The Chevalley (co)homology table of a Lie algebra is often a tremendous beast. Using AMT, we compute the homology of the Lie algebra of all triangular matrices soln over Q or Zp for large enough primes p . We determine the column and row in the table of 750694235" title="Click to view the MathML source">Hk(soln;Z) where the p -torsion first appears. Module Hk(soln;Zp) is expressed by the homology of a chain subcomplex for the Lie algebra of all strictly triangular matrices niln, using the Künneth formula. All conclusions are accompanied by computer experiments.
Then we generalize some results to Lie algebras of (strictly) triangular matrices and
with respect to any partial ordering ⪯ on [n]. We determine the multiplicative structure of
w.r.t. the cup product over fields of zero or sufficiently large characteristic, the result being the exterior algebra.
Matchings used here can be analogously defined for other Lie algebra families and in other (co)homology theories; we collectively call them normalization matchings. They are useful for theoretical as well as computational purposes.