The classification of uniserialsl(2)⋉V(m)-modules and a new interpretation of the Racah–Wigner 6 j -symbol

An algebraic independence result related to a conjecture of Dixmier on binary form invariants

“…By Racah's formula for 6j symbols [41, Appendix B] (see also [4, §7] Light square means positive, dark means negative and white square means zero. In general, zeros of 6j symbols are poorly understood (see, e.g., [43,45,14]). As to the curious (k, n) = (2, 3) exception or white square in the top left corner of the picture, it has been given a representation-theoretic explanation in [38].…”

Classification of finite dimensional uniserial representations of conformal Galilei algebras

“…Since every non-trivial ideal of g contains z, it follows that any non-faithful representation of g is obtained from a representation of sl(2) ⋉ V (m). Therefore, the classification of all non-faithful uniserial g-modules follows from [CS1], while the classification of all faithful uniserial g-modules is given by the following theorem, which is the main result of the paper.…”

“…Previously, we obtained a classification of all uniserial g-modules when g = sl(2)⋉ V (a), a ≥ 1, over the complex numbers (see [CS1]), as well as when g is abelian, over a sufficiently large perfect field (see [CS2]). In the first case the classification turns out to be equivalent to determining all non-trivial zeros of the Racah-Wigner 6j-symbol within certain parameters, while in the second a sharpened version of the Primitive Element Theorem plays a central role, specially over finite fields.…”

Indecomposable modules of a family of solvable Lie algebras

“…The socle series 0 = soc 0 (U) ⊂ soc 1 (U) ⊂ · · · ⊂ soc k (U) = U of U is inductively defined by declaring soc i (U)/soc i−1 (U) to be the socle of U/soc i−1 (U), that is, the sum of all irreducible submodules of U/soc i−1 (U) , for 1 ≤ i ≤ k. Then U is uniserial if and only if the socle series of U has irreducible factors. In the last years, the classification of the uniserial modules of important classes of solvable and perfect Lie algebras has been achieved in various research papers [CGS,CS,CS1,Pi,C]. In particular, [Pi] and [C] classify a wider class of modules, called cyclic in [Pi] and perfect cyclic in [C], over the perfect Lie algebras sl(2) ⋉ F 2 and sl(n + 1) ⋉ F n+1 , for F = C, respectively.…”