This archive is a companion to the paper
"Complex structures on indecomposable 6-dimensional nilpotent real Lie groups".

It contains

0) The directory "6/nilp" with all commutation relations files for the algebras.

1) All the Reduce programs used to get the complex structures in  the paper.
These programs can be run under the computer algebra system "Reduce".
They are in general indexed as
"program*.red"
where * is the section number of the paper where it occurs
or
"program*_#.red"
where * is the section number and # the subsection number
(or the subsection number followed from the subsubsection number)
of the paper where it occurs.

2) All Reduce files containing the matrices J computed by the programs
(with the exception of the very simple matrices in section 13.4.5 of the paper)
These are indexed similarly to the programs, and automatically created when running
the corresponding program.
They can be loaded in "Reduce" for further calculations.
By using the Reduce switch "on factor$" or "off factor$ on exp$", one can modify the
display of the matrix elements J(k,l).

3) All result files "rprogram**.tex" of the foregoing programs

CONTENTS OF THE ARCHIVE :

"6/nilp"
directory with all commutation relations files for the algebras.

"program3_1.red"
computes the complex structures on the algebra g_{6,3} in the case xi(1,6) neq 0.

"matJ3_1.red"  Reduce file containing the matrice J computed by this program.

"rprogram3_1.tex"  result file (output) of the foregoing program

"program3_2.red"
computes the complex structures on the algebra g_{6,3} in the case
xi(1,6) = 0, xi(2,5) neq 0.

"matJ3_2.red"  Reduce file containing the matrice J computed by this program.

"rprogram3_2.tex"  result file (output) of the foregoing program

"program3_3.red"
computes the complex structures on the algebra g_{6,3} in the case
xi(1,6) = 0, xi(2,5) = 0.

"rprogram3_3.tex"  result file (output) of the foregoing program

"matJ3_3.red"  Reduce file containing the matrice J computed by this program.

"program4.red"
computes the complex structures on the algebra g_{6,7}.

"matJ4.red"  Reduce file containing the matrice J computed by this program.

"rprogram4.tex"  result file (output) of the foregoing program

"program5.red"
computes the complex structures on the algebra g_{6,4}.

"matJ5.red"  Reduce file containing the matrice J computed by this program.

"rprogram5.tex"  result file (output) of the foregoing program

"program6_1.red"
computes the complex structures on the algebra g_{6,1} in the case
xi(2,1) neq 0, xi(2,3) neq 0, xi(1,3) neq xi(2,4).

"matJ6_1.red"  Reduce file containing the matrice J computed by this program.

"rprogram6_1.tex"  result file (output) of the foregoing program

"program6_2.red"
computes the complex structures on the algebra g_{6,1} in the case
xi(2,1) neq 0, xi(2,3) = 0, xi(1,3) neq xi(2,4).

"matJ6_2.red"  Reduce file containing the matrice J computed by this program.

"rprogram6_2.tex"  result file (output) of the foregoing program

"program6_3.red"
computes the complex structures on the algebra g_{6,1} in the case
xi(2,1) neq 0,  xi(1,3) = xi(2,4).

"matJ6_3.red"  Reduce file containing the matrice J computed by this program.

"rprogram6_3.tex"  result file (output) of the foregoing program

"program6_4.red"
computes the complex structures on the algebra g_{6,1} in the case
xi(2,1) = 0,  xi(2,3) neq 0.

"matJ6_4.red"  Reduce file containing the matrice J computed by this program.

"rprogram6_4.tex"  result file (output) of the foregoing program

"program6_5.red"
computes the complex structures on the algebra g_{6,1} in the case
xi(2,1) = 0,  xi(2,3) = 0.

"matJ6_5.red"  Reduce file containing the matrice J computed by this program.

"rprogram6_5.tex"  result file (output) of the foregoing program

"program6_6.red"
computes the complex structures on the algebra g_{6,1} in the case
xi(2,3) neq 0.

"matJ6_6.red"  Reduce file containing the matrice J computed by this program.

"rprogram6_6.tex"  result file (output) of the foregoing program

"program7.red"
computes the complex structures on the algebra g_{6,6}

"matJ7.red"  Reduce file containing the matrice J computed by this program.

"rprogram7.tex"  result file (output) of the foregoing program

"program8.red"
computes the complex structures on the algebra g_{6,5}

"matJ8.red"  Reduce file containing the matrice J computed by this program.

"rprogram8.tex"  result file (output) of the foregoing program

"program9.red"
computes the complex structures on the algebra g_{6,8}

"matJ9.red"  Reduce file containing the matrice J computed by this program.

"rprogram9.tex"  result file (output) of the foregoing program

"program10_1.red"
computes the complex structures on the algebra M10 in the case
xi(4,3) neq xi(2,1).

"matJ10_1.red"  Reduce file containing the matrice J computed by this program.

"rprogram10_1.tex"  result file (output) of the foregoing program

"program10_2and3.red"
computes the complex structures on the algebra M10 in the case
xi(4,3) = xi(2,1) and in each of the subcases
case 2.1 : xi(3,3) = xi(1,1)
case 2.2 : xi(3,3) neq xi(1,1).

"matJ10_2.red"  Reduce file containing the matrice J computed by this program.
in the case 2.1 : xi(3,3) = xi(1,1)

"matJ10_3.red"  Reduce file containing the matrice J computed by this program.
in the case 2.2 : xi(3,3) neq xi(1,1)

"rprogram10_2and3.tex"  result file (output) of the foregoing program

"program11_-1.red"
proves there is no complex structures on the algebra M14_{-1}

"rprogram11_-1.tex"  result file (output) of the foregoing program

"program11.red"
computes the  complex structures on the algebra M14_{1}

"matJ11.red"  Reduce file containing the matrice J computed by this program.

"rprogram11.tex"  result file (output) of the foregoing program

"program12_-1.red"
proves there is no complex structures on the algebra M18_{-1}

"rprogram12_-1.tex"  result file (output) of the foregoing program

"program12.red"
computes the  complex structures on the algebra M18_{+1}

matJ12.red  Reduce file containing the matrice J computed by this program.

"rprogram12.tex"  result file (output) of the foregoing program

"program13_1a.red"
computes the complex structures on the algebra M5 in the case
xi(2,4) neq 0.

"matJ13_1a.red"  Reduce file containing the matrice J computed by this program.

"rprogram13_1a.tex"  result file (output) of the foregoing program

"program13_1b.red"
computes the complex structures on the algebra M5 in the case
xi(2,3) neq 0.

"matJ13_1b.red"  Reduce file containing the matrice J computed by this program.

"rprogram13_1b.tex"  result file (output) of the foregoing program

"program13_21.red"
(subsubsection 13.2.1 of the paper)
computes the complex structures on the algebra M5 in the case
xi(2,1)*xi(2,4) neq 0.

"matJ13_21.red"  Reduce file containing the matrice J computed by this program.

"rprogram13_21.tex"  result file (output) of the foregoing program

"program13_22.red"
(subsubsection 13.2.2 of the paper)
computes the complex structures on the algebra M5 in the case
xi(2,1)*xi(2,3) neq 0, xi(2,4)=0.

"matJ13_22.red"  Reduce file containing the matrice J computed by this program.

"rprogram13_22.tex"  result file (output) of the foregoing program

"program13_23.red"
(subsubsection 13.2.3 of the paper)
computes the complex structures on the algebra M5 in the case
xi(2,1) neq 0, xi(2,2)=xi(2,4)=0, xi(2,2) + xi(3,3) neq 0.

"matJ13_23.red"  Reduce file containing the matrice J computed by this program.

"rprogram13_23.tex"  result file (output) of the foregoing program

"program13_24.red"
(subsubsection 13.2.4 of the paper)
computes the complex structures on the algebra M5 in the case
xi(2,1) neq 0, xi(2,2) = xi(2,4) = 0, xi(3,3) = -xi(2,2), xi(3,4) = -xi(2,1).

"matJ13_24.red"  Reduce file containing the matrice J computed by this program.

"rprogram13_24.tex"  result file (output) of the foregoing program

"program13_25.red"
(subsubsection 13.2.5 of the paper)
computes the complex structures on the algebra M5 in the case
xi(2,1) neq 0, xi(2,2) = xi(2,4) = 0, xi(3,3) = -xi(2,2), xi(3,4) neq -xi(2,1).

"matJ13_25.red"  Reduce file containing the matrice J computed by this program.

"rprogram13_25.tex"  result file (output) of the foregoing program

"program13_3.red"
computes a local chart for the differentiable manifold $\mathfrak{X}_{M5}$
in the neighborhood of the canonical complex structure $J_0$

"matJ13_3.red"  Reduce file containing the matrice J computed by this program.

"rprogram13_3.tex"  result file (output) of the foregoing program

"program13_43.red"
(subsubsection 13.4.3 of the paper)
computes the complex structures on the algebra M5
up to equivalence in the case 1 where B and C are not both 0.

"matJ13_43.red"  Reduce file containing the matrice J computed by this program.

"rprogram13_43.tex"  result file (output) of the foregoing program


"program13_45case2_1.red"
(subsubsection 13.4.5 of the paper)
computes the complex structures on the algebra M5
up to equivalence in the case 2 where B = C = 0
and the subcase case2.1 where xi(5,5) = 0.

"rprogram13_45case2_1.tex"  result file (output) of the foregoing program

"program13_45case2_2.red"
(subsubsection 13.4.5 of the paper)
computes the complex structures on the algebra M5
up to equivalence in the case 2 where B = C = 0
and the subcase case2.2 where xi(5,5) neq 0.

"rprogram13_45case2_2.tex"  result file (output) of the foregoing program

Note :
Before processing the result files "rprogram**.tex" with LaTeX, modify first the output
file so as no expression like xi(2,3) or b(5,1) is  cut in any line change.
Then make by hand the following modification :
(the are given below for SEDIT)
 C/$/ /* *
 C/USD/$/* *
 C/xi(/{\xi^/* *
 C/,1)/_1}/* *
 C/,2)/_2}/* *
 C/,3)/_3}/* *
 C/,4)/_4}/* *
 C/,5)/_5}/* *
 C/,6)/_6}/* *
 C/x(1)/x_1/* *
 C/x(2)/x_2/* *
 C/x(3)/x_3/* *
 C/x(4)/x_4/* *
 C/x(5)/x_5/* *
 C/x(6)/x_6/* *
 C/**/^/* *
 C/*/ /**
 C/b(/{b^/* *
 C/J2(/{J2^/* *
 top       %the 5 following commands are for deleting all empty lines with SEDIT)
 all b
 del *
 save
 quit
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