Quantization by cochain twists and nonassociative differentials

Abstract

Abstract. We show that several standard associative quantizations in mathematical physics can be expressed as cochain module-algebra twists in the spirit of Moyal products at least to O( 3 ), but to achieve this we twist not by a 2-cocycle but by a 2-cochain. This implies a hidden nonassociavitity not visible in the algebra itself but present in its deeper noncommutative differential geometry, a phenomenon first seen in our previous work on semiclassicalisation of differential structures. The quantisations are induced by a classical group covariance and include: enveloping algebras U (g) as quantisations of g * , a Fedosov-type quantisation of the sphere S 2 under a Lorentz group covariance, the Mackey quantisation of homogeneous spaces, and the standard quantum groups Cq [G]. We also consider the differential quantisation of R n for a given symplectic connection as part of our semiclassical analysis and we outline a proposal for the Dirac operator.

On κ-deformation and triangular quasibialgebra structure

“…It cannot be the limit of the standard quasi triangular structure on U q (so(n)), whose R matrix indeed diverges in the limit (except in the special case of dimension three [38]). But U q (so(n)) can also be endowed with the structure of a triangular quasibialgebra [47] and it would be interesting to see whether there is a limit of this reproducing the R and Φ above. It would also be very nice to have a more geometrical understanding of the intertwiners of representations, perhaps in the spirit of [48]; this might be another way to obtain exact rather than perturbative results.…”

Twisting and $\kappa $-Poincaré

“…We shall show below that it does not exists such f 2 which provides 1 κ 2 terms in the coproducts (25)(26)(27)). Let us notice that due to (28) we get [f 1 , [f 1 , ∆ 0 (N)]] = 0 and we see from (27) that ∆ 2 (N) contains in left factors of tensor product the terms quadratic in P and in right ones the terms linear in N. Such property due to (29) …”

“…where ∆ 2 are given explicitly by formulae (25)(26)(27). We shall show below that it does not exists such f 2 which provides 1 κ 2 terms in the coproducts (25)(26)(27)).…”

“…Interestingly enough, one can introduce suitable quantum modification of Wigner-Inonu contraction which applied to Drinfeld-Jimbo deformation U q (so(3, 2)) of (A)dS algebra provides in the limit R ∞ (R -AdS radius) the κ-deformed quantum Poincaré Hopf algebra [1,23]. It has been shown however by Young and Zegers [24,25] (see also [26,18]) that there exists as well the quantum contraction of universal Drinfeld twist for U q (O(3, 2)) (or U q (O(4, 1))) 3 what permits to introduce κ-deformed Poincaré-Hopf algebra as belonging to the category of triangular quasi-Hopf algebras. In this paper we shall perform for particular choice of κ-Poincaré coproducts simple calculations demonstrating that a twist generating the κ-deformed Poincaré algebra from classical (undeformed) Poincaré-Hopf algebra does not exist even if one considers general cochain twists allowing noncoassociativity (φ = 1 ⊗ 1 ⊗ 1) 4 .…”

On Quasi-Hopf Smash Products and Twisted Tensor Products of Quasialgebras

“…[1], [5], [24]), regarded as a further extension of noncommutative geometry, with the "coordinate algebra" allowed to be nonassociative. By analogy with the associative situation, the twisted tensor product of quasialgebras might be regarded as a representative for the cartesian product of "nonassociative spaces".…”