Deformation Quasi-Hopf Algebras of Non-semisimple Type from Cochain Twists

Abstract

One way to obtain Quantized Universal Enveloping Algebras (QUEAs) of non-semisimple Lie algebras is by contracting QUEAs of semisimple Lie algebras. We prove that every contracted QUEA in a certain class is a cochain twist of the corresponding undeformed universal envelope. Consequently, these contracted QUEAs possess a triangular quasi-Hopf algebra structure. As examples, we consider κ-Poincaré in 3 and 4 spacetime dimensions.

Beyond Fock space in three-dimensional semiclassical gravity

“…In this work however we focus mainly on the construction of the Hilbert space of the theory in terms of states labelled by momentum eigenvalues. The reason we focus on this basic aspect is that in analogous four dimensional models a consistent formulation of Fock space has proved to be a surprisingly difficult task [5][6][7][8][9][10]. The results we present here show that in three dimensions a formulation of Fock space consistent with the group nature of momenta and their associated deformed symmetries is in fact possible.…”

Twisting and $\kappa $-Poincaré

“…-however the cohomological arguments imply the existence of universal Drinfeld twist for D=4 κ-Poincaré [25] we do not have even an explicit formula for Drinfeld twist describing the U q (o(3, 2)) deformation 8 before quantum κ-contraction.…”

“…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 .…”