A program to study the Nikolayevsky derivation on Lie algebras with a parameter.
The Nikolayevksy (or pre-Einstein) derivation of a Lie algebra g was defined in
Yuri Nikolayevsky. Einstein solvmanifolds and the pre-Einstein derivation. Trans. Amer. Math. Soc. 363 (2011), 3935-3958
It is the unique (up to automorphisms) semisimple derivation N such that
(*) tr(ND)=tr D, D in Der(g)
The nilradical in Der(g) can be identified with the null space of the trace form,
n = {X in Der(g) : tr(ND)=0 for all D in Der(g)}
If the Lie algebra does not contain parameters, N and n can be computed explicitly. Gleipnir is also able to compute the centralizer of N in n. This is useful in order to compute the space of all derivations N verifying (*).
In the presence of parameters, Gleipnir is not always capable of producing the Nikolayevsky derivation N and the null space n. In this case, it only computes a space that is guaranteed to contain N+n.
This program has been used in the calculations leading to Proposition 2.7 in
Diego Conti and Federico A. Rossi. Indefinite nilsolitons and Einstein solvmanifolds. [https://doi.org/10.1007/s12220-021-00850-7](Journal of Geometric Analysis (2022) 32:88 )
The code is based on joint work with Federico A. Rossi, and partly taken from DEMONbLAST
Gleipnir requires cmake and Wedge
To compile, run
mkdir build
cd build
cmake ..
cmake --build .
You may need to set the WEDGE_PATH environment variable to point to your installation of Wedge, e.g.
export WEDGE_PATH=/home/user/wedge
To invoke the program on a single Lie algebra without parameters, pass it as an argument, as in
./gleipnir 0,0,12,0,24+13,14-23,15-26+2*34
Invoke Gleipnir without parameters to run it over all nilpotent Lie algebras of dimension 7, as appearing in
M.P. Gong. Classification of nilpotent Lie algebras of dimension 7 (over algebraically closed fields and R), Thesis (Ph.D.)--University of Waterloo (Canada), 1998.