Finite-N operator algebras and the Hilbert space of bilocal holography

A bstract We give an operator-algebraic and representation-theoretic description of the Hilbert spaces of finite- N bilocal holography. This work is a sequel to the finite- N Hilbert space construction of arXiv:2602.20788 . The central result is the establishment of an invariant dual-pair operator algebra: before imposing the singlet constraint the Fock space carries commuting actions of the color group and of a bilocal Lie algebra, while the projection to the singlet sector selects a single irreducible representation of the invariant Lie algebra, which we call a master algebra . The finite- N reduction has complementary operator-algebraic and invariant-theory manifestations: quadratic identities characterize the selected representation, while the familiar trace relations appear as null-state identities within that representation. We summarize the orthogonal, symplectic and unitary cases, identify the corresponding finite- N constraints, compute the singlet Casimirs, and explain how finite traces and partition functions are obtained through characters of the resulting irreducible representations. This provides a novel, previously unknown mathematical description of the singlet space.