I have in mind a mathematical structure I've never heard of before. Does anyone know what might be?

It is a manifold with vector fields whose Lie brackets have structure coefficients that are invariant.

**Example**: zY - yZ, xZ - zX, yX - xY, wtX + x(vT - U), wtY + y(vT - U), wtZ + z(vT - U) over ℝ⁷ with coordinates (t,u,v,w,x,y,z) and differential operators (T,U,V,W,X,Y,Z). The invariants are v, w, s ≡ t + uv and r² ≡ x² + y² + z² + 2wtu + vwu² (r² < 0 is possible if w ≠ 0, so r may be imaginary). The first two are homogeneous, but not the last two, though the derived invariant wr² - vs² = w(x² + y² + z²) - vt² is. All the invariants are homogeneous in (t,u,x,y,z), and also in (t,v,w,x,y,z), so one can extend the geometry to an "inhomogeneous" form, by adding in the corresponding translation generators (T,U,X,Y,Z) or (T,V,W,X,Y,Z). The invariants remain so, if the coordinates are replaced by differences or differentials (e.g. s = t + uv becomes ds = dt + u dv or ds = dt + v du); and the original action of the fields on the coordinates becomes part of the adjoint action on the inhomogeneous algebra.

The algebra is not a Lie algebra, but its restriction to each orbit is. Consequently, the manifold partitions into "Lie domains"; the interfaces between any two Lie domains is a boundary whose Lie algebra is a contraction of the two domains that meet at that interface. So, in the cited example, the domains are so(3,1) for vw > 0, se(3) for vw = 0 and so(4) for vw < 0, if just looking at the original 6 fields. The interface between the vw > 0 and vw < 0 domains is the domain vw = 0 which has a contraction se(3) of both so(3,1) and so(4) on it.

If including the fields (T,U,X,Y,Z), the domains become iso(3,1) for vw > 0, iso(4) for vw < 0, galileo for v ≠ 0, w = 0, carroll for v = 0, w ≠ 0 and the static group for v = 0, w = 0. Then there are 3 different contractions for vw > 0 → vw = 0 → vw < 0, depending on whether v = 0, w = 0 or both; as well as contractions for v > 0 → v = 0 ← v < 0 for w = 0, and w > 0 → w = 0 ← w < 0 for v = 0.

In Nonlinear Poisson Brackets, Geometry and Quantization (Translations of Mathematical Monographs, volume 119) is a somewhat similar development: a kind of inverse to the process Lie Algebra → Coadjoint Representation → Poisson Manifold that is generally applicable to Poisson Manifolds is discussed and developed here.

Lydia.