Notes on higher-spin algebras: minimal representations and structure constants

TL;DR

This work clarifies higher-spin algebras as symmetry algebras of minimal representations of classical Lie algebras, and provides explicit invariant bilinear and trilinear forms for the $rak{sp}_{2N}$, $rak{sl}_{N}$ (via $hs_{\lambda}$), and $rak{so}_{N}$ families. By realizing HS algebras as quotients of Universal Enveloping Algebras by Joseph ideals and exploiting reductive dual pairs, the authors derive concrete generating-function expressions, trace formulas, and determinant-based evaluations of $n$-linear forms, including the notable $G^{(n)}$ determinants. They demonstrate isomorphisms among the HS algebras (e.g., $hs(\frak{so}_5)\simeq hs(\frak{sp}_4)$ and $hs(\frak{so}_6)\simeq hs_0(\frak{sl}_4)$) and analyze how special values of the deformation parameter $\lambda$ yield ideals and finite-dimensional truncations, attaching a coset interpretation to these truncations. The paper also connects to the well-studied $3D$ HS algebra $hs[\lambda]$ via the $N=2$ case and discusses reduced-oscillator formulations and the Lone-Star product, with outlooks toward applications in Vasiliev's equations, correlation functions, and broader HS theories involving mixed-symmetry and partially-massless fields.

Abstract

The higher-spin (HS) algebras so far known can be interpreted as the symmetries of the minimal representation of the isometry algebra. After discussing this connection briefly, we generalize this concept to any classical Lie algebras and consider the corresponding HS algebras. For sp(2N) and so(N), the minimal representations are unique so we get unique HS algebras. For sl(N), the minimal representation has one-parameter family, so does the corresponding HS algebra. The so(N) HS algebra is what underlies the Vasiliev theory while the sl(2) one coincides with the 3D HS algebra hs[lambda]. Finally, we derive the explicit expression of the structure constant of these algebras --- more precisely, their bilinear and trilinear forms. Several consistency checks are carried out for our results.