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Description_of_contents.txt
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Description_of_contents.txt
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Example Programs.zip
(1) Description_of_contents.mac is this file.
(2) DEF (see 1).
(3) Lie_bracket_operation_and_root_system (see 2).
(4) Operations_and_Converter (see 3).
(5) Decompose_into_248_basis_matrices (see 4).
(6) Relement.txt, element_C.txt and element_F.txt (see 5).
(7) Lie_algebra_f4_e6_g2_define (see 6)
(8) Some Lie subalgebras of r8C (see 7-12)
Take a good look at the programming codes and use them at your own risk.
1. 'DEF'
DEF/*.mac
1.1 contents
The 248 digitalized basis matrices of r8C.
There are 248 *.mac files, which are 248 basis matrices defined in Maxima.
1.2 How to use
Save the DEF folder to a location as you like.
2. 'Lie_bracket_operation_and_root_system'
Lie_bracket_operation_and_root_system/wxMaxima/*.mac
2.1 contents
Program for checking Lie bracket operations and confirming root systems.
There are 9 files.
248_basis_matrices_load.mac is a program for loading 248 basis matrices.
r8C_independent.mac is a program to check that 248 basis matrices are independent.
r4C_bracket_root_operation.mac is a program for r4C,
which checks Lie bracket operations and root system.
The upper left 52 × 52 elements are taken from the 248 × 248 matrix.
r6C_bracket_root_operation.mac is a program for r6C.
The upper left 78 × 78 elements are taken from the 248 × 248 matrix.
Also check R4C(78 x 78) in r6C.
r7C_bracket_root_operation.mac is a program for r7C.
The upper left 133 × 133 elements are taken from the 248 × 248 matrix.
Also check R4C(133 x 133) in r7C and r6C(133 x 133) in r7C.
r8C_bracket_root_operation.mac is a program for R8C.
Also checks r4C (248 x 248) in r8C,r6C (248 x 248) in r8C and
r7C (248 x 248) in r8C.
e8e7e6f4_compact_element_definition .mac is a program for
compact Lie algebra definition and Killing form calculation.
r2C_bracket_root_operation (e8c).mac is a program of r2C.
r2C_bracket_root_operation.mac is a program checks r2C(28 x 28).
The upper left 28 × 28 elements are taken from the 248 × 248 matrix.
2.2 How to use
(1)Change the location where the DEF folder is stored and the check
result storage folder.
(2)Load or copy & paste the contents of 248_basis_matrices_load.mac ffile into Maxima.
(3)Copy & paste the contents of the *.mac file into Maxima.
(4)Select Evalute Cell(s) from Cell menu.
2.3 Usage example
In case of R6C
1st: Start wxMaxima
2nd: Copy & paste 248_basis_matrices_load.mac
3rd: Evalute Cell(s)
4th: Copy & paste R6C_bracket_root_operation.mac
5th: Evalute Cell(s)
Before executing r4C, r6C, r7C, r8C, and e8e7e6f4_compact,
you must copy & paste and Evalute Cell(s) of 248_basis_matrices_load.mac.
2.4 Remark
Running R7C_bracket_root_operation.mac takes about 1 hour.
Running R8C_bracket_root_operation.mac takes several hours.
3. 'Operations_and_Converter'
Operations_and_Converter/Ade8c/*.py
Operations_and_Converter/wxMaxima/*.mac
Operations_and_Converter/Operation_and_data_flow.pdf
3.1 contens
Operations of the Cayley algebra and the Jordan algebra are defined in wxMaxima/1st_step.mac.
We give the image of the adjoint representation of r8c by a 248× 248 dimensional matrix over complex numbers.
Operation_and_data_flow.pdf shows operation and data flow of wxMaxima and Python programs which convert to the above matrix.
wxMaxima/*.mac are programming codes of Maxima.
Ade8c/*.py are programming codes of Python.
3.2 How to use
Please operate according to Operation_and_data_flow.pdf.
4.Decompose_into_248_basis_matrices'
Decompose_into_248_basis_matrices/wxMaxima/Dcomp.mac
4.1 content
Decompose an element of r8C into a linear combination of the 248 basis matrices.
The following are the symbols of the basis matrices.
Urdij,Urmij,Urτi,Urtij,Urαi,Uraij,Urβi,Urbij,Urρ1,
Urχi,Urxij,Urγi,Uryij,Urξ1,Urη1,Urμi,Urzij,Urψi,Urwij,
Urζ1,Urω1,Urr1,Urs1,Uru1 where i and j are indexes.
Decomp.mac is a macro program
4.2 How to use
Copy and paste the contents of Decomp.mac file into Maxima.
4.3 usage example
Substitute a Lie bracket calculation result in R.
R:X.Y-Y.X$
Dcomp();
5. Relement.txt, element_C.txt and element_F.txt
Relement.txtx is a symbol lists of the vector space R8C.
element_C.txt and element_F.txt are basis symboles of a matrix R8C.
6.Lie_algebra_f4_e6_g2_define
Lie_algebra_f4_e6_g2_define/wxMaxima/*.mac
Lie_algebra_f4_e6_g2_define/E6mat/*.mat
6.1 contents
f4_Liealgebra.mac is the digitalization of f4 into 27 X 27 matrices.
e6_Liealgebra.mac is the digitalization of e6 into 27 X 27 matrices.
g2_Liealgebra.mac is the digitalization of g2 into 8 X 8 matrices.
f4_bracket_operation.mac is a bracket operation check program.
e6_bracket_operation.mac is a bracket operation check program.
g2_bracket_operation.mac is a bracket operation check program.
f4_definition_check.mac is a program that checks the definition of f4, checks the orthogonal bases, and checks the transpose with the inner product.
e6_definition_check.mac s a program that checks the definition of e6 and checks the orthogonal bases.
g2_definition_check.mac is a definition of g2 check program.
matrices_of_E6_load.mac is a Lie group E6 elements load program.
F4_definition_check.mac is a program that checks whether the elements of F4 satisfy the definition of F4.
E6_definition_check.mac is a program that checks whether the elements of E6 satisfy the definition of E6.
E6mat/*.mat are exponential matrices of the basis of the Lie algebra e6.
6.2 How to use
Copy and paste the contents of ******.mac file into Maxima.
7. Su3bracketcal.mac
This is a programming code that checks Lie bracket calculations of Rsu(3).
8. Sp3bracketcal.mac
This is a programming code that checks Lie bracket calculations of Rsp1(3) and Rsp2(3).
9. So9bracketcal.mac
This is a programming code that checks Lie bracket calculations of Rso(9).
10. So10bracketcal.mac
This is a programming code that checks Lie bracket calculations of Rso(10).
11. Su6bracketcal.mac
This is a programming code that checks Lie bracket calculations of Rsu(6).
12. Sp4bracketcal.mac
This is a programming code that checks Lie bracket calculations of Rsp(4).