Let g be a Lie algebra and h a subalgebra of g.  Then the vector space A^p(g, h) of h relative p-chains consists of those p-forms alpha on g such that Hook(x, alpha) = 0 and Hook(x, d(alpha)) = 0 for all x in h.  A p-form alpha in A^p(g, h) is closed if d(alpha) = 0 and exact if alpha = d(beta) for some (p - 1)-form beta in A^(p - 1)(g, h).  The Lie algebra cohomology H^p(g, h) of g relative to h in degree p is the vector space of closed p-forms in A^p(g, h), modulo the exact p-forms in A^p(g, h).   The Lie algebra cohomology H^p(g, h) of g relative to h with coefficients in a representation rho of g may also be computed.  The text by D. B. Fuks (Chapter 1) and the papers by Hochschild and Koszul contain the basic material on Lie Algebra cohomology.

The  LieAlgebra package currently contains 3 commands: RelativeChains, Cohomology, and CohomologyDecomposition.  for finding Lie algebra cohomology. Future extensions of this package will include commands for computing the Serre-Hochschild spectral sequence.

The command RelativeChains(h) returns a list C = [C1, C2, C3, ...] of all relative chains for the subalgebra h.  The zero chain is C1 = [] and the 1-chain is C1 = annihilator(h).  If h has dimension m, then the list C of relative chains will have m (possibly empty) elements.

The command Cohomology(C) computes the cohomology of the sequence C = [C1, C2, C3, ...] with respect to the exterior derivative d.  This requires that d(Ci) is contained in Cj, j = i + 1, for all i = 1, 2, 3, ....  If C1 is a list of (p - 1)-forms on g, then H = Cohomology(C) is a list H = [H1, H2, H3, ...], where H1 is a list of p-forms on g which define a basis for the cohomology at C2, H2 is a list of (p + 1)-forms on g which define a basis for the cohomology at C3, and so on.

The command CohomologyDecomposition(alpha, H, h) returns a pair of forms beta, delta such that alpha = beta + d(delta) and beta is a linear combination of the cohomology representatives given by H and delta is a h relative form The form beta is uniquely determined, the form delta is not.  In particular, if the closed form alpha is exact, then beta = 0.

The command CohomologyDecomposition(alpha, H, R) returns a pair of forms beta, delta such that alpha = beta + d(delta) and beta is a linear combination of the cohomology representatives given by H and delta is in R.