcommand{\invariant}[1]{#1\nobreakdash-in\-vari\-ant}
\newcommand{\liealgebra}[1]{#1\nobreakdash-Lie~al\-ge\-bra}
\newcommand{\liealgebras}[1]{#1\nobreakdash-Lie~al\-ge\-bras}
\newcommand{\linear}[1]{#1\nobreakdash-lin\-ear}
\newcommand{\invariance}[1]{#1\nobreakdash-invariance}
\newcommand{\invariant}[1]{#1\nobreakdash-invariant}
\newcommand{\liealgebra}[1]{#1\nobreakdash-Lie~algebra}
\newcommand{\liealgebras}[1]{#1\nobreakdash-Lie~algebras}
\newcommand{\linear}[1]{#1\nobreakdash-linear}
\newcommand{\many}[1]{#1\nobreakdash-many}
\newcommand{\module}[1]{#1\nobreakdash-module}
\newcommand{\modules}[1]{#1\nobreakdash-modules}
\newcommand{\onedimensional}{one\nobreakdash-di\-men\-sion\-al}
\newcommand{\onedimensional}{one\nobreakdash-dimensional}
\newcommand{\onetoone}{one\nobreakdash-to\nobreakdash-one}
\newcommand{\representation}[1]{#1\nobreakdash-rep\-re\-sen\-ta\-tion}
\newcommand{\representations}[1]{#1\nobreakdash-rep\-re\-sen\-ta\-tions}
\newcommand{\subalgebra}[1]{#1\nobreakdash-sub\-al\-ge\-bra}
\newcommand{\subrepresentation}[1]{#1\nobreakdash-sub\-rep\-re\-sen\-ta\-tion}
\newcommand{\subrepresentations}[1]{#1\nobreakdash-sub\-rep\-re\-sen\-ta\-tion}
\newcommand{\twococycle}{$2$\nobreakdash-co\-cy\-cle}
\newcommand{\twococycles}{$2$\nobreakdash-co\-cy\-cles}
\newcommand{\twodimensional}{two\nobreakdash-di\-men\-sion\-al}
\newcommand{\representation}[1]{#1\nobreakdash-representation}
\newcommand{\representations}[1]{#1\nobreakdash-representations}
\newcommand{\subalgebra}[1]{#1\nobreakdash-subalgebra}
\newcommand{\subrepresentation}[1]{#1\nobreakdash-subrepresentation}
\newcommand{\subrepresentations}[1]{#1\nobreakdash-subrepresentation}
\newcommand{\twococycle}{$2$\nobreakdash-cocycle}
\newcommand{\twococycles}{$2$\nobreakdash-cocycles}
\newcommand{\twodimensional}{two\nobreakdash-dimensional}
\newcommand{\ua}{universal enveloping algebra}
\newcommand{\uas}{universal enveloping algebras}
\newcommand{\vectorspace}[1]{#1\nobreakdash-vec\-tor space}
\newcommand{\vectorspaces}[1]{#1\nobreakdash-vec\-tor spaces}
\newcommand{\vectorspace}[1]{#1\nobreakdash-vector space}
\newcommand{\vectorspaces}[1]{#1\nobreakdash-vector spaces}
\newcommand{\weightspace}[1]{#1\nobreakdash-weight space}
\newcommand{\weightvector}[1]{#1\nobreakdash-weight vector}

\newcommand{\kspan}{{\vspan}{}_\kf} % I will go to hell for this


%%%%% MATH STUFF

Expand Down Expand Up @@ -207,15 +213,17 @@
\newcommand{\catname}[1]{\mathbf{#1}}
\newcommand{\cAlg}[1]{#1\text{-}\catname{Alg}}
\newcommand{\cCAlg}[1]{#1\text{-}\catname{CAlg}}
\newcommand{\cFilt}[1]{#1\text{-}\catname{Filt}}
\newcommand{\cGrad}[1]{#1\text{-}\catname{Grad}}
\newcommand{\cfAlg}[1]{#1\text{-}\catname{fAlg}}
\newcommand{\cgAlg}[1]{#1\text{-}\catname{gAlg}}
\newcommand{\cLie}[1]{#1\text{-}\catname{Lie}}
\newcommand{\cMod}[1]{#1\text{-}\catname{Mod}}
\newcommand{\cMon}{\catname{Mon}}
\newcommand{\cRep}[1]{#1\text{-}\catname{Rep}}
\newcommand{\cSet}{\catname{Set}}
\newcommand{\cVect}[1]{#1\text{-}\catname{Vect}}

% spaces and sets
\newcommand{\Aff}{\mathbb{A}}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\BF}{BF}
\DeclareMathOperator{\centerlie}{Z}
Expand All @@ -230,16 +238,16 @@
\DeclareMathOperator{\Symm}{S}
\DeclareMathOperator{\Tensor}{T}
\DeclareMathOperator{\Univ}{U}
\DeclareMathOperator{\vspan}{span}

% functions
\DeclareMathOperator{\ad}{ad}
\DeclareMathOperator{\can}{can}
\DeclareMathOperator{\card}{card}
\DeclareMathOperator{\ringchar}{char}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\flt}{flt}
\DeclareMathOperator{\gr}{gr}
\NewDocumentCommand{\gr}{O{}}{
{\operatorname{gr}}{}_{#1} % hell again
}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\tr}{tr}
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7 changes: 6 additions & 1 deletion glossarylist.tex
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Expand Up @@ -7,18 +7,22 @@
\glsxtrnewsymbol[description={commutator of~$X$ and~$Y$}]{commutator space}{\ensuremath{[X,Y]}}
\glsxtrnewsymbol[description={space of derivations of~$A$}]{derivations}{\ensuremath{\Der(A)}}
\glsxtrnewsymbol[description={$i$-th term of the derived series of~$\glie$}]{derived series}{\ensuremath{\glie^{(i)}}}
\glsxtrnewsymbol[description={exterior algebra of~$V$}]{exterior algebra}{\ensuremath{\Exterior(V)}}
\glsxtrnewsymbol[description={exterior algebra of~$V$}]{exterior algebra}{\ensuremath{\bigwedge(V)}}
\glsxtrnewsymbol[description={category of filtered~{\algebras{$\kf$}}}]{filtered algebras}{\ensuremath{\cfAlg{\kf}}}
\glsxtrnewsymbol[description={general linear Lie algebra}]{general lie matrix}{\ensuremath{\gllie_n(\kf)}}
\glsxtrnewsymbol[description={general linear Lie algebra of~$V$}]{general lie endomorphism}{\ensuremath{\gllie(V)}}
\glsxtrnewsymbol[description={category of graded~{\algebras{$\kf$}}}]{graded algebras}{\ensuremath{\cgAlg{\kf}}}
\glsxtrnewsymbol[description={interal semidirect product}]{internal semidirect product}{\ensuremath{\hlie \ltimes I}}
\glsxtrnewsymbol[description={space of invariants for a representation~$V$ of~$\glie$}]{invariants}{\ensuremath{V^{\glie}}}
\glsxtrnewsymbol[description={Killing form}]{killing form}{\ensuremath{\kappa}}
\glsxtrnewsymbol[description={Lie bracket}]{lie bracket}{\ensuremath{[-,-]}}
\glsxtrnewsymbol[description={Lie ideal}]{lie ideal}{\ensuremath{\ideal}}
\glsxtrnewsymbol[description={loop Lie~algebra of~$\glie$}]{loop lie algebra}{\ensuremath{\looplie(\glie)}}
\glsxtrnewsymbol[description={category of left~\modules{$A$}}]{module category}{\ensuremath{\cMod{A}}}
\glsxtrnewsymbol[description={monoid algabra of~$M$}]{monoid algebra}{\ensuremath{\kf[M]}}
\glsxtrnewsymbol[description={nilpotent part of~$x$}]{nilpotent part}{\ensuremath{x_n}}
\glsxtrnewsymbol[description={normalizer of~$U$ in~$\glie$}]{normalizer}{\ensuremath{\normallie_{\glie}(U)}}
\glsxtrnewsymbol[description={Ore extension}]{ore extension}{\ensuremath{R[t;\sigma,\delta]}}
\glsxtrnewsymbol[description={product of~$\glie$ and~$\hlie$}]{product of lie algebras}{\ensuremath{\glie \times \hlie}}
\glsxtrnewsymbol[description={quotient Lie~algebra of~$\glie$ by~$I$}]{quotient lie algebra}{\ensuremath{\glie/I}}
\glsxtrnewsymbol[description={quotient representation of~$V$ by~$U$}]{quotient representation}{\ensuremath{V/U}}
Expand All @@ -29,6 +33,7 @@
\glsxtrnewsymbol[description={endomorphism algebra of a representation~$V$}]{rep endo}{\ensuremath{\End_{\glie}(V)}}
\glsxtrnewsymbol[description={semidirect product of~$\hlie$ by~$I$ over~$\theta$}]{semidirect product}{\ensuremath{\hlie \ltimes_\theta I}}
\glsxtrnewsymbol[description={semisimple part of~$x$}]{semisimple part}{\ensuremath{x_s}}
\glsxtrnewsymbol[description={skew polynomial algebra}]{skew polynomial algebra}{\ensuremath{R[t;\delta]}}
\glsxtrnewsymbol[description={special linear Lie algebra}]{special lie matrix}{\ensuremath{\sllie_n(\kf)}}
\glsxtrnewsymbol[description={special linear Lie algebra of~$V$}]{special lie endomorphism}{\ensuremath{\sllie_n(V)}}
\glsxtrnewsymbol[description={standard basis vector of~$\sllie_2(\kf)$}]{standard basis e}{\ensuremath{e}}
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34 changes: 27 additions & 7 deletions references.bib
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Expand Up @@ -11,13 +11,33 @@ @book {Humphreys
}

@online {semidirect_jacobi,
TITLE = {Semi-direct product Lie algebra},
AUTHOR = {Dietrich Burde},
HOWPUBLISHED = {Mathematics Stack Exchange},
DATE = {2016-03-05},
URL = {https://math.stackexchange.com/q/1684527},
URLDATE = {2019-04-08},
LABEL = {MS}
TITLE = {Semi-direct product Lie algebra},
AUTHOR = {Dietrich Burde},
HOWPUBLISHED = {Mathematics Stack Exchange},
DATE = {2016-03-05},
URL = {https://math.stackexchange.com/q/1684527},
URLDATE = {2019-04-08},
LABEL = {MS}
}

@book {bourbaki_algebra,
TITLE = {Algebra~I},
SUBTITLE = {Chapters~1--3},
SERIES = {Elements of Mathematics},
AUTHOR = {Bourbaki, Nicolas},
PUBLISHER = {Springer-Verlag Berlin Heidelberg New York},
YEAR = {1989},
EDITON = {2nd printing}
}

@online{associated_generated,
TITLE = {Generators of associated graded algebra},
AUTHOR = {Lewis Topley},
HOWPUBLISHED = {MathOverflow},
DATE = {2015-11-24},
URL = {https://mathoverflow.net/q/224454},
URLDATE = {2019-04-12},
LABEL = {MO}
}

@book {Lectures_on_sl2_modules,
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32 changes: 30 additions & 2 deletions sections/extensions.tex
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Expand Up @@ -401,7 +401,21 @@ \subsubsection{Central Extensions}
We have overall constructed for all Lie~algebras~$\hlie$ and~$I$ a {\onetoone} correspondence between
\begin{itemize}
\item
Lie~brackets~$[-,-]$ on~$\hlie \oplus I$ that makes the standard short exact sequence~$0 \to I \to \hlie \oplus I \to \hlie \to 0$ into a central extension of~$\hlie$ by~$I$ and
Lie~brackets~$[-,-]$ on the vector space~$\hlie \oplus I$ that make the standard short exact sequence
\[
0
\to
I
\to
\hlie
\oplus
I
\to
\hlie
\to
0
\]
into a central extension of~$\hlie$ by~$I$ and
\item
{\twococycles}~$\kappa \colon \hlie \times \hlie \to I$,
\end{itemize}
Expand Down Expand Up @@ -688,7 +702,21 @@ \subsubsection{Split Extensions and Semidirect Products}
We have now overall constructed for all Lie~algebras~$\hlie$ and~$I$ a {\onetoone} correspondence between
\begin{itemize}
\item
Lie~brackets~$[-,-]$ on~$\hlie \oplus I$ that makes the standard short exact sequence~$0 \to I \to \hlie \oplus I \to \hlie \to 0$ into a split extension of~$\hlie$ by~$I$ and
Lie~brackets~$[-,-]$ on the vector space~$\hlie \oplus I$ that makes the standard short exact sequence
\[
0
\to
I
\to
\hlie
\oplus
I
\to
\hlie
\to
0
\]
into a split extension of~$\hlie$ by~$I$ and
\item
Lie~algebra homomorphisms~$\theta \colon \hlie \to \Der(I)$,
\end{itemize}
Expand Down

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