Contraction does not preserve generation: the invariant rings of the Bacry–Lévy-Leblond kinematical family

The invariants of kinematical Lie algebras are customarily obtained by contracting those of the simple algebras they come from. This paper exhibits and characterizes an explicit loss of generation in that procedure: the contraction of a generating system may fail to generate the ring while preserving the algebraic independence of its elements. The Carroll algebra exhibits it. Its quadratic invariant, obtained by contraction in the Cayley-Klein scheme, is H squared, whereas the ring is generated by H; the contracted set generates a proper subalgebra although its elements remain independent and remain two, which is the correct number. No count of invariants can detect that difference.

Establishing this requires determining the full rings. Over the parameter space (a,b,e) organizing the family, with the two remaining coefficients forced by Jacobi, the Poisson center is determined exactly, and not only up to some truncation degree: it is R[C2,C4] when (a,e) is not (0,0), it is R[H,C4] on the Carrollian line a=e=0 with b nonzero, and it is R[H,P^2,K^2,P.K] at the static point, freely generated in all three cases. The Hilbert series are therefore exactly 1/((1-t^2)(1-t^4)), 1/((1-t)(1-t^4)) and 1/((1-t)(1-t^2)^3). By the Duflo isomorphism, the center of the enveloping algebra is likewise a polynomial algebra in all three strata.

The instrument is the converse of the sum rule of Joseph and Shafrir, whose price of admission is the computation of the fundamental semi-invariant. It is computed here directly from the Pfaffians of maximal order in the coordinates of the dual, and turns out to be 1 on the generic stratum and at the static point, and H on the Carrollian line. The degree inequality saturates with equality in all three strata.

As by-products: the stratification of the parameter space by the pair (index, fundamental semi-invariant), and the exact count of ten isomorphism classes in the family, with matching upper and lower bounds.

A reproducibility archive accompanies the deposit. All central computations are in exact arithmetic.