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

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 (a - 1) are defined as the maximal symmetries of free conformal scalar field with 2 a 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 a - d/2 , which coincides with the annihilator of the one-row a-box Young diagram representation of . Hence, the corresponding finite-dimensional coset algebra spanned by massless and PM generators is equivalent to the symmetries of this representation.