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.

For chains of regular injections A p ⊂ A p−1 ⊂ . . . ⊂ A 1 ⊂ A 0 of Hopf algebras the sets of maximal extended Jordanian twists {F E k } are considered. We prove that under certain conditions there exists for A 0 the twist F B k≺0 composed by the factors F E k . The general construction of a chain of twists is applied to the universal envelopings U (g) of classical Lie algebras g . We study the chains for the infinite series A n , B n and D n . The properties of the deformation produced by a chain U (g) F B k≺0 are explicitly demonstrated for the case of g = so(9).

We provide first explicite examples of quantum deformations of D = 4 conformal algebra with mass-like deformation parameters, in applications to quantum gravity effects related with Planck mass. It is shown that one of the classical r-matrices defined on the Borel subalgebra of sl(4) with o(4, 2) reality conditions describes the light-cone κ-deformation of D = 4 Poincaré algebra. We embed this deformation into the three-parameter family of generalized κ-deformations, with r-matrices depending additionally on the dilatation generator. Using the extended Jordanian twists framework we describe these deformations in the form of noncocommutative Hopf algebra. We describe also another four-parameter class of generalized κ-deformations, which is obtained by continuous deformation of distinguished κ-deformation of D = 4 Weyl algebra, called here the standard κ-deformation of Weyl algebra.

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The properties of the set L of extended jordanian twists are studied. It is shown that the boundaries of L contain twists whose characteristics differ considerably from those of internal points. The extension multipliers of these "peripheric" twists are factorizable. This leads to simplifications in the twisted algebra relations and helps to find the explicit form for coproducts. The peripheric twisted algebra U (sl(4)) is obtained to illustrate the construction. It is shown that the corresponding deformation U P (sl(4)) cannot be connected with the Drinfeld-Jimbo one by a smooth limit procedure. All the carrier algebras for the extended and the peripheric extended twists are proved to be Frobenius.

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We study multivariate Chebyshev polynomials associated with root systems. Using properties of specialized singular elements corresponding to a root system , we construct explicitly the measure weight function γ. The latter ensures that these polynomials are orthonormal; it defines the scalar product in the function space where multivariate U-type Chebyshev polynomials constitute a basis. The obtained results are illustrated by constructing and studying 2-variate polynomials for root systems , and .

The nontrivial subspaces with primitive coproducts are found in the deformed universal enveloping algebras. They can form carrier spaces for additional Jordanian twists. The latter can be used to construct sequences of twists for algebras whose root systems contain long series of roots. The corresponding twist for the so(5) algebra is given explicitly.