Triply factorised groups and skew left braces

Ramón Esteban-Romero (Polytechnic University of Valencia)

19-Nov-2020, 16:00-17:00 (3 years ago)

Abstract: The Yang-Baxter equation is a consistency equation of the statistical mechanics proposed by Yang [Yang67] and Baxter [Baxter73] that describes the interaction of many particles in some scattering situations. This equation lays the foundation for the theory of quantum groups and Hopf algebras. During the last years, the study suggested by Drinfeld [Drinfeld92] of the so-called set-theoretic solutions of the Yang-Baxter equation has motivated the appearance of many algebraic structures. Among these structures we find the *skew left braces*, introduced by Guarnieri and Vendramin [GuarnieriVendramin17] as a generalisation of the structure of left brace defined by Rump [Rump07]. It consists of a set $B$ with two operations $+$ and $\cdot$, not necessarily commutative, that give $B$ two structures of group linked by a modified distributive law.

The multiplicative group $C=(B, {\cdot})$ of a skew left brace $(B, {+}, {\cdot})$ acts on the multiplicative group $K=(B, {+})$ by means of an action $\lambda\colon C\longrightarrow \operatorname{Aut}(K)$ given by $\lambda(a)(b)=-a+a\cdot b$, for $a$, $b\in B$. With respect to this action, the identity map $\delta\colon C\longrightarrow K$ becomes a derivation or $1$-cocycle with respect to $\lambda$. In the semidirect product $G=[K]C=\{(k,c)\mid k\in K, c\in C\}$, there is a diagonal-type subgroup $D=\{(\delta(c), c)\mid c\in C\}$ such that $G=KD=CD$, $K\cap D=C\cap D=1$. This approach was presented by Sysak in [Sysak11-PortoCesareo] and motivates the use of techniques of group theory to study skew left braces.

We present in this talk some applications of this approach to obtain some results about skew left braces. These results have been obtained in collaboration with Adolfo Ballester-Bolinches.

This work has been supported by the research grants PGC2018-095140-B-I00 from the Ministerio de Ciencia, Innovaci\'on y Universidades (Spanish Government), the Agencia Estatal de Investigaci\'on (Spain), and FEDER (European Union), and PROMETEO/2017/057 from the Generalitat (Valencian Community, Spain).

References

[Baxter73] R. Baxter. Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. I. Some fundamental eigenvectors. Ann. Physics, 76(1):1–24, 1973.

[Drinfeld92] V. G. Drinfeld. On some unsolved problems in quantum group theory. In P. P. Kulish, editor, Quantum groups. Proceedings of workshops held in the Euler International Mathematical Institute, Leningrad, fall 1990, volume 1510 of Lecture Notes in Mathematics, pages 1–8. Springer-Verlag, Berlin, 1992.

[GuarnieriVendramin17] L. Guarnieri and L. Vendramin. Skew-braces and the Yang-Baxter equation. Math. Comp., 86(307):2519–2534, 2017.

[Rump07] W. Rump. Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra, 307:153–170, 2007.

[Sysak11-PortoCesareo] Y. P. Sysak. Products of groups and quantum Yang-Baxter equation. Notes of a talk in Advances in Group Theory and Applications, Porto Cesareo, Lecce, Italy, 2011.

[Yang67] C. N. Yang. Some exact results for many-body problem in one dimension with repulsive delta-function interaction. Phys. Rev. Lett, 19:1312–1315, 1967

group theory

Audience: researchers in the topic


GOThIC - Ischia Online Group Theory Conference

Series comments: Please send a message to andrea.caranti@unitn.it to receive a link to the Zoom room where the conference (a series of talks, actually) takes place.

Organizer: Andrea Caranti*
*contact for this listing

Export talk to