(Co)homology of Lie algebras via algebraic Morse theory

Abstract

The fundamental theorem of cancellation AMT [4] and [11], which is the algebraic generalization of discrete Morse theory [2] for simplicial complexes and smooth Morse theory [10] for differentiable manifolds, is discussed in the context of general chain complexes of free modules.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 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 gln≺ and gln⪯ with respect to any partial ordering ⪯ on [n]. We determine the multiplicative structure of H⁎(gln⪯) 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.