Typed a solution to Problem 25.

.gitignore

@@ -5,4 +5,8 @@
 *.log
 *.out
 *.run.xml
+*.sage
+*.sage.py
+*.scmd
+*.sout
 *.synctex.gz

sheet_05/exercise_25.tex

@@ -1 +1,137 @@
-\addtocounter{section}{1}
+\section{}
+
+The relations
+\[
+  [h,e] = 2e \,,
+  \qquad
+  [h,f] = -2f \,,
+  \qquad
+  [e,f] = h
+\]
+can be regarded as \emph{rewritting rules}
+\[
+  hf \to fh - 2f \,,
+  \qquad
+  eh \to he - 2e \,,
+  \qquad
+  ef \to fe + h \,.
+\]
+To write a monomial~$e^l h^m f^n$ as a linear combination of the given basis monomials we repeatedly apply this rewritting rules and expand the resulting expressions if needed.
+This is best done using computers.
+We give two solutions for doing so:
+
+The first solution uses \texttt{python} together with the multi-purpose \texttt{sympy} package to do the necessary work:
+
+\begin{pythoncode}
+from sympy import *
+
+e, h, f = symbols('e h f', commutative=False)
+
+def sl2expand(term):
+  while True:
+    newterm = term
+    newterm = newterm.subs(h*f, f*h - 2*f)
+    newterm = newterm.subs(e*h, h*e - 2*e)
+    newterm = newterm.subs(e*f, f*e + h)
+    newterm = expand(newterm)
+    if newterm == term:
+      break
+    else:
+      term = newterm
+  return term
+\end{pythoncode}
+We get from this program the following results:
+\begin{consoleoutput}
+>>> sl2expand(e * h * f)
+-4*f*e + f*h*e - 2*h + h**2
+>>> sl2expand(e**2 * h**2 * f**2)
+-288*f*e + 240*f*h*e - 56*f*h**2*e + 4*f*h**3*e + 64*f**2*e**2 - 16*f**2*h*e**2 + f**2*h**2*e**2 - 32*h + 48*h**2 - 18*h**3 + 2*h**4
+>>> sl2expand(h * f**3)
+-6*f**3 + f**3*h
+>>> sl2expand(h**3 * f)
+-8*f + 12*f*h - 6*f*h**2 + f*h**3
+\end{consoleoutput}
+Hence
+\begin{align*}
+  ehf
+  ={}&
+  - 4 f e + f h e - 2 h + h^2 \,,
+  \\
+  e^2 h^2 f^2
+  ={}&
+  -288 f e + 240 f h e - 56 f h^2 e + 4 f h^3 e + 64 f^2 e^2
+  \\
+  {}&
+  - 16 f^2 h e^2 + f^2 h^2 e^2 - 32 h + 48 h^2 - 18 h^3 + 2 h^4 \,,
+  \\
+  h f^3
+  ={}&
+  -6 f^3 + f^3 h \,,
+  \\
+  h^3 f
+  ={}&
+  -8 f + 12 f h - 6 f h^2 + f  h^3 \,.
+\end{align*}
+
+Another software solution is provided by \texttt{sage}, a powerful computer algebra system.
+Lie algebras and their PBW bases are already implemented and an explicit explanation can be found in \cite{sage_pbw}.
+
+% After a first compiling we get a *.sagetex.sage file, on which we have to run sage.
+% Then we need to compile the document again.
+\begin{sagecommandline}
+sage: g = lie_algebras.three_dimensional_by_rank(QQ, 3, names=['E','F','H'])
+
+sage: def sort_key(x):                                       
+....:   if x == 'F':
+....:     return 0
+....:   if x == 'H':
+....:     return 1
+....:   if x == 'E':
+....:     return 2
+
+
+sage: pbw = g.pbw_basis(basis_key=sort_key)
+sage: E,F,H = pbw.algebra_generators()
+
+An easy test:
+sage: H * E - E * H
+sage: H * F - F * H
+sage: E * F - F * E
+
+Now the real deal:
+sage: E * H * F
+sage: E^2 * H^2 * F^2
+sage: H * F^3
+sage: H^3 * F
+\end{sagecommandline}
+(The above output in the lines 12,~14,~16,~18,~20,~22,~24 is in fact autogenerated by \texttt{sage} when compiling this document.)
+
+We want to point out that~$h f^3$ can also be computed smartly, and more generally~$h f^n$ for every~$n \geq 0$:
+If~$A$ is any~{\algebra{$k$}} and~$a \in A$ then the map~$[a,-] \colon A \to A$ is a derivation of~$A$, i.e.
+\[
+  [a, xy]
+  =
+  [a,x] y + x [a,y] \,.
+\]
+Hence
+\[
+  [h, f^n]
+  =
+  [h, f f \dotsm f]
+  =
+    [h,f] f \dotsm f
+  + f [h, f] \dotsm f
+  + \dotsb
+  + f f \dotsm [h,f] \,.
+\]
+With~$[h,f] = -2f$ we find that
+\[
+  [h, f^n]
+  =
+  -2 n f^n \,.
+\]
+With~$[h, f^n] = h f^n - f^n h$ we find by rearranging that
+\[
+  h f^n = f^n h - 2 n f^n \,.
+\]
+

sheet_05/references.bib

@@ -9,3 +9,12 @@ @book{noncommutative_noetherian
   ISBN = {978-0-8218-2169-5},
   ISSN = {1065-7339}
 }
+
+@online{sage_pbw,
+  TITLE = {The Poincaré--Birkhoff--Witt Basis For A Universal Enveloping Algebra},
+  AUTHOR = {Scrimshaw, Travis},
+  DATE = {2013-11-03},
+  URL = {http://doc.sagemath.org/html/en/reference/algebras/sage/algebras/lie_algebras/poincare_birkhoff_witt.html},
+  URLDATE = {2019-05-23},
+  LABEL = {Sag}
+}

sheet_05/specificstyle.sty

@@ -4,4 +4,32 @@
 \DeclareFieldFormat{postnote}{#1}
 \DeclareFieldFormat{multipostnote}{#1}
 
+\usepackage{listings}
+\lstnewenvironment{pythoncode}{
+  \lstset{
+    numbers           = left,
+    stepnumber        = 1,
+    numbersep         = 5pt,
+    language          = Python,
+    basicstyle        = \footnotesize\listingsfont,
+    showstringspaces  = false,
+    frame             = single,
+    breaklines        = true
+  }
+}{}
+\lstnewenvironment{consoleoutput}{
+  \lstset{
+    basicstyle        = \footnotesize\listingsfont,
+    showstringspaces  = false,
+    frame             = single,
+    breaklines        = true
+  }
+}{}
+
+\usepackage{sagetex}
+
+\usepackage{fontspec}
+\newfontfamily\listingsfont{DejaVu Sans Mono}
+\setmonofont[Scale=MatchLowercase]{DejaVu Sans Mono}
+
 \DeclareMathOperator{\irr}{V}