A fuzzy bipolar celestial sphere

Abstract

We introduce a non-commutative deformation of the algebra of bipolar spherical harmonics supporting the action of the full Lorentz algebra. Our construction is close in spirit to the one of the non-commutative spherical harmonics associated to the fuzzy sphere and, as such, it leads to a maximal value of the angular momentum. We derive the action of Lorentz boost generators on such non-commutative spherical harmonics and show that it is compatible with the existence of a maximal angular momentum.

BMS algebras in 4 and 3 dimensions, their quantum deformations and duals

“…Concerning bialgebra duals also the case of W 1 is better understood. All bialgebra structures are triangular and fully classified by means of the following Jordanian rmatrices: 16 r n = l 0 ∧ l n . The dual family of Lie algebra brackets in W • 1 reads now [56,58] [ε 0 , ε m ] n = (m − 2n)ε m−i , for m = 0 (B.8)…”

“…In the recent paper [15] we constructed the κ-deformation of the BMS algebra (quantum deformations of BMS algebra were also considered in [16]) B 4 proposed by Barnich and Troessaert [5], which in addition to the infinitely many supertranslations contains an infinite number of superrotations, generalizing the Lorentz sector of the original BMS algebra B 4 BMS . The interest in such a deformation is at least threefold.…”

On localized and coherent states on some new fuzzy spheres

“…where π Λ is the N -dimensional unitary irreducible representation of U so (3). The latter is characterized by the condition π Λ (C) = Λ(Λ + 1), where C = E a E −a is the Casimir (sum over a ∈ {+, 0, −}), and E a make up the Cartan-Weyl basis E a of so (3).…”

“…where π Λ is the N -dimensional unitary irreducible representation of U so (3). The latter is characterized by the condition π Λ (C) = Λ(Λ + 1), where C = E a E −a is the Casimir (sum over a ∈ {+, 0, −}), and E a make up the Cartan-Weyl basis E a of so (3). A set of generators of A Λ alternative to {x + , x − } is therefore {E + , E − } in the π Λ -representation, see [12] for details.…”

“…In particular, such algebras may describe spacetimes at microscopic scales, regularizing ultraviolet divergencies in quantum field theory (QFT) and/or allowing the quantization of gravity, or may help to unify fundamental interactions (see e.g. [1,2,3,4,5]). Noncommutative geometry [6,7,8,9] develops the needed machinery of differential geometry on such algebras.…”