We propose a counterexample to the Gröbner ring conjecture in the irrational case. More precisely, we construct a valuation domain V of Krull dimension 1 and a finitely generated ideal J of V[X,Y] equipped with some irrational monomial order <, such that the ideal LT<(J) generated by the leading terms of the elements of J is not finitely generated.

By work of Belyĭ [2], the absolute Galois group GQ=Gal(Q‾/Q) of the field Q of rational numbers can be embedded into A=Aut(Fˆ2), the automorphism group of the free profinite group Fˆ2 on two generators. The image of GQ lies inside GTˆ, the Grothendieck-Teichmüller group. While it is known that every abelian representation of GQ can be extended to GTˆ, Lochak and Schneps [13] put forward the challenge

Let V be a finite-dimensional vector space over a finite field, and suppose G≤ΓL(V) is a group with a unique subnormal quasisimple subgroup E(G) that is absolutely irreducible on V. A base for G is a set of vectors B⊆V with pointwise stabiliser GB=1. If G has a base of size 1, we say that it has a regular orbit on V. In this paper we investigate the minimal base size of groups G with E(G)/Z(E(G))≅PSLn(q)

Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, SupLat, of complete lattices and join-preserving morphisms. We connect the two methods by showing that any Hopf algebra, H in SupLat, has a corresponding group, R(H), which we call its

Work of Linnell shows that the space of left-orderings of a group is either finite or uncountable, and in the case that the space is finite, the isomorphism type of the group is known—it is what is known as a Tararin group. By defining semiconjugacy of circular orderings in a general setting (that is, for arbitrary circular orderings of groups that may not act on S1), we can view the subspace of left-orderings

Let F be a field of characteristic different from two. Suppose that N2 is the category of finite-dimensional nilpotent Lie algebras of class two over the field F and that ALT is the category of alternating bilinear maps of F-vector spaces. We establish a relation between the category N2 and the category ALT. Then we show that the problem of determining the capability of these Lie algebras reduces to

For an irreducible complex character χ of the finite group G, let π(χ) denote the set of prime divisors of the degree χ(1) of χ. Denote then by ρ(G) the union of all the sets π(χ) and by σ(G) the largest value of |π(χ)|, as χ runs in Irr(G). The ρ-σ conjecture, formulated by Bertram Huppert in the 80's, predicts that |ρ(G)|≤3σ(G) always holds, whereas |ρ(G)|≤2σ(G) holds if G is solvable; moreover,

Let V be a valuation domain with quotient field K. We show how to describe all extensions of V to K(X) when the V-adic completion Kˆ is algebraically closed, generalizing a similar result obtained by Ostrowski in the case of one-dimensional valuation domains. This is accomplished by realizing such extensions by means of pseudo-monotone sequences, a generalization of pseudo-convergent sequences introduced

We take advantage of the correspondence between pseudogroups and inverse quantal frames, and of the recent description of Morita equivalence for inverse quantal frames in terms of biprincipal bisheaves, to define Morita equivalence for pseudogroups and to investigate its applications. In particular, two pseudogroups are Morita equivalent if and only if their corresponding localic étale groupoids are

Let G=SL(2,5) be the special linear group of 2×2-matrices with coefficients in the field with 5 elements. We show that the principal block over a splitting field K of characteristic two of the group algebra KG has a 3-cluster tilting module. This gives the first example of a representation-infinite block of a group algebra having a cluster tilting module and answers a question by Erdmann and Holm.

If k is a field and R is a commutative k-algebra, we explore the question of when the ring DR|k of k-linear differential operators on R is isomorphic to its opposite ring. Under mild hypotheses, we prove this is the case whenever R Gorenstein local or when R is a ring of invariants. As a key step in the proof we show that in many cases of interest canonical modules admit right D-module structures.

The article deals with profinite groups in which centralizers are virtually procyclic. Suppose that G is a profinite group such that the centralizer of every nontrivial element is virtually torsion-free while the centralizer of every element of infinite order is virtually procyclic. We show that G is either virtually pro-p for some prime p or virtually torsion-free procyclic (Theorem 1.1). The same

We first investigate the algebraic structure of vertex algebroids B when B are simple Leibniz algebras. Next, we use these vertex algebroids B to construct indecomposable non-simple C2-cofinite N-graded vertex algebras VB‾. In addition, we classify N-graded irreducible VB‾-modules and examine conformal vectors of these N-graded vertex algebras VB‾.

The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying

In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative

Let A be the path algebra of a Dynkin quiver Q over a finite field, and P be the category of projective A-modules. Denote by C1(P) the category of 1-cyclic complexes over P, and n˜+ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in C1(P). In this paper, we prove the existence of Hall polynomials in C1(P), and then establish a relationship between the Hall

Given a commutative field K of characteristic zero we study in this paper a class of K-algebras which contains the coordinate rings of the classical Danielewski surfaces. We describe their locally nilpotent K-derivations and compute their Makar-Limanov invariant. We also determine the K-algebras in our class that have a trivial Makar-Limanov invariant.

We continue our work, started in [9], on the program of classifying triples (X,Y,V), where X,Y are simple algebraic groups over an algebraically closed field of characteristic zero with X

Given a coalgebra D, a commutative algebra A, a commutative A-algebra B and a measuring ψ:D⊗A→B, we define an algebra [D](A,ψ)B that generalizes the symmetric algebra over the module of Kähler differentials SymB(ΩB/A1). We show that the spectrum of [D](A,ψ)B is isomorphic to a prolongation space as defined by Moosa and Scanlon, providing a direct construction of the latter. These prolongation spaces

In this article we study two classes of integral domains. The first is characterized by having a finite intersection of principal ideals being finitely generated only when it is principal. The second class consists of the integral domains in which a finite intersection of principal ideals is always non-finitely generated except in the case of containment of one of the principal ideals in all the others

Let m be a positive integer and let Ω be a finite set. The m-closure of G≤Sym(Ω) is the largest permutation group on Ω having the same orbits as G in its induced action on the Cartesian product Ωm. The 1-closure and 2-closure of a solvable permutation group need not be solvable. We prove that the m-closure of a solvable permutation group is always solvable for m≥3.

Let R=k[x,y,z] be a standard graded 3-variable polynomial ring, where k denotes any field. We study grade 3 homogeneous ideals I⊆R defining compressed rings with socle k(−s)ℓ⊕k(−2s+1), where s⩾3 and ℓ⩾1 are integers. The case for ℓ=1 was previously studied in [8]; a generically minimal resolution was constructed for all such ideals. The paper [7] generalizes this resolution in the guise of (iterated)

We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over Z, and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie algebra which is flat as a module. As an auxiliary result we prove that the Koszul

We show that the quantized flag manifold at a root of unity has natural affine open covering parametrized by the elements of the Weyl group. In particular, the quantized flag manifold turns out to be a quasi-scheme in the sense of Rosenberg [12].

Let R be a prime ring, Q the Utumi quotient ring of R and Ω the expansion closed set of products of automorphisms and skew derivations of Q. Let Λ⊆Ω∖Ωm, where m≥0, be such that for any w∈Λ(g,h), where g,h are automorphisms of Q, w(xy)−g(x)w(y)−w(x)h(y) is a sum of terms with δ(x) and δ′(y), where δ∈Λ|w|−1 or δ′∈Λ|w|−1 or δ,δ′∈Ωm (Definition 3). Suppose that no monic linear identities with words in

We examine two different m-traces in the category of representations over the quantum Lie superalgebra associated to sl(m|n) at root of unity. The first m-trace is on the ideal of projective modules and leads to new Extended Topological Quantum Field Theories. The second m-trace is on the ideal of perturbative typical modules. We consider the quotient with respect to negligible morphisms coming from

We construct a categorification of the modular data associated with every family of unipotent characters of the spetsial complex reflection group G(d,1,n). The construction of the category follows the decomposition of the Fourier matrix as a Kronecker tensor product of exterior powers of the character table S of the cyclic group of order d. The representations of the quantum universal enveloping algebra

We here both unify and generalize nonassociative structures on typed binary trees, that is to say plane binary trees which edges are decorated by elements of a set Ω. We prove that we obtain such a structure, called an Ω-dendriform structure, if Ω has four products satisfying certain axioms (EDS axioms), including the axioms of a diassociative semigroup. This includes matching dendriform algebras introduced

Let n be a locally nilpotent infinite-dimensional Lie algebra over C. Let U(n) and S(n) be its respective universal enveloping algebra and symmetric algebra. Consider the Jacobson topology on the primitive spectrum of U(n), and the Poisson topology on the primitive Poisson spectrum of S(n). We provide a homeomorphism between the corresponding topological spaces (at the level of points, it gives a bijection

We construct orbifolds of holomorphic lattice vertex operator algebras for non-abelian finite automorphism groups G. To this end, we construct twisted modules for automorphisms g together with the projective representation of the centralizer of g on the twisted module. This allows us to extract the irreducible modules of the fixed-point VOA VG, and to compute their characters and modular transformation

In the paper we describe the class of all solvable extensions of an infinite-dimensional filiform Leibniz algebra. The filiform Leibniz algebra is taken as a maximal pro-nilpotent ideal of a residually solvable Leibniz algebra. It is proved that the second cohomology group of the extension is trivial.

A quandle is an algebraic system with a binary operation satisfying three axioms modelled on the three Reidemeister moves of planar diagrams of links in the 3-space. The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian

For any integers m,n with m≠0 and n>0, let Gm,n denote the group presented by 〈x,y,z|x=[zm,x][zn,y]〉; for any integers m,n>0, let Hm,n denote the group presented by 〈x,y,z|x=[xm,zn][y,z]〉. By investigating cohomology jump loci of irreducible GL(2,C)-character varieties, we show: if m,m′≠0, n,n′>0 and Gm′,n′≅Gm,n, then m=m′,n=n′; if m,m′,n,n′>0 and Hm′,n′≅Hm,n, then m′=m,n′=n.

We introduce a new framework for solving an important class of computational problems involving finite permutation groups, which includes calculating set stabilisers, intersections of subgroups, and isomorphisms of combinatorial structures. Our techniques are inspired by and generalise ‘partition backtrack’, which is the current state-of-the-art algorithm introduced by Jeffrey Leon in 1991. But, instead

The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant γG(M) for a finite dimensional module M of a finite group G which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant

We say that a finite group G acting on a set Ω has Property (⁎)p for a prime p if Pω is a Sylow p-subgroup of Gω for all ω∈Ω and Sylow p-subgroups P of G. Property (⁎)p arose in the recent work of Tornier (2018) on local Sylow p-subgroups of Burger-Mozes groups, and he determined the values of p for which the alternating group An and symmetric group Sn acting on n points has Property (⁎)p. In this

A subset of a group invariably generates the group if it generates even when we replace the elements by any of their conjugates. In a 2016 paper, Pemantle, Peres and Rivin show that the probability that four randomly selected elements invariably generate Sn is bounded away from zero by an absolute constant for all n. Subsequently, Eberhard, Ford and Green have shown that the probability that three

The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the 195826352 conjugacy classes of transitive subgroups of the symmetric group S48 of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive

We study those pre-Hilbert absolute-valued algebras satisfying the identity (x2,y,x2)=0. We prove that such an algebra A is finite-dimensional in each one of the following two cases: (1) A satisfies the additional identity (x,x2,x)=0, (2) A contains a weak left-unit. In the first case A is flexible and isomorphic to either R, C, C⁎, H, H⁎, O, O⁎ or P. In the second one A has a left-unit and is isomorphic

Let A be a nondegenerate dimer (or ghor) algebra on a torus, and let Z be its center. Using cyclic contractions, we show the following are equivalent: A is noetherian; Z is noetherian; A is a noncommutative crepant resolution; each arrow of A is contained in a perfect matching whose complement supports a simple module; and the vertex corner rings eiAei are pairwise isomorphic.

We introduce and discuss an elementary tool from representation theory of finite groups for constructing linear codes invariant under a given permutation group G. The tool gives theoretical insight as well as a recipe for computations of generator matrices and weight distributions. In some interesting cases a classification of code vectors under the action of G can be obtained. As an explicit example

Given a sequence d→=(d1,…,dk) of natural numbers, we consider the Lie subalgebra h of gl(d,F), where d=d1+⋯+dk and F is a field of characteristic 0, generated by two block upper triangular matrices D and E partitioned according to d→, and study the problem of computing the nilpotency degree m of the nilradical n of h. We obtain a complete answer when D and E belong to a certain family of matrices that

Let F be a finite field. Consider a direct sum V of an infinite number of copies of F, consider the dual space V⋄, i.e., the direct product of an infinite number of copies of F. Consider the direct sum V=V⋄⊕V. The object of the paper is the group GL‾ of continuous linear operators in V. We reduce the theory of unitary representations of GL‾ to projective representations of a certain category whose

We prove that the quantum toroidal algebras Es associated with different root systems s of glm|n type are isomorphic. We also show the existence of Miki automorphism of Es, which exchanges the vertical and horizontal subalgebras. To obtain these results, we establish an action of the toroidal braid group on the direct sum ⊕sEs of all such algebras.

We show that if the principal p-block of a finite group G contains exactly 5 irreducible ordinary characters, then a Sylow p-subgroup of G has order 5, 7 or is isomorphic to one of the non-abelian 2-groups of order 8.

We introduce the notions of τ-exceptional and signed τ-exceptional sequences for any finite dimensional algebra. We prove that for a fixed algebra of rank n, and for any positive integer t≤n, there is a bijection between the set of signed τ-exceptional sequences of length t, and (basic) ordered support τ-rigid objects with t indecomposable direct summands. If the algebra is hereditary, our notions

In this paper, we explore a canonical connection between the algebra of q-difference operators V˜q, affine Lie algebras and affine vertex algebras associated to certain subalgebra A of the Lie algebra gl∞. We also introduce and study a category R of V˜q-modules. More precisely, we obtain a realization of V˜q as a covariant algebra of the affine Lie algebra A⁎ˆ, where A⁎ is a 1-dimensional central extension

Inspired by the work of Amitsur [1] on finite groups whose irreducible characters all have degree (multiplicity) 1 or 2, in this paper we study association schemes whose irreducible characters all have multiplicity 1 or 2. We will first show that the general case can be reduced to commutative association schemes. Then for commutative association schemes with multiplicities 1 or 2, we prove that their

Let R be a commutative ring with identity and let I be a two-generated ideal of R. We denote by SL2(R) the group of 2×2 matrices over R with determinant 1. We study the action of SL2(R) by matrix right-multiplication on V2(I), the set of generating pairs of I. Let Fitt1(I) be the second Fitting ideal of I. Our main result asserts that V2(I)/SL2(R) identifies with a group of units of R/Fitt1(I) via

We develop a version of Bass-Serre theory for Lie algebras (over a field k) via a homological approach. We define the notion of fundamental Lie algebra of a graph of Lie algebras and show that this construction yields Mayer-Vietoris sequences. We extend some well known results in group theory to N-graded Lie algebras: for example, we show that one relator N-graded Lie algebras are iterated HNN extensions

We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let S be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite right restriction Ehresmann semigroups whose categories are

Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau–Ginzburg models dual to punctured Riemann surfaces. For a consistent dimer embedded in a torus, we explicitly compute the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics

Throughout this paper, all groups are finite and G always denotes a finite group. Let σ={σi|i∈I} be a partition of the set of all primes P. The group G is said to be: σ-primary if G is a σi-group for some i=i(G); σ-nilpotent if G=G1×…×Gn for some σ-primary groups G1,…,Gn; σ-soluble if every chief factor of G is σ-primary; σ-full if G possesses a Hall σi-subgroup for all i such that σi∩π(G)≠∅. A subgroup

For an infinite field k of positive characteristic, we determine all homomorphisms between Weyl modules for GLn(k), where one of the partitions is a hook. As a consequence we obtain a non-vanishing result concerning homomorphisms between Weyl modules for algebraic groups of type B, C and D when one of the partitions is a hook.

Let π1,…,πk be smooth irreducible representations of p-adic general linear groups. We prove that the parabolic induction product π1×⋯×πk has a unique irreducible quotient whose Langlands parameter is the sum of the parameters of all factors (cyclicity property), assuming that the same property holds for each of the products πi×πj (i

Let G be a finite group, and T(G) be the sum of all complex irreducible character degrees of G. In this paper, we aim to characterize the structure of finite groups in terms of T(G). We show that if |G|/T(G)<(p+1)/2, then G has a normal Sylow p-subgroup; and that if |G|/T(G)<3p2/(p2+2), then G is p-supersolvable, where p is a prime.

We show that for a given exact category, there exists a bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed length abelian subcategories). In particular, we show that a length exact category is abelian if and only if simple objects form a semibrick, that is, Schur's lemma holds.

In this paper, we define the notion of Hopf crossed squares for cocommutative Hopf algebras extending the notions of crossed squares of groups and of Lie algebras. We prove the equivalence between the category of Hopf crossed squares and the category of double internal groupoids in the category of cocommutative Hopf algebras. The Hopf crossed squares turn out to be the internal crossed modules in the

A pseudo-Riemannian Lie group (G,〈⋅,⋅〉) is a connected and simply connected Lie group with a left-invariant pseudo-Riemannian metric of type (p,q). This paper is to study pseudo-Riemannian Lie groups with non-Killing conformal vector fields induced by derivations which is an extension from non-Killing left-invariant conformal vector fields. First we prove that a Riemannian (i.e. type (n,0)), Lorentzian

We study Gorenstein flat objects in the category Rep(Q,R) of representations of a left rooted quiver Q with values in Mod(R), the category of all left R-modules, where R is an arbitrary associative ring. We show that a representation X in Rep(Q,R) is Gorenstein flat if and only if for each vertex i the canonical homomorphism φiX:⊕a:j→iX(j)→X(i) is injective, and the left R-modules X(i) and CokerφiX