(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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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[ 16779, 581]*) (*NotebookOutlinePosition[ 17523, 607]*) (* CellTagsIndexPosition[ 17479, 603]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Heisenberg group calculations", "Subtitle", Background->RGBColor[0, 1, 0]], Cell[CellGroupData[{ Cell[BoxData[ \(heisen[{x_, y_, t_}, {z_, w_, s_}] = {x + z, y + w, \((t + s)\)\ + \((x\ \ w - y\ z)\)/2}\)], "Input"], Cell[BoxData[ \({x + z, w + y, s + t + 1\/2\ \((w\ x - y\ z)\)}\)], "Output"] }, Open ]], Cell[BoxData[{ \(\(a = {x0, y0, t0};\)\), "\n", \(\(b = {x1, y1, t1};\)\), "\n", \(\(c = {x2, y2, t2};\)\)}], "Input"], Cell["This operation is associative:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(heisen[heisen[a, b], c] - 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x\ y0)\)}\)], "Output"] }, Open ]], Cell["Its differential is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(difflefta = {D[lefta[{x, y, t}], x], D[lefta[{x, y, t}], y], D[lefta[{x, y, t}], t]} // MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"1", "0", \(-\(y0\/2\)\)}, {"0", "1", \(x0\/2\)}, {"0", "0", "1"} }], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output"] }, Open ]], Cell["\<\ Note that this differential is a constant function of x, y and \ t.\ \>", "Text"], Cell[TextData[{ "Since it has determinant 1, the Lebesgue measure\nis invariant by \ left-translations:\n ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(R\^3\)\(f( y)\) \[DifferentialD]y = \(\[Integral]\_\(R\^3\)\%\ \ \)\(f(\(l\_a\)(x))\) \[DifferentialD]x\)]] }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[""], "Input"], Cell[TextData[{ "Here is an arbitrary vector field:\n \ v=A(x,y,t)", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\/\[PartialD]x\)]], "+B", Cell[BoxData[ \(TraditionalForm\`\((x, y, t)\) \[PartialD]\/\[PartialD]y\)]], "+C", "(x,y,t)", Cell[BoxData[ \(TraditionalForm\`\[PartialD]\/\[PartialD]t\)]] }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[CellGroupData[{ Cell[BoxData[ \(vectorfield[{x_, y_, t_}] = {A[{x, y, t}], B[{x, y, t}], C[{x, y, t}]}\)], "Input"], Cell[BoxData[ \({A[{x, y, t}], B[{x, y, t}], C[{x, y, t}]}\)], "Output"] }, Open ]], Cell[TextData[{ "Vector fields act on functions:\n ", "v(f)=A(x,y,t)", Cell[BoxData[ \(TraditionalForm\`\[PartialD]f\/\[PartialD]x\)]], "+B", Cell[BoxData[ \(TraditionalForm\`\((x, y, t)\) \[PartialD]f\/\[PartialD]y\)]], "+C(x,y,t)", Cell[BoxData[ \(TraditionalForm\`\[PartialD]f\/\[PartialD]t\)]] }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(vectorfield[{x, y, t}] . {D[f[x, y, t], x], D[f[x, y, t], y], D[f[x, y, t], t]}\)\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{\(C[{x, y, t}]\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "[", \(x, y, t\), "]"}]}], "+", RowBox[{\(B[{x, y, t}]\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((0, 1, 0)\), Derivative], MultilineFunction->None], "[", \(x, y, t\), "]"}]}], "+", RowBox[{\(A[{x, y, t}]\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "[", \(x, y, t\), "]"}]}]}]], "Output"] }, Open ]], Cell[TextData[{ "Here is the left-translation of the function f by a:\n \ f \[EmptySmallCircle] ", Cell[BoxData[ \(TraditionalForm\`l\_\(\(a\)\(\ \)\)\)]], "\n" }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[CellGroupData[{ Cell[BoxData[ \(leftfbya[x_, y_, t_] = f[\ \(lefta[{x, y, t}]\)[\([1]\)], \(lefta[{x, y, t}]\)[\([2]\)], \(lefta[{x, y, t}]\)[\([3]\)]]\)], "Input"], Cell[BoxData[ \(f[x + x0, y + y0, t + t0 + 1\/2\ \((x0\ y - x\ y0)\)]\)], "Output"] }, Open ]], Cell[TextData[{ "vectorfield acting on the left-translation of f by a:\n \ v(f \[EmptySmallCircle] ", Cell[BoxData[ \(TraditionalForm\`l\_\(\(a\)\(\ \)\)\)]], ")" }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\ \)\(\(uno = vectorfield[{x, y, t}] . {D[leftfbya[x, y, t], x], D[leftfbya[x, y, t], y], D[leftfbya[x, y, t], t]}\)\(\n\) \)\)\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{\(C[{x, y, t}]\), " ", RowBox[{ SuperscriptBox["f", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "[", \(x + x0, y + y0, t + t0 + 1\/2\ \((x0\ y - x\ y0)\)\), "]"}]}], "+", RowBox[{\(B[{x, y, t}]\), " ", RowBox[{"(", RowBox[{ RowBox[{\(1\/2\), " ", "x0", " ", RowBox[{ SuperscriptBox["f", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "[", \(x + x0, y + y0, t + t0 + 1\/2\ \((x0\ y - x\ y0)\)\), "]"}]}], "+", RowBox[{ SuperscriptBox["f", TagBox[\((0, 1, 0)\), Derivative], MultilineFunction->None], "[", \(x + x0, y + y0, t + t0 + 1\/2\ \((x0\ y - x\ y0)\)\), "]"}]}], ")"}]}], "+", RowBox[{\(A[{x, y, t}]\), " ", RowBox[{"(", RowBox[{ RowBox[{\(-\(1\/2\)\), " ", "y0", " ", RowBox[{ SuperscriptBox["f", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "[", \(x + x0, y + y0, t + t0 + 1\/2\ \((x0\ y - x\ y0)\)\), "]"}]}], "+", RowBox[{ SuperscriptBox["f", TagBox[\((1, 0, 0)\), Derivative], MultilineFunction->None], "[", \(x + x0, y + y0, t + t0 + 1\/2\ \((x0\ y - x\ y0)\)\), "]"}]}], ")"}]}]}]], "Output"] }, Open ]], Cell[TextData[{ "left translation by of the action of vector field on a function:\n \ v(f) \[EmptySmallCircle] ", Cell[BoxData[ \(TraditionalForm\`l\_a\)]] }], "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[CellGroupData[{ Cell[BoxData[ \(dos = vectorfield[{x, y, t}] . {D[f[x, y, t], x], D[f[x, y, t], y], D[f[x, y, t], t]} /. {x -> \(lefta[{x, y, t}]\)[\([1]\)], \ y -> \(lefta[{x, y, t}]\)[\([2]\)], \ t -> \(lefta[{x, y, t}]\)[\([3]\)]}\)], "Input"], Cell[BoxData[ RowBox[{ RowBox[{\(C[{x + x0, y + y0, t + t0 + 1\/2\ \((x0\ y - 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x\ y0)\)\), " ", "]"}]}], "]"}], "==", RowBox[{"Coefficient", "[", RowBox[{"dos", ",", " ", RowBox[{ SuperscriptBox["f", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "[", \(x + x0, y + y0, t + t0 + 1\/2\ \((x0\ y - x\ y0)\)\), " ", "]"}]}], "]"}]}], "/.", \({x -> 0, y -> 0, t -> 0}\)}]], "Input"], Cell[BoxData[ \(\(-\(1\/2\)\)\ y0\ A[{0, 0, 0}] + 1\/2\ x0\ B[{0, 0, 0}] + C[{0, 0, 0}] \[Equal] C[{x0, y0, t0}]\)], "Output"] }, Open ]], Cell["A left invariant vector field is then of the form:", "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell[BoxData[ \(TextForm\`\(\(\ \)\(A \((0)\) \[PartialD]\/\[PartialD]x + B \((0)\) \[PartialD]\/\[PartialD]y + \((\ \(-\(1\/2\)\) y\ \ A \((0)\) + \(1\/2\) x\ \ B \((0)\) + C \((0)\))\) \[PartialD]\/\(\(\[PartialD]t\)\(\ \)\)\)\)\)], \ "Text", CellFrame->True, Background->GrayLevel[0.833326]], Cell["\<\ The usual basis is obtained taking A(0)=1, B(0)=0, C(0)=0,etc\ \>", \ "Text"], Cell[BoxData[ \(TextForm\`XX = \ \ \[PartialD]\/\[PartialD]x\ - 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YY[XX[f[x, y, t]]] // Simplify\)], "Input"], Cell[BoxData[ RowBox[{ SuperscriptBox["f", TagBox[\((0, 0, 1)\), Derivative], MultilineFunction->None], "[", \(x, y, t\), "]"}]], "Output"] }, Open ]], Cell["That is, we have checked that [XX,YY]= T", "Text", CellFrame->True, Background->GrayLevel[0.833326]] }, Open ]] }, FrontEndVersion->"5.0 for Macintosh", ScreenRectangle->{{0, 800}, {0, 535}}, ScreenStyleEnvironment->"Presentation", CellGrouping->Manual, WindowSize->{784, 508}, WindowMargins->{{4, Automatic}, {Automatic, 1}}, ShowClosedCellArea->False, Magnification->1.25 ] (******************************************************************* Cached data follows. 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