# cionx/representation-theory-1-notes-ss-15

cionx committed Apr 15, 2019
 @@ -132,7 +132,7 @@ \setlist[enumerate, 1]{leftmargin=*, align=left, label=\arabic*)} \setlist[enumerate, 2]{leftmargin=*, align=left, label=\alph*)} \setlist[description, 1]{leftmargin=\labelwidth, font=\normalfont\itshape} \setlist[description, 2]{leftmargin=\labelwidth} \setlist[description, 2]{leftmargin=\labelwidth, font=\normalfont\itshape} \setlist[itemize, 1]{leftmargin=*} \setlist[itemize, 2]{leftmargin=*,label={\textopenbullet}} @@ -150,6 +150,7 @@ \newcommand{\adsemisimple}{$\ad$\nobreakdash-semisimple} \newcommand{\algebra}[1]{#1\nobreakdash-algebra} \newcommand{\algebras}[1]{#1\nobreakdash-algebras} \newcommand{\basis}[1]{#1\nobreakdash-basis} \newcommand{\bilinear}[1]{#1\nobreakdash-bilinear} \newcommand{\derivation}[1]{#1\nobreakdash-derivation} \newcommand{\dimensional}[1]{#1\nobreakdash-dimensional} @@ -173,6 +174,7 @@ \newcommand{\subalgebra}[1]{#1\nobreakdash-subalgebra} \newcommand{\subrepresentation}[1]{#1\nobreakdash-subrepresentation} \newcommand{\subrepresentations}[1]{#1\nobreakdash-subrepresentation} \newcommand{\threedimensional}{three\nobreakdash-dimensional} \newcommand{\twococycle}{$2$\nobreakdash-cocycle} \newcommand{\twococycles}{$2$\nobreakdash-cocycles} \newcommand{\twodimensional}{two\nobreakdash-dimensional}
 @@ -1,4 +1,5 @@ \glsxtrnewsymbol[description={adjoint representation}]{adjoint representation}{\ensuremath{\ad}} \glsxtrnewsymbol[description={associated graded algebra of~$\glie$}]{associated graded}{\ensuremath{\gr(A)}} \glsxtrnewsymbol[description={the space of bilinear forms~$V \times W \to \kf$}]{bilinear forms}{\ensuremath{\BF(V,W)}} \glsxtrnewsymbol[description={center of~$\glie$}]{center}{\ensuremath{\centerlie(\glie)}} \glsxtrnewsymbol[description={$i$-th term of the central series of~$\glie$}]{central series}{\ensuremath{\glie^i}} @@ -22,6 +23,7 @@ \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={ordered monomial in~$(x_i)_{i \in I}$}]{ordered monomial}{\ensuremath{x_\alpha}} \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}}
 @@ -40,6 +40,38 @@ @online{associated_generated LABEL = {MO} } @book {trees, TITLE = {Trees}, AUTHOR = {Serre, Jean-Pierre}, ORIGLANGUAGE = {french}, TRANSLATOR = {Stillwell, John}, YEAR = 1980, PAGES = {ix+142}, PUBLISHER = {Springer-Verlag Berlin Heidelberg}, DOI = {10.1007/978-3-642-61856-7} } @article {diamond_lemma, TITLE = {The Diamond Lemma for Ring Theory}, AUTHOR = {Bergman, George M.}, Journal = {Advances in Mathematics}, VOLUME = {29}, NUMBER = {2}, PAGES = {178--218}, YEAR = {1978}, DOI = {10.1016/0001-8708(78)90010-5}} } @online {pbw_deformation, TITLE = {Nice proofs of the Poincar\'{e}--Birkhoff--Witt theorem}, AUTHOR = {user25309}, HOWPUBLISHED = {MathOverflow}, DATE = {2017-05-07}, URL = {https://mathoverflow.net/q/269192}, URLDATE = {2019-04-15}, LABEL = {MO} } @book {Lectures_on_sl2_modules, AUTHOR = {Mazorchuk, Volodymyr}, TITLE = {Lectures on {$\mathfrak{sl}_2(\mathbb{C})$}-modules},