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.
Highlights
For the term HS algebra which will be specified later in the text
Besides the explicit construction through oscillators, the aforementioned HS algebras can be obtained as a quotient of universal enveloping algebra (UEA) of the isometry algebra
HS algebra as a coset of UEA has a deep relation to the theory of minimal representations
Summary
In order to keep the current paper as self-complete as possible, we review the definition and the construction of HS algebras collecting knowledge from the physics and mathematics literature. Our focus is on providing the precise definition and role of HS algebras in physics and introducing the notion of minimal representations
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