# “Signature Changing” Generalization of Lie Algebra?

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.

• By invariant, do you mean invariant under the flows of the vector fields? When you call these vector fields an algebra, what is the multiplication operation in the algebra? – Ben McKay Mar 14 '18 at 7:40
• What is a Lie domain? What is the Lie algebra of the boundary of a Lie domain? – Ben McKay Mar 14 '18 at 7:44
• Vector fields on a manifold form a Lie algebra by their Lie bracket. The interesting case, alluded to here, is where the restriction of the algebra to an orbit produces a finite dimensional Lie algebra by virtue of the structure coefficients being constant on the orbit. – Lydia Marie Williamson Mar 17 '18 at 18:25
• Contining on the comment: in such a case, the manifold may be partitioned into "Lie domains", one for each Lie algebra consisting of the union of all the orbits that have that Lie algebra. These issues you can see more clearly with the example posed, which has Lie domains: for so(3,1), se(3) and so(4); the se(3) domain on the boundary of other 2. More interesting for this example is what happens when accounting for the actions on the coordinates, adding in the translation generators T,U,X,Y,Z. Now there are 5 Lie algebras; the se(3) case (vw=0) becoming 3, based on which of v or w is 0. – Lydia Marie Williamson Mar 17 '18 at 18:35
• It's possible to generalize this example by adding in another coordinate q; and deforming the translation operators to aX,aY,aZ,aT+bU,cU, where a,b are functions only of the invariants rr=xx+yy+zz+2wtu+vwuu,s=t+uv given by aa+q(vrr-wss) = 1, cc-qwss=1, b = qrr/(a+c). The 5 original algebra then become 3*5-1, the triplication based on the sign of q (for v=0=w, the q>0 and q<0 algebra are equivalent). Included in this are the (anti-)deSitter algebras and (anti-)Newton-Hooke, as well as the Lie algebras for 4D hyperspherical and 4D hyperbolic geometries. Now there are 14 Lie domains. – Lydia Marie Williamson Mar 17 '18 at 18:44