Given a finite bijective non-degenerate set-theoretic solution $(X,r)$ of the Yang--Baxter equation we characterize when its structure monoid $M(X,r)$ is Malcev nilpotent. Applying this characterization to solutions coming from racks, we rediscover s
ome results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of finite abelian racks and quandles. We also investigate bijective non-degenerate multipermutation (not necessarily finite) solutions $(X,r)$ and show, for example, that this property is equivalent to the solution associated to the structure monoid $M(X,r)$ (respectively structure group $G(X,r)$) being a multipermuation solution and that $G=G(X,r)$ is solvable of derived length not exceeding the multipermutation level of $(X,r)$ enlarged by one, generalizing results of Gateva-Ivanova and Cameron obtained in the involutive case. Moreover, we also prove that if $X$ is finite and $G=G(X,r)$ is nilpotent, then the torsion part of the group $G$ is finite, it coincides with the commutator subgroup $[G,G]_+$ of the additive structure of the skew left brace $G$ and $G/[G,G]_+$ is a trivial left brace.
To every involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation on a finite set $X$ there is a naturally associated finite solvable permutation group ${mathcal G}(X,r)$ acting on $X$. We prove that every primitive permut
ation group of this type is of prime order $p$. Moreover, $(X,r)$ is then a so called permutation solution determined by a cycle of length $p$. This solves a problem recently asked by A. Ballester-Bolinches. The result opens a new perspective on a possible approach to the classification problem of all involutive non-degenerate set-theoretic solutions.
Left braces, introduced by Rump, have turned out to provide an important tool in the study of set theoretic solutions of the quantum Yang-Baxter equation. In particular, they have allowed to construct several new families of solutions. A left brace $
(B,+,cdot )$ is a structure determined by two group structures on a set $B$: an abelian group $(B,+)$ and a group $(B,cdot)$, satisfying certain compatibility conditions. The main result of this paper shows that every finite abelian group $A$ is a subgroup of the additive group of a finite simple left brace $B$ with metabelian multiplicative group with abelian Sylow subgroups. This result complements earlier unexpected results of the authors on an abundance of finite simple left braces.
Given a finite non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation and a field $K$, the structure $K$-algebra of $(X,r)$ is $A=A(K,X,r)=Klangle Xmid xy=uv mbox{ whenever }r(x,y)=(u,v)rangle$. Note that $A=oplus_{ngeq 0} A_n$ is
a graded algebra, where $A_n$ is the linear span of all the elements $x_1cdots x_n$, for $x_1,dots ,x_nin X$. One of the known results asserts that the maximal possible value of $dim (A_2)$ corresponds to involutive solutions and implies several deep and important properties of $A(K,X,r)$. Following recent ideas of Gateva-Ivanova cite{GI2018}, we focus on the minimal possible values of the dimension of $A_2$. We determine lower bounds and completely classify solutions $(X,r)$ for which these bounds are attained in the general case and also in the square-free case. This is done in terms of the so called derived solution, introduced by Soloviev and closely related with racks and quandles. Several problems posed in cite{GI2018} are solved.
We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate co
nsequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring $Z G$ of a finite nilpotent group $G$ has a subgroup of finite index that is generated by three nilpotent groups for which we have an explicit description of their generators. Another application is a new construction of free subgroups in the unit group. In all the constructions dealt with, pairs of subgroups $(H,K)$, called strong Shoda pairs, and explicit constructed central elements $e(G,H,K)$ play a crucial role. For arbitrary finite groups we prove that the primitive central idempotents of the rational group algebras are rational linear combinations of such $e(G,H,K)$, with $(H,K)$ strong Shoda pairs in subgroups of $G$.
