Chains of Frobenius subalgebras of so\((M)\) and the corresponding twists (Q2774706)

Chains of extended Jordanian twists are studied for the universal enveloping algebras \(U(\text{so}(M))\). The carrier subalgebra of a canonical chain \({\mathcal F}_{{\mathcal B}_{0<p\max}}\) cannot cover the maximal nilpotent subalgebra \(N^+(\text{so} (M))\). We demonstrate that there exist other types of Frobenius subalgebras in \(\text{so}(M)\) that can be large enough to include \(N^+(\text{so}(M))\). The problem is that the canonical chains \({\mathcal F}_{{\mathcal B}_{0<p}}\) 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 \(g=\text{so}(5)\) and its defining representation we present the corresponding matrix solution of the Yang-Baxter equation.