## Seminar (Ring Theory)

Ferran Cedó (Universitat Autònoma de Barcelona)

Construcció de noves braces finites simples

Resum: Aquest és un treball conjunt amb l’Eric Jespers i el Jan Okninski. Donat un grup abelià finit A qualsevol, explicaré com construir braces simples finites amb grup multiplicatiu metabelià (és a dir, amb longitud derivada 2) tals que $A$ és isomorf a un subgrup del seu grup additiu. Abans d’aquest treball, cap de les braces simple finites conegudes contenia elements amb ordre additiu  4. En un treball anterior (junt amb David Bachiller, Eric Jespers i Jan Okninski), s’havien construït braces finites simples tals que el seu grup additiu contenia qualsevol grup abelià prefixat d’ordre senar, però el grup multiplicatiu d’aquestes braces era de longitud derivada 3.

## Seminar (Ring Theory)

Eric Jespers (Vrije Universiteit Brussel)

Associative structures associated to set-theoretic solutions of the Yang–Baxter equation

Abstract: Let $(X,r)$ be a set-theoretic solution of the YBE, that is $X$ is a set and $r\colon X\times X \to X\times X$ satisfies
$$(r \times \mathrm{id})\circ (\mathrm{id} \times r)\circ (r \times \mathrm{id}) = (\mathrm{id} \times r)\circ (r \times \mathrm{id})\circ (\mathrm{id} \times r)$$ on $X^{3}$. Write $r(x,y)=(\lambda_x (y), \rho_y (x))$, for $x,y\in X$. Gateva-Ivanova and Majid showed that the study of such solutions is determined by solutions $(M,r_M)$, where
$M=M(X,r) =\langle x\in X\mid xy=\lambda_x(y) \rho_y(x), \text{ for all } x,y\in X \rangle$
is the structure monoid of  $(X,r)$, and $r_M$ restricts to $r$ on $X^2$. For left non-degenerate solutions, i.e. all $\sigma_x$ are bijective, it has been shown that $M(X,r)$ is a regular submonoid of $A(X,r)\rtimes \mathcal{G}(X,r)$, where $\mathcal{G}(X,r)=\langle \lambda_x\mid x\in X\rangle$ is the permutation group of $(X,r)$, and
$A(X,r) =\langle x\in X \mid x\lambda_{x}(y) =\lambda_{x}(y) \lambda_{\sigma_{x}(y)}(\rho_{y}(x) \rangle$
is the derived monoid of $(X,r)$. It also is the structure monoid of the rack solution $(X,r’)$ with
$r'(x,y)=(y,\lambda_y\rho_{\lambda^{-1}_x(y)}(x)).$
This solution “encodes”  the relations determined by the map $r^{2} \colon X^{2} \to X^{2}$. The elements of $A=A(X,r)$ are normal, i.e.  $aA=Aa$ for all $a\in A$. It is this “richer structure” that has been exploited by several authors to obtain information on the structure monoid $M(X,r)$ and the structure algebra $kM(X,r)$.

In this talk  we report on some  recent investigations for arbitrary solutions, i.e. not necessarily left non-degenerate nor bijective.
This is joint work with F. Ced\’o and C. Verwimp.  We prove that there is a  unique $1$-cocycle $M(X,r)\to A(X,r)$ and we determine when this mapping is injective, surjective, respectively bijective. One then obtains a monoid homomorphism $M(X,r) \to A(X,r)\rtimes \langle \sigma_x \mid x\in X\rangle$. This mapping is injective when all $\sigma_x$ are injective. Further we determine the left cancellative congruence $\eta$ on $M(X,r)$ and show that $(X,r)$ induces a set-theoretic solution in $M(X,r)/\eta$ provided $(X,r)$ is left non-degenerate.
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