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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 29752, 750]*) (*NotebookOutlinePosition[ 30883, 785]*) (* CellTagsIndexPosition[ 30839, 781]*) (*WindowFrame->Normal*) Notebook[{ Cell[BoxData[ RowBox[{ StyleBox["\[IndentingNewLine]", FontWeight->"Plain"], RowBox[{ StyleBox[\(Lecture\ 9\ - \ October\ 7, \ 2004\), FontFamily->"Times", FontSize->16, FontWeight->"Plain"], StyleBox["\[IndentingNewLine]", FontFamily->"Times", FontSize->16, FontWeight->"Plain"], StyleBox["\[IndentingNewLine]", FontFamily->"Times", FontSize->16, FontWeight->"Plain"], StyleBox[\(Prof . \ Victor\ Ka\[CHacek]\), FontFamily->"Times", FontWeight->"Plain"], StyleBox["\[IndentingNewLine]", FontFamily->"Times", FontWeight->"Plain"], StyleBox[\(Scribe : \ Yaim\ Cooper\), FontFamily->"Times", FontWeight->"Plain"], StyleBox["\[IndentingNewLine]", FontWeight->"Plain"]}]}]], "Input", PageWidth->PaperWidth, TextAlignment->Center], Cell[BoxData[ RowBox[{\(Definition . \ \ A\ Polynomial\ map\ \(f : \ \[DoubleStruckCapitalF]\^m\) \[Rule] \ \(\[DoubleStruckCapitalF]\^\(\(n\)\(\ \ \)\)\) is\ a\ map\ of\ the\ following\ form\), ",", " ", RowBox[{ RowBox[{"f", RowBox[{"(", GridBox[{ {\(x\_1\)}, {"."}, {"."}, {"."}, {\(x\_m\)} }], ")"}]}], " ", "\[Rule]", " ", RowBox[{ RowBox[{"(", GridBox[{ {\(\(P\_1\) \((\(x\_\(1. .. \)\) x\_m)\)\)}, {"."}, {"."}, {"."}, {\(\(P\_n\) \((\(x\_\(1. .. \)\) x\_m)\)\)} }], ")"}], " ", "where", " ", \(P\_\(\(i\)\(\ \)\)\), "are", " ", \(\(polynomials\)\(.\)\(\ \ \)\)}]}]}]], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ StyleBox["Exercise", FontVariations->{"Underline"->True}], StyleBox[" ", FontVariations->{"Underline"->True}], StyleBox["9.1", FontVariations->{"Underline"->True}], " ", "Let", " ", "A", " ", "be", " ", "a", " ", "nilpotent", " ", "operator", " ", "in", " ", "a", " ", "finite", " ", "dimensional", " ", "vector", " ", "space", " ", "V", " ", "over", " ", "a", " ", "field", " ", "\[DoubleStruckCapitalF]", " ", "of", " ", "charachteristic", " ", "0.", " ", "Let", " ", "Exp", \((A)\)}], "=", \(\[Sum]\+\(j = 1\)\%\[Infinity] A\^j\/\(j!\)\)}], " "}], "\[IndentingNewLine]", RowBox[{ " ", \(Show\ that\ Exp \((A)\) : V \[Rule] V\ is\ an\ invertible\ linear\ map\ with\ inverse\ Exp \(\((\(-A\))\)\ \(.\)\)\)}]}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ RowBox[{\(Solution . \ \ A\ is\ nilpotent\ so\ A\^m = \(0\ for\ some\ m \ \[Element] \[DoubleStruckCapitalN] . \[IndentingNewLine]Let\ n\ be\ the\ \ smallest\ natural\ number\ for\ which\ A\^n = 0. \ Then\)\), " "}], "\[IndentingNewLine]", RowBox[{\(Exp \((A)\)\), " ", "=", RowBox[{ StyleBox[\(\[Sum]\+\(j = 1\)\%\(n - 1\)\), FontSize->14], RowBox[{ StyleBox[\(A\^j\/\(j!\)\), FontSize->9], StyleBox[".", FontSize->14], " "}]}]}]}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(\((1)\)\ \ Exp \((A)\)\ is\ \(\(linear\)\(:\)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ RowBox[{"\t", RowBox[{ RowBox[{\(\((Exp \((A)\))\) \((\(c\_1\) v\_1 + \(c\_2\) v\_2\ )\)\), "=", RowBox[{ RowBox[{"(", StyleBox[\(\[Sum]\+\(j = 1\)\%\(n - 1\)A\^j\/\(j!\)\), FontSize->9], StyleBox[")", FontSize->9]}], \((\(c\_1\) v\_1 + \(c\_2\) v\_2\ )\)}]}], "\[IndentingNewLine]", \(\(=\)\(\[Sum]\+\(j = 1\)\%\(n - \ 1\)\(\(A\^j\) \((\(c\_1\) v\_1 + \(c\_2\) v\_2)\)\)\/\(j!\) = \[Sum]\+\(j = 1\ \)\%\(n - 1\)\((\(c\_1\) \(A\^j\) v\_1 + \(c\_2\) \(A\^j\) \ v\_2)\)\/\(j!\)\)\), "\[IndentingNewLine]", \(\(=\)\(\(c\_1\) \(\[Sum]\+\(j = 1\)\%\(n - 1\ \)\(\(A\^j\) v\_1\)\/\(j!\)\) + \(c\_2\) \(\[Sum]\+\(j = 1\)\%\(n - \ 1\)\(\(A\^j\) v\_2\)\/\(j!\)\) = \(c\_1\) Exp \((A)\) v\_1 + \(c\_2\) Exp \((A)\) \(\(v\_2\)\(.\)\(\ \ \)\)\)\)}]}]], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(\((2)\)\ \ Exp \((\(-A\))\) is\ the\ inverse\ of\ Exp \((A)\) . \ \ Note\ that\ if\ we\ show\ Exp \ \((\(-A\))\) Exp \((A)\) = I, \ \[IndentingNewLine]it\ follows\ that\ Exp \((A)\) Exp \((\(-A\))\) = \(I\ \ since\ - \((\(-A\))\) = A\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Consider\ Exp \((xA)\) Exp \((cA)\)\ and\ Exp \((\((x + c)\) A)\), \ where\ x\ and\ c\ are\ real\), "\[IndentingNewLine]", \(numbers . \ \ \ Since\ differentiation\ with\ respect\ to\ x\ gives\ \ the\ same\ result\ on\ both\ sides, \ and\ the\ expressions\ are\ equal\ when\ x\ = \ 0, \ these\ two\ expressions\ are\ equal\ \ \), "\[IndentingNewLine]", \(Using\ x\ = \(1\ and\ a = \(-1\)\ \ gives\ the\ desired\ \ \(\(result\)\(.\)\(\ \ \)\)\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox[\(Exercise\ 9.2\), FontVariations->{"Underline"->True}], ":", " ", \(If\ in\ addition\)}], ",", " ", \(V\ is\ an\ algebra\ and\ A\ is\ a\ nilpotent\ derivation\ of\ \ V\), ",", "\[IndentingNewLine]", \(then\ Exp \((A)\)\ is\ an\ automorphism\ of\ \ the\ \(\(algebra\)\(.\)\)\)}]], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Solution . \ \ We' ve\ shown\ in\ 9.1 that\ Exp \((A)\)\ is\ an\ invertivble\ linear\ map\ so\ we\ only\ have\ \ \), "\[IndentingNewLine]", \(left\ to\ show\ that\ Exp \((A)\)\ preserves\ multiplication\ in\ \(\(V\ \)\(.\)\(\ \ \)\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ RowBox[{\(First\ we\ show\ by\ induction\ that\ \(A\^j\) \((\(v\_1\) v\_2)\)\), "=", RowBox[{ RowBox[{\(\[Sum]\+\(k = 0\)\%j\), RowBox[{ RowBox[{"(", FractionBox["j", "k", FractionLine->None], ")"}], \(A\^k\), \(v\_1\), \(A\^\(j - k\)\), \(v\_2 . \ \ \[IndentingNewLine]Base\), " ", \(case : \ A \((\(v\_1\) v\_2)\)\)}]}], "=", \(\((Av\_1)\) v\_2 + \(v\_1\) \((Av\_2)\)\ by\ the\ definition\ of\ a\ \ derivation\)}]}], "\[IndentingNewLine]", RowBox[{\(Inductive\ \(step : \ \ \(A\^j\) \((\(v\_1\) v\_2)\)\)\), "=", RowBox[{"A", RowBox[{"(", RowBox[{\(\[Sum]\+\(k = 0\)\%\(j - 1\)\), RowBox[{ RowBox[{"(", FractionBox[\(j - 1\), "k", FractionLine->None], ")"}], \(A\^k\), \(v\_1\), \(A\^\(j - k - 1\)\), \(v\_2\)}]}], ")"}]}]}], "\[IndentingNewLine]", RowBox[{"=", RowBox[{ RowBox[{\(\[Sum]\+\(k = 0\)\%\(j - 1\)\), RowBox[{ RowBox[{"(", FractionBox[\(j - 1\), "k", FractionLine->None], ")"}], \((\(A\^\(k + 1\)\) \(v\_1\) \(A\^\(j - k - 1\)\) v\_2 + \(A\^k\) \(v\_1\) \(A\^\(j - k\)\) v\_2)\)}]}], "=", RowBox[{\(\[Sum]\+\(k = 0\)\%j\), RowBox[{ RowBox[{"(", FractionBox["j", "k", FractionLine->None], ")"}], \(A\^k\), \(v\_1\), \(A\^\(j - k\)\), \(v\_2\)}]}]}]}]}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ RowBox[{\(So\ Exp \((A)\) \((\(v\_1\) v\_2)\)\), "=", RowBox[{\(\((\[Sum]\+\(j = 1\)\%\(n - 1\)A\^j\/\(j!\))\) \((\(v\_1\) v\_2)\)\), "=", RowBox[{\(\[Sum]\+\(j = 1\)\%\(n - 1\)\), FractionBox[ RowBox[{\(\[Sum]\+\(k = 0\)\%j\), RowBox[{ RowBox[{"(", FractionBox["j", "k", FractionLine->None], ")"}], \(A\^k\), \(v\_1\), \(A\^\(j - k\)\), \(v\_2\)}]}], \(j!\)]}]}]}], \ "\[IndentingNewLine]", RowBox[{"=", RowBox[{ RowBox[{\(\[Sum]\+\(j = 1\)\%\(n - 1\)\), RowBox[{\((1\/\(j!\))\), RowBox[{ UnderoverscriptBox[ StyleBox["\[Sum]", FontSize->12], \(i = 0\), \(n - 1\)], \(\((1\/\(i!\))\) \(A\^j\) \(v\_1\) \(A\^i\) v\_2\)}]}]}], "=", RowBox[{\((\[Sum]\+\(j = 1\)\%\(n - 1\)\((1\/\(j!\))\) \(A\^j\) v\_1)\), RowBox[{"(", RowBox[{ UnderoverscriptBox[ StyleBox["\[Sum]", FontSize->12], \(i = 0\), \(n - 1\)], \(\((1\/\(i!\))\) \(A\^i\) v\_2\)}], ")"}]}]}]}], "\[IndentingNewLine]", \(\(=\)\(\((Exp \((A)\) v\_1)\) \(\((Exp \((A)\) v\_2)\)\(.\)\(\ \ \)\)\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(Thus\ Exp \((A)\)\ is\ an\ automorphism\ of\ \(\(V\)\(.\)\(\ \ \)\)\)], \ "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Lemma\ 3. \ \ If\ \(f : \[DoubleStruckCapitalF]\^\(\(n\)\(\ \)\)\) \ \[Rule] \ \(\(\(\(\[DoubleStruckCapitalF]\^\(\(n\)\(\ \)\)\) is\ a\ polynomial\ map\ such\ that\ df\)\( | \_\(\(a\)\(\ \ \)\)\)\) : \ \[DoubleStruckCapitalF]\^n \[Rule] \ \[DoubleStruckCapitalF]\^\(\ \(n\)\(\ \)\)\ is\ a\)\), "\[IndentingNewLine]", \(nonsingular\ linear\ operator\ for\ some\ a \[Element] \(\ \[DoubleStruckCapitalF]\^n\) then\ f \((\[DoubleStruckCapitalF]\^n)\)\ contains\ a\ nonempty\ \ Zariski\), "\[IndentingNewLine]", \(\ open\ subset\ containing\ f \((a)\) . \[IndentingNewLine]\ \((Note : \ \ this\ is\ like\ the\ implicit\ function\ theorem\ with\ Zariski\ open\ sets\ \ replacing\ open\ sets)\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(Lemma\ 4. \ \ Take\ \[GothicH]\ to\ be\ a\ Cartan\ subalgebra\ of\ \ \[GothicG]\ with\ a \[Element] \[GothicH]\ a\ regular\ element . \ \ Then\ \ \[GothicH] \[Element] \(\(\[GothicG]\_0\^a\)\(.\)\(\ \)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Proof . \ \ \[GothicH]\ is\ a\ nilpotent\ subalgebra\ so\ \ ad \ \((a)\)\( | \_\[GothicH]\)\ is\ nilpotent\ \ so\ \[GothicH]\ \[Subset] \ \[GothicG]\_0\^a\ \ But\ \ \[GothicG]\_0\^a\ is\ a\), "\[IndentingNewLine]", \(\ nilpotent\ Lie\ Algebra\ and\ \[GothicH]\ is\ a\ maximal\ nilpotent\ \ subalgebra . \ \ Thus\ \[GothicH]\ = \(\(\[GothicG]\_0\^a\)\(.\)\)\)}], \ "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Proof\ of\ Theorem . \ \ Take\ \[GothicH]\ a\ Cartan\ subalgebra\ of\ \ \[GothicG] . \ \ Then\ the\ root\ space\), "\[IndentingNewLine]", \(decomposition\ is\ given\ by\ \[GothicG] = \(\[CirclePlus]\_\(\[Alpha] \ \[Element] \(\[GothicH]\^*\)\)\ \ \[GothicG]\_\[Alpha]\), \ \[IndentingNewLine]where\ \[GothicG]\_\[Alpha]\ = \ {a \[Element] \[GothicG] \ | \(\((ad \((h)\) - \[Alpha] \((h)\))\)\^n\) a = 0\ for\ n\ sufficiently\ large}\), "\[IndentingNewLine]", \(Moreover, \ \([\[GothicG]\_\[Alpha], \ \[GothicG]\_\[Beta]]\) \[Subset] \ \[GothicG]\_\(\[Alpha] + \[Beta]\)\ \ and\ \[GothicG]\_0 = \ \(\(\[GothicH]\)\(.\)\(\ \ \)\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{\(Note\ that\ if\ a \[Element] \[GothicG]\_\[Alpha]\ and\ \ \[Alpha] \[NotEqual] 0, \ then\ ad \((a)\) is\ a\ nilpotent\ operator\), "\[IndentingNewLine]", \((since\ \(\((ad \ \((a)\))\)\^N\) \[GothicG]\_\[Beta] \[Subset] \[GothicG]\_\(\[Beta] + N\ \[Alpha]\)\ and\ \[GothicG]\_\[Alpha] \[NotEqual] 0 for\ a\ finite\ number\ of\ \[IndentingNewLine]distinct\ \ \[Alpha]' s\ since\ the\ dimension\ of\ \[GothicG]\ is\ finite)\), "\ \[IndentingNewLine]", RowBox[{\(\(Let\ b\_1 ... \) b\_n\ be\ the\ union\ of\ \ bases\ of\ \[GothicG]\_\[Alpha], \ \[GothicG]\_\[Alpha] \[NotEqual] 0. \ \ Define\ the\ polynomial\ map\), " "}], "\[IndentingNewLine]", \(f : \[GothicG] \[Rule] \[GothicG]\ \ by\), "\[IndentingNewLine]", RowBox[{\(f \((\[Sum]\+\(i = 1\)\%m\( x\_i\) b\_i + h)\)\), " ", "=", " ", RowBox[{ StyleBox["(", FontSize->18], RowBox[{"Exp", " ", RowBox[{"(", RowBox[{ RowBox[{\(\(x\_1\) \((ad \((b\_1)\))\) ... \), "Exp", RowBox[{ StyleBox["(", FontSize->14], \(\(x\_m\) \((ad \((b\_m)\))\)\), StyleBox[")", FontSize->18]}], \((h)\), "\[IndentingNewLine]", "where", " ", \(x\_i\), \(b\_i\)}], "\[Element]", \(\(\[CirclePlus]\_\(\[Alpha] \[Element] \(\ \[GothicH]\^*\)\)\ \ \[GothicG]\_\[Alpha]\ for\ \[Alpha]\) \[NotEqual] \ 0\ \ and\ h\), " ", "\[Element]", RowBox[{ "\[GothicH]", " ", "Note", " ", "that", "\[IndentingNewLine]", " ", RowBox[{ StyleBox["(", FontSize->18], RowBox[{"Exp", " ", RowBox[{"(", RowBox[{ RowBox[{\(\(x\_1\) \((ad \((b\_1)\))\) ... \), "Exp", RowBox[{ StyleBox["(", FontSize->14], \(\(x\_m\) \((ad \((b\_m)\))\)\), StyleBox[")", FontSize->18]}], "is", " ", "an", " ", "automorphism", " ", "of", " ", "\[GothicG]"}], ",", " ", \(by\ Lemma\ 2. \ \ Moreover\), ",", " ", \(f\ is\ a\ polynomial\ map\ in\ the\ entries\ h\ \ and\ \(\(x\_i\)\(.\)\(\ \ \)\)\)}]}]}]}]}]}]}]}]}]}]}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Now, \ apply\ Lemma\ 3. \ \ Take\ a\ \[Element] \ \[GothicH]\ such\ that\ \ \[Alpha] \((a)\) \[NotEqual] 0\ for\ all\ nonzero\ \[Alpha]\ for\ which\ \[GothicG]\_\[Alpha]\ is\ \ \), "\[IndentingNewLine]", \(\(\(nonzero\)\(.\)\(\ \ \)\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(Compute\ df\( | \_a\)\((b + h)\) = d\/dt\( | \_\(t = 0\)\)\((f \((t \((\[Sum]\+\(i = 1\)\%m\( x\_i\) b\_i + h)\) + a)\))\) . \ \ Taylor\ expanding\ we\ get\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(\(\(=\)\(\((\(\((I + \(tx\_1\) adb\_1 + o \((t\^2)\))\) ... \) \((I\ + \(tx\_m\) adb\_m)\) + o \((t\^2)\))\) \((a + th)\)\)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(\(\(=\)\(d\/dt\( | \_\(t = 0\)\)\ \ \ t[b, a] + Ith\)\)\), "\[IndentingNewLine]", \(\(\(=\)\(d\/dt\( | \_\(t = 0\)\)\ \ \ t \((\([b, a]\) + h)\)\)\)\), "\[IndentingNewLine]", \(\(\(=\)\(\([b, a]\) + h\)\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(So\ df\( | \_a\)\((b + h)\) = \([b, a]\) + h\ which\ is\ nonsingular\ since\ it\ is\ the\ identity\ on\ \(\ \[GothicG]\_0\) and\ on\ \[GothicG]\_\[Alpha]\), "\[IndentingNewLine]", \(for\ all\ nonzero\ \[Alpha]\ it\ is\ - \(\(ad\)\(\ \)\(a\)\(\ \ \)\(which\)\(\ \)\(is\)\(\ \)\(invertible\)\(\ \)\(because\)\(\ \)\(ad\)\(\ \ \)\(a\)\(\ \)\(has\)\(\ \)\(the\)\(\ \)\(form\)\(\ \)\)\), "\ \[IndentingNewLine]", \(\[Alpha] \((a)\)* Identity\ + \ \(\(nilpotent\)\(\ \)\(part\)\(\ \ \)\(and\)\(\ \)\(\ \[Alpha]\) \((a)\)\(\ \)\(is\)\(\ \)\(nonzero\)\(\ \)\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(By\ Lemma\ 3, \ (*\()\) \((Exp \((\(\(x\_1\) \((ad \((b\_1)\))\) \ ... \) Exp \((\(x\_m\) \((ad \((b\_m)\))\))\) \[GothicH]\ contains\ a\ \ Zariski\ open\ subset\ \(\[CapitalOmega]\_h\) of\ \[GothicG], \ \(since\ x\_1 ... \) x\_m \[Element] \(\(\[DoubleStruckCapitalF]\)\(.\)\(\ \ \)\)\)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Let\ \[CapitalOmega]\_r\ be\ the\ set\ of\ regular\ elements\ of\ \ \[GothicG] . \ \ We\ know\ it\ is\ a\ nonempty\ Zariski\ open\ set . \ \ Let\ \ \[CapitalOmega]\_i = \[CapitalOmega]\_\(\[GothicH]\_i\)\), "\ \[IndentingNewLine]", \(for\ i\ = \ 1, 2. \)}], "Input", PageWidth->Infinity, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(Since\ the\ intersection\ of\ finitely\ many\ Zariski\ open\ sets\ is\ \ nonempty, \ \[CapitalOmega]\_\(\[GothicH]\_1\) \[Intersection] \ \[CapitalOmega]\_\(\[GothicH]\_2\) \[Intersection] \[CapitalOmega]\_r\ is\ \(\ \(nonempty\)\(.\)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Take\ b \[Element] \ \[CapitalOmega]\_\(\[GothicH]\_1\) \[Intersection] \ \[CapitalOmega]\_\(\[GothicH]\_2\) \[Intersection] \[CapitalOmega]\_r . \ \ b\ \ is\ regular\ and\ contained\ in\ \(\[Sigma]\_1\) \((\[GothicH]\_1)\)\ and\ \ in\ \(\[Sigma]\_2\) \((\[GothicH]\_2)\)\), "\[IndentingNewLine]", \(for\ some\ automorphisms\ \[Sigma]\_1\ and\ \[Sigma]\_2\ due\ to\ \ (*\()\)\(.\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{\(Hence\ \(\[Sigma]\_1\^\(-1\)\) \((b)\) \[Element] \ \[GothicH]\_1\ and\ \(\[Sigma]\_2\^\(-1\)\) \((b)\) \[Element] \[GothicH]\_2 \ . \ \ These\ are\ regular\ elements\ in\ \[GothicH]\_1\ and\ \[GothicH]\_2\), \ "\[IndentingNewLine]", RowBox[{"respectively", ",", " ", \(hence\ by\ Lemma\ 4\), ",", " ", RowBox[{\(\[GothicH]\_1\), StyleBox["=", FontSize->16], StyleBox[\(\[GothicG]\_0\^\(\(\[Sigma]\_1\^\(-1\)\) \((b)\)\)\), FontSize->9]}], StyleBox[",", FontSize->9], StyleBox[" ", FontSize->9], RowBox[{\(\[GothicH]\_2\), StyleBox["=", FontSize->16], StyleBox[\(\(\[GothicG]\_0\^\(\(\[Sigma]\_2\^\(-1\)\) \ \((b)\)\)\)\(.\)\), FontSize->9]}]}]}], "Input", PageWidth->Infinity, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ RowBox[{\(Take\ \[Sigma]\), "=", RowBox[{\(\[Sigma]\_2\^\(-1\)\[SmallCircle]\[Sigma]\_1 . \ \ Then\ \ \[Sigma] \((\(\[Sigma]\_1\^\(-1\)\) \((b)\))\)\), "=", RowBox[{\(\[Sigma]\_2\^\(-1\)\), \((b)\), " ", "maps", " ", \(\[GothicH]\_1\), "to", " ", RowBox[{\(\[GothicH]\_2\), StyleBox[".", FontSize->9], StyleBox[" ", FontSize->16]}]}]}]}]], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Note : \ \ We\ reduced\ this\ theorem\ to\ the\ construction\ of\ a\ \ certain\ map . \ \ This\), "\[IndentingNewLine]", \(idea\ was\ developed\ further\ by\ Grothendieck\ who\ realized\ that\ \ maps\ between\), "\[IndentingNewLine]", \(objects\ are\ often\ more\ important\ than\ the\ objects\ \ \(\(themselves\)\(.\)\(\ \ \)\)\)}], "Input", PageWidth->Infinity, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ RowBox[{ RowBox[{ StyleBox["Exercise", FontVariations->{"Underline"->True}], StyleBox[" ", FontVariations->{"Underline"->True}], RowBox[{ StyleBox["9.3", FontVariations->{"Underline"->True}], StyleBox[".", FontVariations->{"Underline"->True}], " ", "Prove"}], " ", "the", " ", "second", " ", "part", " ", "of", " ", "the", " ", "theorem"}], ",", " ", \(ie . \ that\ any\ \[GothicH] = \[GothicG]\_0\^a\), ",", " ", \(for\ \[DoubleStruckCapitalF] = \[DoubleStruckCapitalC]\), ",", " ", \(\(using\)\(\ \)\(the\)\(\ \)\(implicit\)\(\ \)\(function\)\(\ \)\ \(theorem\)\(\ \)\(instead\)\(\ \)\(of\)\(\ \)\(Lemma\)\(\ \)\(3.\)\(\ \ \ \)\)}]], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{\(Solution . \ \ The\ proof\ holds\ as\ above\ until\ the\ line\ \ where\ Lemma\ 3\ is\ used . \ \ In\ place\ of\), "\[IndentingNewLine]", RowBox[{\(Lemma\ 3\ we\ use\ the\ implicit\ function\ theorem\ which\ \ shows\ that\ because\ df\( | \_a\)is\ nonsingular\ f \((\[GothicG])\)\ \ contains\ an\ open\ neighborhood\ \[CapitalOmega]\_\[GothicH]\ of\ f \((a)\) \ . \ \ We\ need\ only\), " "}], "\[IndentingNewLine]", RowBox[{\(to\ show\ that\ \[CapitalOmega]\_\[GothicH] \[Intersection] \ \[CapitalOmega]\_r\ is\ nonempty . \ \ After\ this\ take\ b \[Element] \ \[CapitalOmega]\_\[GothicH] \[Intersection] \[CapitalOmega]\_r\ . \ \ Since\ \ b\ is\ regular\), " "}], "\[IndentingNewLine]", RowBox[{\(and\ contained\ in\ the\ image\ of\ \[GothicH]\ under\ some\ \ automorphism\ \[Sigma], \ \(\[Sigma]\^\(-1\)\) \((b)\) \[Element] \[GothicH]\ \ and\ is\ a\), " "}], "\[IndentingNewLine]", RowBox[{\(regular\ element\ since\ it\ is\ the\ image\ under\ \ automorphism\ of\ a\ regular\ element\), ",", " ", RowBox[{\(and\ thus\ \[GothicH]\), StyleBox["=", FontSize->16], RowBox[{ StyleBox[\(\[GothicG]\_0\^\(\(\[Sigma]\^\(-1\)\) \((b)\)\)\), FontSize->12], StyleBox[".", FontSize->9], StyleBox[" ", FontSize->9]}]}]}]}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Finally, \ we\ show\ that\ the\ intersection\ of\ an\ open\ set\ with\ a\ nonempty\ \ Zariski\), "\[IndentingNewLine]", \(open\ set\ is\ nonempty . \ \ We\ use\ the\ fact\ that\ if\ a\ \ polynomial\ vanishes\ on\ an\ open\ \), "\[IndentingNewLine]", \(nonempty\ subset\ of\ \[DoubleStruckCapitalC]\^n, \ the\ polynomial\ is\ identically\ zero . \ \ The\ corresponding\), "\ \[IndentingNewLine]", \(Zariski\ open\ set\ is\ the\ empty\ set . \ \ So\ if\ a\ Zariski\ open\ \ set\ does\ not\ intersect\ an\ open\ \), "\[IndentingNewLine]", \(neighborhood\ in\ \[DoubleStruckCapitalC]\^n, \ it\ is\ the\ empty\ set . \ \ The\ contrapositive\ of\ this\ statement\), \ "\[IndentingNewLine]", \(gives\ the\ desired\ \(\(result\)\(.\)\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(--\(\ --\(--\(--\(--\(--\(--\(--\(----\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\)\ \)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(Trace\ \(\(form\)\(.\)\(\ \ \)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(Let\ \[Pi]\ be\ a\ representation\ of\ a\ Lie\ Algebra\ \[GothicG]\ in\ \ a\ finite\ dimensional\ vector\ space\ \(\(V\)\(.\)\(\ \ \)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(Definition . \ \ A\ trace\ form\ on\ \[GothicG]\ is\ the\ following\ \ bilinear\ \(form : \ \((a, b)\)\_V\) = Tr \(\((\[Pi] \((a)\) \[Pi] \((b)\))\)\(.\)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Note\ the\ following\ properties\ of\ the\ trace\ \(form : \ \ \[IndentingNewLine]\((1)\) Bilinearity\)\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \((2)\) Symmetry\), "\[IndentingNewLine]", \(\ \ \ \ \ \ \ \ \ \((3)\)\ Invariance\ \ \((ie . \ \((\([a, b]\), \ \ c)\)\_V + \((b, \([a, \ c]\))\)\_V = 0, which\ is\ equivalent\ to\ \ \((\([a, b]\), \ c)\)\_V = \ \((a, \([b, \ \ c]\))\)\_V)\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(\(\(Proof\)\(.\)\(\ \)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(\((1)\)\ Follows\ from\ bilinearity\ of\ matrix\ multiplication\ and\ \ the\ linearity\ of\ the\ trace\ \(\(operation\)\(\ \)\(.\)\)\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ \(\((2)\)\ Clear, \ as\ Tr \((\([A, B]\))\) = 0\)], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(\((3)\) Tr \((\([\[Pi] \((a)\), \[Pi] \((b)\)]\) \[Pi] \((c)\))\) + Tr \((\[Pi] \((b)\)[\[Pi] \((a)\), \[Pi] \((c)\)])\)\), "\ \[IndentingNewLine]", \(\(\(=\)\(Tr \((\[Pi] \((a)\) \[Pi] \((b)\) \[Pi] \((c)\)\ - \ \[Pi] \ \((b)\) \[Pi] \((a)\) \[Pi] \((c)\) + \[Pi] \((b)\) \[Pi] \((a)\) \[Pi] \((c)\ \) - \[Pi] \((b)\) \[Pi] \((c)\) \[Pi] \((a)\))\)\)\)\), \ "\[IndentingNewLine]", \(\(\(=\)\(Tr \((\([\[Pi] \((a)\), \ \[Pi] \((b)\) \[Pi] \ \((c)\)]\))\)\)\)\), "\[IndentingNewLine]", \(\(\(=\)\(0\)\)\), "\[IndentingNewLine]", \(Exchanging\ a\ and\ b\ gives\ \((\([b, a]\), \ c)\)\_V + \ \((a, \([b, \ \ c]\))\)\_V = \(0\ \[Implies] \((\([a, b]\), \ c)\)\_V = \ \((a, \([b, \ c]\ \))\)\_V\)\)}], "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[{ \(Proposition . \ \ If\ \[GothicG]\ is\ a\ Lie\ Algebra\ and\ \ \((\(.\)\(\(,\)\(.\)\))\)\ is\ an\ invariant\ symmetric\ bilinear\ form, \ then\ \(M\^ \[UpTee] \) is\ an\ ideal\ of\ \[GothicG]\ if\ M\ is\ an\ ideal\ of\ \(\(\[GothicG] \ . \ \ \((Where\ \(M\^ \[UpTee] \) = \(\({a \[Element] \[GothicG] | \((a, M)\) = 0}\)\(.\)\))\)\)\(\ \)\)\), \ "\[IndentingNewLine]", \(In\ particular, \ \(\[GothicG]\^ \[UpTee] \) = ker \((\(.\)\(\(,\)\(.\)\))\)\ is\ an\ ideal\ of\ \(\(G\)\(.\)\)\)}], \ "Input", PageWidth->PaperWidth, FontFamily->"Times New Roman", 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