论文标题
作为$ \ mathrm {suplat} $的共同质量hopf代数的残余物作为偏斜的牙套
Skew Braces as Remnants of Co-quasitriangular Hopf Algebras in $\mathrm{SupLat}$
论文作者
论文摘要
偏斜的牙套最近引起了人们的关注，作为研究Yang-Baxter方程的设定理论解决方案的一种方法。在这里，我们通过研究类别中的Hopf代数，即$ \ Mathrm {suplat} $，介绍了这些解决方案的新方法，该方法是完整晶格的$ \ mathrm {suplat} $，并加入了保留的形态。我们通过证明任何Hopf代数，$ \ Mathcal {h} $中的$ \ MATHRM {suplat} $都有一个相应的组，$ r（\ Mathcal {h}）$来连接这两种方法，我们称之为$ \ \ m nation $ \ nathcal in Ind y y y y y y y y y y y， $ r（\ Mathcal {h}）$，与其组结构兼容。相反，任何具有兼容YBE解决方案的组都可以通过这种方式实现。此外，众所周知，任何此类群体都具有诱导的二级组结构，使其成为左支架的偏斜。通过将小组视为共同Qualiangular Hopf代数的残余，$ \ Mathcal {H} $，该辅助组结构出现为$ \ Mathcal {H h} $的trans变的投影。最后，对于任何YBE解决方案，我们在$ \ mathrm {suplat} $中获得了FRT型Hopf代数，其残余物恢复了解决方案的通用偏斜支撑。
Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, $\mathrm{SupLat}$, of complete lattices and join-preserving morphisms. We connect the two methods by showing that any Hopf algebra, $\mathcal{H}$ in $\mathrm{SupLat}$, has a corresponding group, $R(\mathcal{H})$, which we call its remnant and a co-quasitriangular structure on $\mathcal{H}$ induces a YBE solution on $R(\mathcal{H})$, which is compatible with its group structure. Conversely, any group with a compatible YBE solution can be realised in this way. Additionally, it is well-known that any such group has an induced secondary group structure, making it a skew left brace. By realising the group as the remnant of a co-quasitriangular Hopf algebra, $\mathcal{H}$, this secondary group structure appears as the projection of the transmutation of $\mathcal{H}$. Finally, for any YBE solution, we obtain a FRT-type Hopf algebra in $\mathrm{SupLat}$, whose remnant recovers the universal skew brace of the solution.
