Algebras and groups defined by permutation relations of alternating type

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (1)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$ runs through \Alt_{n}, the alternating group, is considered. The associated group, defined by the same (group) presentation, is described. A description of the radical of the algebra is found. It turns out that the radical is a finitely generated ideal that is nilpotent and it is determined by a congruence on the underlying monoid, defined by the same presentation

Finitely Presented Monoids and Algebras defined by Permutation Relations of Abelian Type, II

The class of finitely presented algebras A over a field K with a set of generators x_{1},...,x_{n} and defined by homogeneous relations of the form x_{i_1}x_{i_2}...x_{i_l}=x_{sigma(i_1)}x_{sigma(i_2)}...x_{sigma(i_l)}, where l geq 2 is a given integer and sigma runs through a subgroup H of Sym_n, is considered. It is shown that the underlying monoid S_{n,l}(H)= <x_1,x_2,...,x_n|x_{i_1}x_{i_2}...x_{i_l}=x_{sigma(i_1)}x_{sigma(i_2)}...x_{\sigma (i_l)}, sigma in H, i_1,...,i_l in {1,...,n}> is cancellative if and only if H is semiregular and abelian. In this case S_{n,l}(H) is a submonoid of its universal group G. If, furthermore, H is transitive then the periodic elements T(G) of G form a finite abelian subgroup, G is periodic-by-cyclic and it is a central localization of S_{n,l}(H), and the Jacobson radical of the algebra A is determined by the Jacobson radical of the group algebra K[T(G)]. Finally, it is shown that if H is an arbitrary group that is transitive then K[S_{n,l}(H)] is a Noetherian PI-algebra of Gelfand-Kirillov dimension one; if furthermore H is abelian then often K[G] is a principal ideal ring. In case H is not transitive then K[S_{n,l}(H)] is of exponential growth.Comment: 8 page

Structure monoids of set-theoretic solutions of the Yang-Baxter equation

Given a set-theoretic solution $(X,r)$ of the Yang--Baxter equation, we denote by $M=M(X,r)$ the structure monoid and by $A=A(X,r)$, respectively $A'=A'(X,r)$, the left, respectively right, derived structure monoid of $(X,r)$. It is shown that there exist a left action of $M$ on $A$ and a right action of $M$ on $A'$ and 1-cocycles $\pi$ and $\pi'$ of $M$ with coefficients in $A$ and in $A'$ with respect to these actions respectively. We investigate when the 1-cocycles are injective, surjective or bijective. In case $X$ is finite, it turns out that $\pi$ is bijective if and only if $(X,r)$ is left non-degenerate, and $\pi'$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r)$ is left non-degenerate, in particular $\pi$ is bijective, we define a semi-truss structure on $M(X,r)$ and then we show that this naturally induces a set-theoretic solution $(\bar M, \bar r)$ on the least cancellative image $\bar M= M(X,r)/\eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/\eta$, for example when $(X,r)$ is irretractable, then $\bar r$ is an extension of $r$. It also is shown that non-degenerate irretractable solutions necessarily are bijective.Comment: 21 pages. Some minor changes have been implemented. To appear in Publicacions Matem\`atiques Some additional minor changes were implemente

Braces and symmetric groups with special conditions

We study symmetric groups and left braces satisfying special conditions, or identities. We are particularly interested in the impact of conditions like $\textbf{Raut}$ and $\textbf{lri}$ on the properties of the symmetric group and its associated brace. We show that the symmetric group $G=G(X,r)$ associated to a nontrivial solution $(X,r)$ has multipermutation level $2$ if and only if $G$ satisfies $\textbf{lri}$. In the special case of a two-sided brace we express each of the conditions $\textbf{lri}$ and $\textbf{Raut}$ as identities on the associated radical ring $G_*$. We apply these to construct examples of two-sided braces satisfying some prescribed conditions. In particular we construct a finite two-sided brace with condition $\textbf{Raut}$ which does not satisfy $\textbf{lri}$. (It is known that condition $\textbf{lri}$ implies $\textbf{Raut}$). We show that a finitely generated two-sided brace which satisfies \textbf{lri} has a finite multipermutation level which is bounded by the number of its generators.Comment: 20 page