#DEMONbLAST 0.3

A program to compute Diagonal Einstein Metrics On Nice Lie Algebras of Surjective Type. Based on the following papers:

• [CR19] D. Conti and F. A. Rossi, Construction of nice nilpotent Lie groups, Journal of Algebra, Volume 525, 2019, Pages 311-340, ISSN 0021-8693.
• [CR19b] D. Conti and F. A. Rossi, Ricci-flat and Einstein pseudoriemannian nilmanifolds, Complex Manifolds 2019; 6:170-193.
• [CR20] D. Conti and F. A. Rossi, Indefinite Einstein metrics on nice Lie groups, Forum Mathematicum, vol. 32, no. 6, 2020, pp. 1599-1619.
• [CR22] D. Conti and F. A. Rossi, Nice pseudo-Riemannian nilsolitons. Journal of Geometry and Physics 173:104433 (2022)

Copyright Diego Conti 2018-2022 diego.conti@unimib.it. Latest version available at https://github.com/diego-conti/DEMONbLAST.

##How to build In order to build demonblast, you need cmake and Wedge 0.4. Run

mkdir build
cd build
cmake ..
cmake --build .

This will create the executable demonblast in the subdirectory build.

demonblast has been developed and tested under Ubuntu Linux 21.10 with gcc 11.2.0

##How to test

You can run some tests by running

cd build/test
ctest .

(or ctest --test-dir build/test with newer versions of cmake)

##Running demonblast You should run demonblast from the project root directory, e.g.

build/demonblast --help

This command will print a list of command-line options, some of which are self-explanatory.

Diagram selection

There are three ways to run demonblast.

• --all-partitions n will process all nice diagrams in dimension n. For example,

  build/demonblast --all-partitions 6

will classify all nice diagrams of dimension 6 and write the output in the directory output/6.

• --partition n1 n2 ... nk will process the partition (n1 n2 ... nk). For example,

  build/demonblast --partition 4 2

will classify all nice diagrams with partition (4,2) and write the output to the file output/6/part4_2.dot

• --digraph diag will process the single nice diagram indicate by diag. For example,

  build/demonblast --digraph "6:4->[6]1,5->[6]2"

will process the diagram with six nodes and four arrows corresponding to the Lie algebra (46,56,0,0,0,0). Notice that the input format for this option is the same as the format used in the output (indeed the file output/6/part4_2.dot contains this string). Notice that the quotation marks are necessary to avoid the character > from being interpreted from the shell as redirection.

Notice that diagrams are cached in the directory diagrams. All paths used by demonblast are relative to the directory from which it is invoked, which is why invocation from the project root directory is recommended.

This will allow demonblast to employ the structure constants stored in the directory coefficients. This is necessary to ensure that the resulting output is consistent through different runs and with the literature quoted above.

Lie algebra mode

By default, demonblast runs in Lie algebra mode. This means that it attempts to classify all Lie algebras for every diagram. This is only possible when the quadratic equations corresponding to the Jacobi identity reduce to linear equations (which is very often the case in low dimensions, since most structure constants can be assumed to be ±1 by exploiting the action of diagonal matrices, see [CR2019]).

Since GiNaC does not produce canonical output, the resulting structure constants may differ in subsequent runs. Additionally, there are cases in which demonblast is not able to solve all equations, and they must be handled manually.

To address this, demonblast allows caching the structure constants by means of the option --coefficients store. This has the effect of storing the structure constants in files contained in the directory coefficients. One can then instruct demonblast to retrieve structure constants from these files by means of the option --coefficients load.

Notice that the coefficient files can be edited manually, which allows one to plugin explicit solutions for the quadratic equations, or simply to rename the parameters.

demonblast is currently shipped with coefficient files up to dimension 9, consistently with the tables of [CR19].

Diagram mode

Invoking demonblast with the option --mode diagram has the effect of classifying nice diagrams without regard for the underlying Lie algebra structures.

Table mode

Invoking demonblast with the option --mode table has the effect of printing out a table of Lie algebras corresponding to the indicated partitions.

BUG: this is printed to a file in --partition mode.

Metrics

The option --diagonal-ricci-flat-metrics instructs demonblast to print out the signatures of diagonal Ricci-flat metrics as in [CR19b].

The option --diagonal-nilsoliton-metrics instructs demonblast to print out the signatures of diagonal nilsoliton metrics as in [CR22]. In this context, this means that the Ricci operator takes the form Ric=lambda Id + D, with lambda a nonzero constant and D a derivation. In particular, one recovers diagonal Einstein metrics of nonzero scalar curvature in the case where D is zero. Notice that a nilsoliton metric as above is Einstein if and only if all derivations have zero trace, i.e. the Nikolayevsky derivation is zero. Thus, the diagonal Einstein metrics of [CR20] can be computed by invoking

build/demonblast  --all-partitions 8 --mode table --diagonal-nilsoliton-metrics --only-traceless-derivations

Restrictions

The following options have the effect of restricting the diagrams that are considered

• --only-traceless-derivations only include diagrams for which the Nikolayevsky derivation is zero
• --simple-Nikolayevsky true only include diagrams for which the Nikolayevsky derivation is simple (i.e. has distinct eigenvalues)
• --simple-Nikolayevsky false only include diagrams for which the Nikolayevsky derivation is simple (i.e. has distinct eigenvalues)
• --kernel-root-matrix-dimension arg only include diagrams for which the dimension of the kernel of the root matrix satisfies the indicated equality or inequality. For instance, --kernel-root-matrix-dimension ">0" selects diagrams for which the kernel has positive dimension, and --kernel-root-matrix-dimension "=0" is equivalent to --only-traceless-derivations
• --cokernel-root-matrix-dimension arg only include diagrams for which the dimension of the cokernel of the root matrix satisfies the indicated equality or inequality.
• --irreducible only include connected nice diagrams, i.e. irreducible nice Lie algebras.
• --only-with-metric if either --diagonal-ricci-flat-metrics or --diagonal-nilsoliton-metrics is specified, omit nice diagrams where the indicated type of metric does not appear because of the linear obstructions.

Other output

The following options control output:

• --derivations include Lie algebra derivations in output.
• --lcs-and-ucs in table mode, write the dimensions of the lower and upper central series.
• --invert inverts the nodes, so that e.g. the Heisenberg Lie algebra appears as (0,0,12) instead of (23,0,0).
• --matrix-data prints out information associated to the root matrix.