Let g be a Lie algebra and let Phi: g -> g be a derivation on g.  Then the right extension of g by Phi is the Lie algebra k = g + R (R = real numbers) with Lie bracket [(x, a), (y, b)] = [[x, y] + b*Phi(x) - a*Phi(y), 0], where x, y in g and a, b in R.  The extension k is said to be trivial if k splits as a Lie algebra direct sum k = g' + R, where g' is isomorphic to g.  The extension k is trivial precisely when Phi is an inner derivation.

Let g be a Lie algebra and let beta be a closed 2-form on g.  Then the central extension of g by Phi is the Lie algebra k = g + R (R = real numbers) with Lie bracket [(x, a), (y, b)] = [[x, y], beta(x, y)], where x, y in g and a, b in R.  The extension k is said to be trivial if k splits as a Lie algebra direct sum k = g' + R, where g' is isomorphic to g.  The extension k is trivial precisely when beta is exact, that is, beta  = d(alpha).

LieAlgebraExtension computes a right extension when its second argument is a matrix and a central extension when the second argument is a 2-form.  The procedure returns the Lie algebra data structure for the extended algebra.

The command Extension is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Extension(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Extension(...).