Ferran Cedo

Autonomous University of Barcelona | UAB ·  Departamento de Matemáticas

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 some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of fini...

Involutive non-degenerate set theoretic solutions of the Yang-Baxter equation are considered, with a focus on finite solutions. A rich class of indecomposable and irretractable solutions is determined and necessary and sufficient conditions are found in order that these solutions are simple. Then a link between simple solutions and simple left brac...

To every involutive non-degenerate set-theoretic solution [Formula: see text] of the Yang–Baxter equation on a finite set [Formula: see text] there is a naturally associated finite solvable permutation group [Formula: see text] acting on [Formula: see text]. We prove that every primitive permutation group of this type is of prime order [Formula: se...

We study involutive non-degenerate set-theoretic solutions (X,r) of the Yang-Baxter equation on a finite set X. The emphasis is on the case where (X,r) is indecomposable, so the associated permutation group G(X,r) acts transitively on X. One of the major problems is to determine how such solutions are built from the imprimitivity blocks; and also h...

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 some results obtained recently by Lebed and Mortier, and by Lebed and Vendramin on the description of...

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,+,⋅) is a structure determined by two group structures on a set B: an abelian group (B,+) and a gr...

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 permutation group of this type is of prime order $p$. Moreover, $(X,r)$ is then a so called permutation s...

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,+...

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 $A=A(K,X,r)=K\langle X\mid xy=uv \mbox{ whenever }r(x,y)=(u,v)\rangle$. Note that $A=\oplus_{n\geq 0} A_n$ is a graded algebra, where $A_n$ is the linear span of all the elements $x_1\cdots x_n$, for $x_1...

We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang–Baxter equation are explored...

Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been recently discovered. It is thus a fundamental problem to construct and classify all simple left braces, as they can...

We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang--Baxter equation are explore...

The problem of constructing all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation recently has been reduced to the problem of describing all the left braces. In particular, the classification of all finite left braces is fundamental in order to describe all finite such solutions of the Yang-Baxter equation. In this p...

We prove that a finite non-degenerate involutive set-theoretic solution (X,r) of the Yang-Baxter equation is a multipermutation solution if and only if its structure group G(X,r) admits a left ordering or equivalently it is poly-(infinite cyclic).

We prove that a finite non-degenerate involutive set-theoretic solution (X,r) of the Yang-Baxter equation is a multipermutation solution if and only if its structure group G(X,r) admits a left ordering or equivalently it is poly-(infinite cyclic).

Braces were introduced by Rump as a promising tool in the study of the set-theoretic solutions of the Yang-Baxter equation. It has been recently proved that, given a left brace $B$, one can construct explicitly all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation such that the associated permutation group is isomorp...

Braces were introduced by Rump as a promising tool in the study of the set-theoretic solutions of the Yang-Baxter equation. It has been recently proved that, given a left brace $B$, one can construct explicitly all the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation such that the associated permutation group is isomorp...

We show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

Irreducible representations of the plactic monoid M of rank four are studied. Certain concrete families of simple modules over the plactic algebra over a field K are constructed. It is shown that the Jacobson radical of is nilpotent. Moreover, the congruence ρ on M determined by coincides with the intersection of the congruences determined by the p...

We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang-Baxter equation, we call them simply solutions. We show that the structure group G(X,r) of a finite non-trivial solution (X,r) cannot be an Engel group. It is known that the structure group G(X,r) of a finite multipermutation solution (X,r) is a poly-Z group, thus our resu...

New constructions of braces on finite nilpotent groups are given and hence this leads to new solutions of the Yang-Baxter equation. In particular, it follows that if a group G of odd order is nilpotent of class three, then it is a homomorphic image of the multiplicative group of a finite left brace (i.e. an involutive Yang-Baxter group) which also...

A new family of non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation is constructed. All these solutions are strong twisted unions of multipermutation solutions of multipermutation level at most two. A large subfamily consists of irretractable and square-free solutions. This subfamily includes a recent example of Vendramin....

Given a left brace $G$, a method to construct all the involutive, non-degenerate set-theoretic solutions $(Y,s)$ of the YBE, such that $\mathcal{G}(Y,s)\cong G$ is given. This method depends entirely on the brace structure of $G$.

Let $H$ be a subgroup of $\text{Sym}_n$, the symmetric group of degree $n$. For a fixed integer $l \geq 2$, the group $G$ presented with generators $x_1, x_2, \ldots ,x_n$ and with relations $x_{i_1}x_{i_2}\cdots x_{i_l} =x_{\sigma (i_1)} x_{\sigma (i_2)} \cdots x_{\sigma (i_l)}$, where $\sigma$ runs through $H$, is considered. It is shown that $G$...

For a regular representation $H \subseteq \text{Sym}_n$ of the generalized quaternion group of order $n=4k$, with $k\geq 2$, the monoid $S_n(H)$ presented with generators $a_1,a_2,\dots ,a_n$ and with relations $a_1a_2\cdots a_n=a_{\sigma(1)}a_{\sigma(2)}\cdots a_{\sigma(n)}$, for all $\sigma\in H$, is investigated. It is shown that $S_n(H)$ has th...

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},\ldots , a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}\cdots a_{n} =a_{\sigma (1)} a_{\sigma (2)} \cdots a_{\sigma (n)}$, where $\sigma$ runs through a subset $H$ of the symmetric group $\text{Sym}_{n}$ of degree $n$, is investigated. Gro...

A new method to construct involutive non-degenerate set-theoretic solutions $(X^n,r^{(n)})$ of the Yang-Baxter equation from an initial solution $(X,r)$ is given. Furthermore, the permutation group $\mathcal{G}(X^n,r^{(n)})$ associated to the solution $(X^n,r^{(n)})$ is isomorphic to a subgroup of $\mathcal{G}(X,r)$, and in many cases $\mathcal{G}(...

It is shown that every minimal prime ideal of the Chinese algebra of any finite rank is generated by a finite set of homogeneous elements of degree 2 or 3. A constructive way of producing minimal generating sets of all such ideals is found. As a consequence, it is shown that the Jacobson radical of the Chinese algebra is nilpotent. Moreover, the ra...

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is sho...

Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation are discussed and many consequences are derived. In particular, for each positive integer $n$ a finite square-free multipermutation solution of the Yang-Baxter equation with multipermutation level $n$ and an abelian involuti...

We consider algebras over a field K with a presentation K<x1,…,xn:R>, where R consists of square-free relations of the form xixj=xkxl with every monomial xixj, i≠j, appearing in one of the relations. The description of all four generated algebras of this type that satisfy a certain non-degeneracy condition is given. The structure of one of these al...

A band is a semigroup whose elements are idempotents. It is proved that for any field K the commutative K-algebra, constructed in [22. Cedó , F. , Okniński , J. ( 2009 ). Faithful linear representations of bands . Publ. Mat. 53 : 119 – 140 . [Web of Science ®]View all references], associated to a band S with two components E, F such that EFE = F, i...

A band is a semigroup whose elements are idempotents. The paper is motivated by some open problems concerning linear representations of bands. A counterexample to a conjecture stated in F. Cedó and J. Okniński, "Faithful linear representations of bands," Publ. Mat. 53 (1) (2009), 119–140 is given. On the other hand, it is proved that there exist li...

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_1a_2...a_n = a_{sigma(1)}a_{sigma(2)}...a_{sigma(n)}, where sigma runs through an abelian subgroup H of Sym_{n}, the symmetric group, is considered. It is proved that the Jacobson radical of such algebras i...

The class of finitely presented algebras over a field K with a set of generators a1,…,an and defined by homogeneous relations of the form a1a2⋯an=aσ(1)aσ(2)⋯aσ(n), where σ runs through Altn, the alternating group of degree n, is considered. The associated group, defined by the same (group) presentation, is described. A description of the Jacobson r...

It is shown that square free set theoretic involutive non-degenerate solutions of the Yang–Baxter equation whose associated permutation group (referred to as an involutive Yang–Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also i...

A band is a semigroup consisting of idempotents. It is proved that for any field $K$ and any band $S$ with finitely many components, the semigroup algebra $K[S]$ can be embedded in upper triangular matrices over a commutative $K$-algebra. The proof of a theorem of Malcev on embeddability of algebras into matrix algebras over a field is corrected an...

En aquest article s�exposen els conceptes bàsics per entendre la conjectura de Köthe, el context històric en què va sorgir, els avenços més importants sobre la conjectura al llarg de la història i altres conjectures i problemes relacionats.

The class of finitely presented algebras over a field K with a set of generators a1,…,an and defined by homogeneous relations of the form a1a2⋯an=aσ(1)aσ(2)⋯aσ(n), where σ runs through a subset H of the symmetric group Symn of degree n, is introduced. The emphasis is on the case of a cyclic subgroup H of Symn of order n. A normal form of elements o...

In 1992 Drinfeld posed the question of finding the set theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group theoretical interpretation of involutive non-degenerate solutions. Namely, there is a one-to-one correspondence between involutive non-degenerate sol...

Finitely generated linear semigroups over a field K that have intermediate growth are considered. New classes of such semigroups are found and a conjecture on the equivalence of the subexponential growth of a finitely generated linear semigroup S and the nonexistence of free noncommutative subsemigroups in S, or equivalently the existence of a nont...

We give a general model of partially asynchronous, distributed load-balancing algorithms for the discrete load model in parallel computers, where the processor loads are treated as non-negative integers. We prove that all load-balancing algorithms in this model are finite. This means that all load-balancing algorithms based on this model are guaran...

We consider algebras over a field K presented by generators x 1,..., xn and subject to (n2) square-free relations of the form xixj = xkxl with every monomial xixj, i ≠ j, appearing in one of the relations. It is shown that for n > 1 the Gelfand-Kirillov dimension of such an algebra is at least two if the algebra satisfies the so-called cyclic condi...

The structure of the plactic algebra K[M] of rank 2 or 3 is studied. It is shown that these algebras are semiprimitive. Moreover, the prime spectrum is completely described in the rank 2 case. Also it is proved that the plactic algebra of rank n⩾4 is not semiprime and the plactic algebra of rank 3 is not prime.

It is proved that the monoids between an archimedean, irreflexive and rigid monoid Mand its universal group G, obtained by adjoining inverses to M, are archimedean, irreflexive and rigid monoids.Dedicated to the memory of Professor Ahmad Shamsuddin.

Diffusion algorithms are some of the most popular algorithms for dynamic load balancing in which loads move from heavily loaded processors to lightly loaded neighbor processors. To achieve a global load balance in a parallel computer, the algorithm is iterated until the load difference between any two processors is smaller than a specified value. T...

For each finite field K, we construct a commutative Goldie K-algebra R such that the polynomial ring R[x] is not a Goldie ring. This generalizes a construction of Kerr.

We present a new fully distributed dynamic load balancing algorithm called DASUD (Diffusion Algorithm Searching Unbalanced Domains). Since DASUD is iterative and runs in an asynchronous way, a mathematical model that describes DASUD behaviour has been proposed and has been used to prove DASUD's convergence. DASUD has been evaluated by comparison wi...

We present a new fully distributed dynamic load-balancing algorithm called DASUD (Diffusion Algorithm Searching Unbalanced Domains). Since DASUD is iterative and runs in an asynchronous way, a mathematical model that describes DASUD behaviour has been proposed and has been used to prove DASUD's convergence. DASUD has been evaluated by comparison wi...

We fill a gap in [4], and provide a rigorous example of a local ringR whose Jacobson radical is locally nilpotent, butM 2(R) is not strongly π-regular.

We give a new proof of the main result of \cite{1} which does not use the classification of the finite simple groups.

Let α be an ordinal number. It is proved that there exists a monoid M with factorization depth τ(M) = α and the monoid ring R[M]; over any skew field R is a left fir. Furthermore a method for constructing all monoids M such that R[M] is a left fir is given.

Let k be a commutative field. Let G be a locally finite group without elements of order p in case char k = p > 0. In this paper it is proved that the type I8 part of the maximal right quotient ring of the group algebra kG is zero.

It is proved that the universal group of a torsion free rigid monoid is torsion free. As a consequence, a new condition on a monoid M for the monoid ring R[M] to be a 2-fir is given. Furthermore, the monoids between a rigid monoid and its universal group are studied.

By giving new examples of Mal'cev domains, i.e. domains that can not be embedded in any skew field, we answer in the negative some questions of Faith on zip rings.

We characterize the non-planar central configurations of the spatial n-body problem with equal masses which are orbits of a finite group of isometries of R 3. As a corollary we obtain that the spatial n-body problem with equal masses and n > 5 has at least two equivalence classes of non-planar central configurations modulo homotheties and rotations...

We give a new condition on a monoid M for the monoid ring F[M] to be a 2-fir. Furthermore, we construct a monoid M that satisfies all the currently known necessary conditions for F[M] to be a semifir and that the group of units of M is trivial, but M is not a directed union of free monoids.

Le;……ommutative field. Le;……ocally finite group without elements of orde;…n case cha;… …. In this paper it is proved that for some large classes of group;…Δ-hypercentral, residually finite and groups satisfying Min-g for all primes q) the Type I∞ part of the maximal quotient ring of the group ring K[G] is zero.

Let $K$ be a commutative field. Let $G$ be a locally finite group without elements of order $p$ in case char $K = p > 0$. In this paper it is proved that for some large classes of groups $G$ ($\Delta$-hypercentral, residually finite and groups satisfying Min-$q$ for all primes $q$) the Type $mathrm{I}_\infty$ part of the maximal quotient ring of th...

We construct a conical rigid monoid M such that the monoid ring R[M] is not a semifir for any ring R. Thus answering in the negative a question of cohn.

We characterize the regular group algebras whose maximal right and left quotient rings coincide. In fact we prove that if K[G] is a regular group algebra, then Qr(K[G]) = Q1(K[G]) if and only if G is abelian-by-finite. This completes the result of Goursaud and Valette, that prove some special cases, namely when K either has positive characteristic...