Quasi-bialgebras from set-theoretic type solutions of the Yang–Baxter equation

Abstract

We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang–Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists, we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-deformed case of set-theoretic solutions, we also construct admissible Drinfeld twists similar to the set-theoretic ones, subject to certain extra constraints dictated by the q-deformation. These findings greatly generalize recent relevant results on set-theoretic solutions and their q-deformed analogues.

Set-theoretic Yang–Baxter & reflection equations and quantum group symmetries

“…Although we will not extend our discussion further on Drinfeld's twist, it is worth noting that the admissibility of the twist was shown in [14], whereas in [14,15] it was proven that the quantum algebra coming from set-theoretic Baxterized solutions is in fact a quasi-bialgebra, and the twist turns the quasi-bialgebra to the Yangian Hopf algebra, as expected from Proposition 1.16. For a detailed discussion on these fundamental issues we refer the interested reader to [14,15]. We should also note that the discovery of the twist provides important information regarding the derivation of the spectrum of the associated quantum integrable systems, especially the ones with special open boundary conditions.…”

“…The next important step is the diagonalizaton of the constructed spin chain like systems [12,13] for open and periodic system. This is a challenging problem, however the discovery of the associated Drinfeld twist [14,15] is a first important step towards this direction. The deeper understanding of the associated Drinfeld twist and the properties of set-theoretic solutions, especially the involutive ones, will provide the necessary background for the derivation of the universal R-matrix in this context.…”

“…Other solutions were also considered and used within the context of cellular automata [31], whereas in [41,30] the theory of braces was used to produce families of new solutions to the reflection equation, and in [8] skew braces were used to produce reflections. Moreover, key connections between set-theoretic solutions, quantum integrable systems and the associated quantum algebras were uncovered in [12,13,14] and [15,16]. Here, we present some of the fundamental recent results, which have opened new paths on the study of quantum integrable systems coming from settheoretic solutions of the Yang-Baxter and reflection equations.…”

Quantum groups, discrete Magnus expansion, pre-Lie & tridendriform algebras

“…Further study on the quantum algebras associated to the Examples 4.9 and 4.10 will follow in future investigations. Example 4.9 in particular is of special interest given recent findings on the characterization of the quantum algebra associated to involutive set-theoretic solutions of the YBE as quasi-bialgebras [15,16]. These quasi-bialgebras naturally emerge after suitably twisting the Yangian [15].…”

Near braces and p$p$‐deformed braided groups

“…The key observation is that by relaxing more and more conditions on the underlying algebraic structures one identifies more general classes of solutions (see, e.g., [16,12,[17][18][19][20]21] and [22][23][24]). It is also worth noting that interesting links with quantum integrable systems [25,26] as well the quasi-triangular quasi-bialgebras [27][28][29][30] have been recently established, opening up new intriguing paths of investigations.…”