V. D. Lyakhovsky

Extended Jordanian twists for Lie algebras

Jordanian quantizations of Lie algebras are studied using the factorizable twists. For a restricted Borel subalgebra 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 UF(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.

Chains of twists for classical Lie algebras

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

Jordanian twists on deformed carrier subspaces

The nontrivial subspaces with primitive coproducts are found in the deformed universal enveloping algebras. They can form carrier subspaces 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.

Chains of Frobenius subalgebras of so(M) and the corresponding twists

Chains of extended Jordanian twists are studied for the universal enveloping algebras U(so(M)). The carrier subalgebra of a canonical chain FB0≺pmax cannot cover the maximal nilpotent subalgebra N+(so(M)). We demonstrate that there exist other types of Frobenius subalgebras in so(M) that can be large enough to include N+(so(M)). The problem is that the canonical chains FB0≺p do not preserve the primitivity on these new carrier spaces. We show that this difficulty can be overcome and the primitivity can be restored if one changes the basis and passes to the deformed carrier spaces. Finally, the twisting elements for the new Frobenius subalgebras are explicitly constructed. This gives rise to a new family of universal R-matrices for orthogonal algebras. For a special case of g=so(5) and its defining representation we present the corresponding matrix solution of the Yang–Baxter equation.

Recursive algorithm and branching for nonmaximal embeddings

Recurrent relations for branching coefficients in affine Lie algebras integrable highest weight modules are studied. The decomposition algorithm based on the injection fan technique is developed for the case of an arbitrary reductive subalgebra. In particular, we consider the situation where the Weyl denominator becomes singular with respect to the subalgebra. We demonstrate that for any reductive subalgebra it is possible to define the injection fan and the analogue of the Weyl numerator—the tools that describe explicitly the recurrent properties of branching coefficients. Possible applications of the fan technique in conformal field theory models are considered.

Recursion relations and branching rules for simple Lie algebras

The branching rules between simple Lie algebras and their regular (maximal) simple subalgebras are studied. Two types of recursion relations for anomalous relative multiplicities are obtained. One of them is proved to be the factorized version of the other. The factorization property is based on the existence of the set of weights specific for each injection. The structure of is easily deduced from the correspondence between the root systems of the algebra and subalgebra. The recursion relations thus obtained give rise to a simple and effective algorithm for branching rules. The details are illustrated by performing the explicit decomposition procedure for the injection .

Tensor powers for non-simply laced Lie algebras B2-case

We study the decomposition problem for tensor powers of

Quantum Jordanian twist

B_2

On certain properties of branching coefficients for affine Lie algebras

-fundamental modules. To solve this problem singular weight technique and injection fan algorithms are applied. Properties of multiplicity coefficients are formulated in terms of multiplicity functions. These functions are constructed showing explicitly the dependence of multiplicity coefficients on the highest weight coordinates and the tensor power parameter. It is thus possible to study general properties of multiplicity coefficients for powers of the fundamental

Twists in and their quantizations

B_2