Partially-massless higher-spin algebras and their finite-dimensional truncations

Abstract: The global symmetry algebras of partially-massless (PM) higher-spin (HS) fields in (A)dS_d+1 are studied. The algebras involving PM generators up to depth 2 (ℓ − 1) are defined as the maximal symmetries of free conformal scalar field with 2 ℓ order wave equation in d dimensions. We review the construction of these algebras by quotienting certain ideals in the universal enveloping algebra of (A)dS_d+1 isometries. We discuss another description in terms of Howe duality and derive the formula for computing trace in these algebras. This enables us to explicitly calculate the bilinear form for this one-parameter family of algebras. In particular, the bilinear form shows the appearance of additional ideal for any non-negative integer values of ℓ − d/2 , which coincides with the annihilator of the one-row ℓ-box Young diagram representation of sod+2$$ \mathfrak{s}{\mathfrak{o}}_{d+2} $$. Hence, the corresponding finite-dimensional coset algebra spanned by massless and PM generators is equivalent to the symmetries of this representation.