Ferran Cedó

Involutive Yang-Baxter groups

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 solutions on finite sets and groups of I-type. A group G of I-type is a group isomorphic to a subgroup of Fa n ⋊ Sym n so that the projection onto the first component is a bijective map, where Fa n is the free abelian group of rank n and Sym n is the symmetric group of degree n. The projection of G onto the second component Sym n we call an involutive Yang-Baxter group (IYB group). This suggests the following strategy to attack Drinfelds problem for involutive non-degenerate set-theoretic solutions. First classify the IYB groups and second, for a given IYB group G, classify the groups of I-type with G as associated IYB group. It is known that every IYB group is solvable. In this paper some results supporting the converse of this property are obtained. More precisely, we show that some classes of groups are IYB groups. We also give a non-obvious method to construct infinitely many groups of I-type (and hence infinitely many involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation) with a prescribed associated IYB group.

Symmetric central configurations of the spatial n-body problem

Abstract 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.

A family of irretractable square-free solutions of the Yang–Baxter equation

Abstract 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 [38, Example 3.9], who first gave a counterexample to Gateva-Ivanova’s Strong Conjecture [19, Strong Conjecture 2.28 (I)]. All the solutions in this subfamily are new counterexamples to Gateva-Ivanova’s Strong Conjecture and also they answer a question of Cameron and Gateva-Ivanova [21, Open Questions 6.13 (II)(4)]. It is proved that the natural left brace structure on the permutation group of the solutions in this family has trivial socle. Properties of the permutation group and of the structure group associated to these solutions are also investigated. In particular, it is proved that the structure groups of finite solutions in this subfamily are not poly-(infinite cyclic) groups.

The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition

We consider algebras over a field K presented by generators x 1 ,...,x n and subject to ( n 2 ) square-free relations of the form x i x j = x k x l with every monomial x i x j , 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 condition. It is known that this dimension is an integer not exceeding n. For n ≥ 4, we construct a family of examples of Gelfand-Kirillov dimension two. We prove that an algebra with the cyclic condition with generators x 1 ,...,x n has Gelfand-Kirillov dimension n if and only if it is of I-type, and this occurs if and only if the multiplicative submonoid generated by x 1 ,...,x n is cancellative.

Nilpotent Groups of Class Three and Braces

There are fourteen ne gradings on the exceptional Lie algebra e6 over an algebraically closed eld of zero characteristic. We provide their descriptions and a proof that any ne grading is equivalent to one of them. 2010 Mathematics Subject Classication: 17B25, 17B70.There are fourteenfine gradings on the exceptional Lie algebra e6 over an algebraically closed field of zero characteristic. We provide their descriptions and a proof that any fine grading is equivalent to one of them.In this paper we prove mixed norm estimates for Riesz transforms on the group SU(2). From these results vector valued inequalities for sequences of Riesz transforms associated to Jacobi differential operators of different types are deduced.In this paper we introduce new techniques in order to deepen into the structure of a Leavitt path algebra with the aim of giving a description of the center. Extreme cycles appear for the first time; they concentrate the purely infinite part of a Leavitt path algebra and, jointly with the line points and vertices in cycles without exits, are the key ingredients in order to determine the center of a Leavitt path algebra. Our work will rely on our previous approach to the center of a prime Leavitt path algebra.In this paper we present a reformulation of the Galois correspondence theorem of Hopf Galois theory in terms of groups carrying farther the description of Greither and Pareigis. We prove that the class of Hopf Galois extensions for which the Galois correspondence is bijective is larger than the class of almost classically Galois extensions but not equal to the whole class. We show as well that the image of the Galois correspondence does not determine the Hopf Galois structure.We show that the product BMO space can be characterized by iterated commutators of a large class of Calderon-Zygmund operators. This result followsfrom a new proof of boundedness of iterated commutators in terms of the BMO norm of their symbol functions, using Hytonens representation theorem of Calderon-Zygmund operators as averages of dyadic shifts. The proof introduces some new paraproducts which have BMO estimates.In this paper we survey some results on the Dirichlet problem for nonlocal operators of the form. We start from the very basics, proving existence of solutions, maximum principles, and constructing some useful barriers. Then, we focus on the regularity properties of solutions, both in the interior and on the boundary of the domain. In order to include some natural operators L in the regularity theory, we do not assume any regularity on the kernels. This leads to some interesting features that are purely nonlocal, in the sense that they have no analogue for local equations. We hope that this survey will be useful for both novel and more experienced researchers in the field.

Gröbner bases for quadratic algebras of skew type

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 shown that under any degree-lexicographic order on the associated free monoid FM n . of rank n the set of normal forms of elements of S is a regular language in FM n . As one of the key ingredients of the proof, it is shown that an identity of the form x N y N = y N x N holds in S . The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K [ S ] is a finite module over a finitely generated commutative subalgebra of the form K [ A ] for a submonoid A of S .

Finitely presented monoids and algebras defined by permutation relations of abelian type

Abstract The class of finitely presented algebras A over a field K with a set of generators x 1 , … , x n defined by homogeneous relations of the form x i 1 x i 2 ⋯ x i l = x σ ( i 1 ) x σ ( i 2 ) ⋯ x σ ( i l ) , where l ≥ 2 is a given integer and σ 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 σ ( i 1 ) x σ ( i 2 ) ⋯ x σ ( i l ) , σ ∈ H , i 1 , … , i l ∈ { 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.

ON SEMIFIR MONOID RINGS

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.

Faithful linear representations of bands

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 [4, Theorem 10] on embeddability of algebras into matrix algebras over a field is corrected and it is proved that if S = F ∪ E is a band with two components E, F such that F is an ideal of S and E is finite, then S is a linear semigroup. Certain sufficient conditions for linearity of a band S, expressed in terms of annihilators associated to S, are also obtained.

SEMIPRIME QUADRATIC ALGEBRAS OF GELFAND–KIRILLOV DIMENSION ONE

We consider algebras over a field K with a presentation K , 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 algebras is described in detail. In particular, we prove that the Gelfand–Kirillov dimension is one while the algebra is noetherian PI and semiprime in case when the field K has characteristic zero. All minimal prime ideals of the algebra are described. It is also shown that the underlying monoid is a semilattice of cancellative semigroups and its structure is described. For any positive integer m, we construct non-degenerate algebras of the considered type on 4m generators that have Gelfand–Kirillov dimension one and are semiprime noetherian PI algebras.