This is a package of functions for solving Killing-type equations
on homogeneous spaces in Maxima computer algebra system:

http://maxima.sourceforge.net

It is based on prolongation of the respective equations, see [1].
Following types of Killing equations are supported:

- Killing(-Yano) forms -> kfsolve()
- Killing spinors -> kssolve()
- Killing spinor-valued forms -> stdksfsolve()

It works at infinitesimal level, the homogeneous space is represented
as a pair (l,k) of a Lie algebra and its subalgebra, together with
the metric and spin representation:

- (pseudo-) Riemannian space -> riemhom()
- spin space -> spinhom()

The input consists of a list of matrices which form a basis of the Lie
algebra l, and a matrix representing the metric g.  The dimension n of
the space is determined from the size of g.  The bases of the tangent
space and the Lie algebra k are given by splitting the basis of l to the
first n elements and the rest respectively.  The spin space requires
additionally an arbitrary complex or contact structure compatible with g,
which is used to construct the spin representation explicitly.  More or
less complete validation of inputs is automatically performed.

Some Lie algebras and homogeneous spaces are preprogrammed:

- orthogonal LA -> sola()
- Euclidean LA -> eucla()
- Lorentzian LA -> lorla()
- unitary LA -> ula()
- special unitary LA -> sula()
- symplectic LA -> spla1(), spla2()

- standard sphere -> sphere()
- Euclidean space -> euclid()
- hyperbolic space -> hyperb()
- Berger sphere -> bergs()
- quaternionic sphere S^7 -> quats2()
- Aloff-Wallach space W^{1,1} -> w11()

For more information consult the source code or contact the author:

Petr Zima
Mathematical Institute of Charles University, Prague
E-mail: zima@karlin.mff.cuni.cz

Acknowledgment:

The author gratefully acknowledges the support of the grants GAUK 700217
and SVV-2017-260456.

References:

[1] Somberg, P., Zima P.  Killing spinor-valued forms and their
    integrability conditions.  Preprint (2020).  Available online at:
    https://arxiv.org/abs/2003.12431


SIMPLE EXAMPLE OF USAGE
=======================

load ("diag");
load ("grobner");
load ("amat.mac");
load ("homkil.mac");

/* 5-dimensional Berger sphere with family of metrics
   parametrized by s */
m : bergs (3, s);

/* set s as a parameter for solving */
setgcd2vars (s);

/* solve the equation for Killing 1-forms */
kfsolve (m, 1);
/* returns a list [p, c, s, r], where:
   p - number of passes needed for computation
   c - polynomial whose roots are singularities,
       i.e. additional solutions can exist if c = 0
   s - matrix whose columns are solutions (at the origin)
   r - Lie algebra representation on solutions whose
       exponential determines values at other points */