Extended Jordanian twists for Lie algebras

Abstract

Jordanian quantizations of Lie algebras are studied using the factorizable twists. For a restricted Borel subalgebras B ∨ of sl(N ) the explicit expressions are obtained for the twist element F, universal R-matrix and the corresponding canonical element T . It is shown that the twisted Hopf algebra U F (B ∨ ) is self dual. The cohomological properties of the involved Lie bialgebras are studied to justify the existence of a contraction from the Dinfeld-Jimbo quantization to the jordanian one. The construction of the twist is generalized to a certain type of inhomogenious Lie algebras.

Anisotropic Landau-Lifshitz sigma models from q-deformed AdS5×S5 superstrings

“…On the other hand, there is another kind of q-deformations called Jordanian deformations [49][50][51][52]. Jordanian deformed AdS 5 ×S 5 superstring actions have been constructed with linear R-operators satisfying classical Yang-Baxter equation [53].…”

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“…We do not present here the description of the complete κ-deformed algebra. Using the results of the work [11] one can obtain the coproduct applying to the ∆ 0 the twist F of the form: ∆ F (x) = F • ∆ 0 (x) • F −1 , where F = exp (h 6 ⊗ σ(e 6 )) · exp (2λe 1 ⊗ e 5 · e −2σ(e 6 ) ) · exp (2λe 4 ⊗ e 3 e −2σ(e 6 ) ) and σ(e 6 ) = − 1 2 ln (1 − 2λe 6 ) ∼ ln (1 − i κ (P 0 + P 3 )) . It will be given in a forthcoming paper of the present authors.…”

“…In Sect. 4 we shall present some remarks and conclusions (in particular, concerning the structure relations of the κ-deformed d = 4 conformal algebra by applying the twist transformation proposed by Kulish, Lyakhovsky and Mudrov [11]). …”

BRAIDED STATISTICS FROM ABELIAN TWIST IN Κ-Minkowski SPACETIME

“…To understand κ-MST more deeply, twist formalism such as Jordanian twist [13,14,15], κ-like deformation of quantum Weyl and conformal algebra [16] and the light-cone κ-deformation of Poincaré algebra [17] has been tried. In the following, abelian twist in [18,19,20] will be used.…”