Let G be a generalized Baumslag–Solitar group, and let C be a class of groups containing at least one non-trivial group and closed under taking subgroups, extensions, and Cartesian products of the form ∏y∈YXy, where X, Y∈C and Xy is an isomorphic copy of X for every y∈Y. We give a criterion for G to be residually a C-group provided C consists only of periodic groups. We also prove that G is residually

We prove that the isomorphism type of the subrack lattice of a finite group determines the nilpotence class. We analyze the problem of estimating the orders of the group elements corresponding to the atoms of the subrack lattice. As a result, we show that the subrack lattice determines p-nilpotence of the group if a certain condition is met.

Let S be an unramified regular local ring of mixed characteristic two and R the integral closure of S in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements f,g∈S. Let S2 denote the subring of S obtained by lifting to S the image of the Frobenius map on S/2S. When at least one of f,g∈S2, we characterize the Cohen-Macaulayness of R and

In this paper, we introduce and study the notion of a Rota-Baxter operator on a cocommutative Hopf algebra that generalizes notions of a Rota-Baxter operator on a group and a Rota-Baxter operator of weight 1 on a Lie algebra. We show that every Rota-Baxter operator (of weight 1) on a Lie algebra g (resp., on a group G) can be uniquely extended to a Rota-Baxter operator on the universal enveloping algebra

We study the two dual notions of prime avoidance and prime absorbance. We generalize the classical prime avoidance lemma to radical ideals. A number of new criteria are provided for an abstract ring to be compactly packed (C.P., every set of primes satisfies avoidance) or properly zipped (P.Z., every set of primes satisfies absorbance). Special consideration is given to the interaction with chain conditions

We consider Donovan's conjecture in the context of blocks of groups G with defect group D and normal subgroups N◁G such that G=CD(D∩N)N, extending similar results for blocks with abelian defect groups. As an application we show that Donovan's conjecture holds for blocks with defect groups of the form Q8×C2n or Q8×Q8 defined over a discrete valuation ring.

We begin by recalling the role that weak proregularity of an ideal in a commutative ring has in derived completion and adic flatness. We also introduce the new concepts of idealistic and sequential derived completion, and prove a few results about them, including the fact that these two concepts agree iff the ideal is weakly proregular. Next we study the local nature of weak proregularity, and its

Let K be an algebraically closed field of characteristic zero and let KC〚x1,⋯,xe〛 be the ring of formal power series in several variables with exponents in a line free cone C. We consider irreducible polynomials f=yn+a1(x_)yn−1+⋯+an(x_) in KC〚x1,⋯,xe〛[y] whose roots are in KC〚x11n,⋯,xe1n〛. We generalize to these polynomials the theory of Abhyankar-Moh. In particular we associate with any such polynomial

Let V be a valuation domain and let E be a subset of V. For a rank-one valuation domain V, there is a characterization of when Int(E,V) is a Prüfer domain. For a general valuation domain V, we show that Int(E,V) is a Prüfer domain if and only if E is precompact, or there exists a rank-one prime ideal P of V and Int(E,VP) is a Prüfer domain. Then we show that the following statements are equivalent:

The main focus of this paper is on determining the highest non-vanishing local cohomology modules of ΩB/R,ΩB/V(ΩB/k) where R is either a complete regular local ring or a complete local normal domain with coefficient ring V (field k) and B is its integral closure in an algebraic extension of Q(R). Similar problem is also studied over a normal domain R containing a field k of characteristic 0. In this

We define a class of numerical semigroups S, which we call Castelnuovo semigroups, and study the subvariety Mg,1S of Mg,1 consisting of marked smooth curves with Weierstrass semigroup S. We determine the number of irreducible components of these loci and determine their dimensions. Curves with these Weierstrass semigroups are always Castelnuovo curves, which provides the basic tool for our argument

For a partition λ of n∈N, let IλSp be the ideal of R=K[x1,…,xn] generated by all Specht polynomials of shape λ. In the previous paper, the second author showed that if R/IλSp is Cohen-Macaulay, then λ is either (n−d,1,…,1),(n−d,d), or (d,d,1), and the converse is true if char(K)=0. In this paper, we compute the Hilbert series of R/IλSp for λ=(n−d,d) or (d,d,1). Hence, we get the Castelnuovo-Mumford

We find a condition on the underlying graph of an Artin group that fully determines if it is subgroup separable. As a consequence, an Artin group is subgroup separable if and only if it can be obtained from the irreducible spherical Artin groups of ranks 1 and 2 via a finite sequence of free products and direct products with the infinite cyclic group. This result generalizes the Metaftsis-Raptis criterion

In this paper we verify Navarro's refinement of the McKay conjecture for quasi-simple groups of Lie type in their defining characteristic. Navarro's refinement takes into account the action of specific Galois automorphisms on the characters present in the McKay conjecture [13]. Our proof of this case of the conjecture relies on a character correspondence constructed by Maslowski in [12]. Building on

Sets of zero-dimensional ideals in the polynomial ring k[x,y] that share the same leading term ideal with respect to a given term ordering are known to be affine spaces called Gröbner cells. Conca-Valla and Constantinescu parametrize such Gröbner cells in terms of certain canonical Hilbert-Burch matrices for the lexicographical and degree-lexicographical term orderings, respectively. In this paper

We study the existence of a Batalin-Vilkovisky differential on the complete cohomology ring of a Frobenius algebra. We construct a Batalin-Vilkovisky differential on the complete cohomology ring in the case of Frobenius algebras with diagonalizable Nakayama automorphisms.

On construit des homomorphismes entre l'algèbre de Hecke générique Hn(q) de type A et l'algèbre de Hecke affine dégénérée DH(Sn) au moyen d'un élément générique Rqν(X,S) qui satisfait la relation de Hecke quadratique. Par spécialisation des DH(Sn) avec les éléments de Jucys-Murphy associés au groupe symétrique Sn, on obtient des isomorphismes explicites, après extension des scalaires à C(q), entre

Let O be the ring of integers of a number field, and let n≥3. This paper studies bi-interpretability of the ring of integers Z with the special linear group SLn(O), the general linear group GLn(O) and the subgroup Tn(O) of GLn(O) consisting of all the uppertriangular matrices. For each of these groups we provide a complete characterization of arbitrary models of their complete first-order theories

Recently ([17]) we have shown a structure theorem for locally compact groups of polynomial growth. We give now some applications on various growth functions and relations to FCG− - series. In addition, we show some results on related classes of groups.

A group G possesses the Magnus property if for every two elements u, v∈G with the same normal closure, u is conjugate to v or v−1. O. Bogopolski and J. Howie proved independently that the fundamental groups of all closed orientable surfaces possess the Magnus property. The analogous result for closed non-orientable surfaces was proved by O. Bogopolski and K. Sviridov except for one case that was later

We study the elliptic algebras Qn,k(E,τ) introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form a family of quadratic algebras parametrized by coprime integers n>k≥1, an elliptic curve E, and a point τ∈E. We consider and compare several different definitions of the algebras and provide proofs of various statements about them made by Feigin and Odesskii. For example, we

The submonoid of the 3-strand braid group B3 generated by σ1 and σ1σ2 is known to yield an exotic Garside structure on B3. We introduce and study an infinite family (Mn)n≥1 of Garside monoids generalizing this exotic Garside structure, i.e., such that M2 is isomorphic to the above monoid. The corresponding Garside group G(Mn) is isomorphic to the (n,n+1)-torus knot group–which is isomorphic to B3 for

We give a short proof of Green's formula on Hall numbers. By using rotation of triangles, we find that Green's formula is just the associativity of derived Hall numbers.

Following the literature, a group G is called a group of central type if G has an irreducible character that vanishes on G∖Z(G). Motivated by this definition, we say that a character χ∈Irr(G) has central type if χ vanishes on G∖Z(χ), where Z(χ) is the center of χ. Groups where every irreducible character has central type have been studied previously under the name GVZ-groups (and several other names)

Dendriform algebras are certain splitting of associative algebras and arise naturally from Rota-Baxter operators, shuffle algebras and planar binary trees. In this paper, we first consider involutive dendriform algebras, their cohomology and homotopy analogs. The cohomology of an involutive dendriform algebra splits the Hochschild cohomology of an involutive associative algebra. In the next, we introduce

A Lie algebra L is said to be (Θn,sln)-graded if it contains a simple subalgebra g isomorphic to sln such that the g-module L decomposes into copies of the adjoint module, the trivial module, the natural module V, its symmetric and exterior squares S2V and ∧2V and their duals. We describe the multiplicative structures and the coordinate algebras of (Θn,sln)-graded Lie algebras for n≥5, classify these

In this article, we study the relationship among maximal green sequences, complete forward hom-orthogonal sequences and stability functions in abelian length categories. Mainly, we firstly give a one-to-one correspondence between maximal green sequences and complete forward hom-orthogonal sequences via mutual constructions, and then prove that a maximal green sequence can be induced by a central charge

We prove that the Diophantine problem for orientable quadratic equations in free metabelian groups is decidable and furthermore, NP-complete. In the case when the number of variables in the equation is bounded, the problem is decidable in polynomial time.

In this paper we give full classification of rank 3 line arrangements in P2 (over a field of characteristic 0) that have a minimal logarithmic derivation of degree 3. The classification presents their defining polynomials, up to a change of variables, with their corresponding affine pictures. We also analyze the shape of such a logarithmic derivation, towards obtaining criteria for a line arrangement

Kawashima's relation is conjecturally one of the largest classes of relations among multiple zeta values. Gaku Kawashima introduced and studied a certain Newton series, which we call the Kawashima function, and deduced his relation by establishing several properties of this function. We present a new approach to the Kawashima function without using Newton series. We first establish a generalization

We determine the Gerstenhaber structure on the Hochschild cohomology ring of a class of self-injective special biserial algebras. Each of these algebras is presented as a quotient of the path algebra of a certain quiver. In degree one, we show that the cohomology is isomorphic, as a Lie algebra, to a direct sum of copies of a subquotient of the Virasoro algebra. These copies share Virasoro degree 0

In this paper, using Bourbaki's convention, we consider a simple Lie algebra g⊂glm of type B, C or D and a parabolic subalgebra p of g associated with a Levi factor composed essentially, on each side of the second diagonal, by successive blocks of size two, except possibly for the first and the last ones. Extending the notion of a Weierstrass section introduced by Popov to the coadjoint action of the

The extended plus (epf) closure and the rank 1 (r1f) closure are two closure operations introduced by Raymond C. Heitmann for rings of mixed characteristic. Recently, he and Linquan Ma proved that the epf closure satisfies the usual colon-capturing property under mild conditions. In this paper, we extend their result and prove that the epf closure satisfies what we call the p-colon-capturing property

We study fibers of the commutator map on general linear maps using representation theory. Especially, we give detailed information on dimension and geometric connectedness of those fibers over regular semisimple and central elements.

The quantum plane is the non-commutative polynomial algebra in variables x and y with xy=qyx. In this paper, we study the module variety of n-dimensional modules over the quantum plane, and provide an explicit description of its irreducible components and their dimensions. We also describe the irreducible components and their dimensions of the GIT quotient of the module variety with respect to the

Let G be a simple, simply connected linear algebraic group of exceptional type defined over Fq with Frobenius endomorphism F:G→G. In this work we give upper bounds for the number of irreducible Brauer characters in the quasi-isolated ℓ-blocks of GF and GF/Z(GF) when the prime ℓ is bad for G.

Building on previous results concerning hyperbolicity of groups of Fibonacci type, we give an almost complete classification of the (non-elementary) hyperbolic groups within this class. We are unable to determine the hyperbolicity status of precisely two groups, namely the Gilbert-Howie groups H(9,4),H(9,7). We show that if H(9,4) is torsion-free then it is not hyperbolic. We consider the class of

Let Λ be an Artin algebra. In this paper, the notion of nZ-Gorenstein cluster tilting subcategories will be introduced. It is shown that every nZ-cluster tilting subcategory of mod-Λ is nZ-Gorenstein if and only if Λ is an Iwanaga-Gorenstein algebra. Moreover, it will be shown that an nZ-Gorenstein cluster tilting subcategory of mod-Λ is an nZ-cluster tilting subcategory of the exact category Gprj-Λ

Support τ-tilting modules are in bijection with two-term tilting complexes for any finite-dimensional symmetric algebra. This fact motivates us to classify support τ-tilting modules over blocks of finite groups. We classify support τ-tilting modules over particular blocks of finite groups using modular representation theoretical approaches including Clifford theory and Green's indecomposability theorem

For a finite group G and an inverse-closed generating set C of G, let Aut(G;C) consist of those automorphisms of G which leave C invariant. We define an Aut(G;C)-invariant normal subgroup Φ(G;C) of G which has the property that, for any Aut(G;C)-invariant normal set of generators for G, if we remove from it all the elements of Φ(G;C), then the remaining set is still an Aut(G;C)-invariant normal generating

An extremal element in a Lie algebra g over a field of characteristic not 2 is an element x∈g such that [x,[x,g]] is contained in the linear span of x. The linear span of an extremal element, called an extremal point, is an inner ideal of g, i.e. a subspace I satisfying [I,[I,g]]≤I. We show that in characteristic different from 2,3 the geometry with point set the set of extremal points and as lines

We define the notions of a free fusion of structures and a weakly stationary independence relation and use them to prove simplicity for the automorphism groups of order and tournament expansions of homogeneous structures like the rational bounded Urysohn space, the random graph, and the random poset.

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of

We build on the correspondence between abstract key polynomials and minimal pairs made by Novacoski and show how to relate the valuations that are generated by each object. We can then give a geometric interpretation of valuations built in this fashion. To do so we employ an object called diskoid, which is a generalisation of the classical concept of ball in non-archimedian valued fields.

In this paper we study the relations between numerical characteristics of finite dimensional algebras and such classical combinatorial objects as additive chains. We study the behavior of the length function via so-called characteristic sequences of quadratic algebras. As one of our main results, we prove the sharp upper bound for the length of quadratic algebras in terms of the Fibonacci numbers depending

A univariate trace polynomial is a polynomial in a variable x and formal trace symbols Tr(xj). Such an expression can be naturally evaluated on matrices, where the trace symbols are evaluated as normalized traces. This paper addresses global and constrained positivity of univariate trace polynomials on symmetric matrices of all finite sizes. A tracial analog of Artin's solution to Hilbert's 17th problem

This article concerns Ehresmann structures in the partition monoid PX. Since PX contains the symmetric and dual symmetric inverse monoids on the same base set X, it naturally contains the semilattices of idempotents of both submonoids. We show that one of these semilattices leads to an Ehresmann structure on PX while the other does not. We explore some consequences of this (structural/combinatorial

There has been some interest on how the average character degree affects the structure of a finite group. We define, and denote by anz(G), the average number of zeros of characters of a finite group G as the number of zeros in the character table of G divided by the number of irreducible characters of G. We show that if anz(G)<1, then the group G is solvable and also that if anz(G)<12, then G is supersolvable

We show that the derived equivalence between full defect blocks of SL(2,q) in defining characteristic and their Brauer correspondants can be presented as a composition of perverse equivalences.

Ladders of recollements of abelian categories are introduced, and used to address three general problems. Ladders of a certain height allow to construct recollements of triangulated categories, involving derived categories and singularity categories, from abelian ones. Ladders also allow to tilt abelian recollements, and ladders guarantee that properties like Gorenstein projective or injective are

Ultragraphs give rise to labelled graphs. We realize algebras associated to such labelled graphs as groupoid algebras, generalizing a known groupoid algebra realization of ultragraph C*-algebras to any ultragraph. Then, we characterize the shift space of an ultragraph as the tight spectrum of the inverse semigroup associated with an ultragraph via its labelled graph. In the purely algebraic setting

Recently, Brzeziński, Koenig and Külshammer have introduced the notion of normal exact Borel subalgebra of a quasihereditary algebra. They have shown that there exists a one-to-one correspondence between normal directed bocses and quasihereditary algebras with a normal and homological exact Borel subalgebra. In this short note, we prove that every exact Borel subalgebra is automatically normal. As

Let A be a Koszul Artin-Schelter regular algebra, σ a graded automorphism of A and δ a degree-one σ-derivation of A. We introduce an invariant for δ called the σ-divergence of δ. We describe the Nakayama automorphism of the graded Ore extension B=A[z;σ,δ] explicitly using the σ-divergence of δ, and construct a twisted superpotential ωˆ for B so that it is a derivation quotient algebra defined by ωˆ

The main goal of this paper is to characterize limit key polynomials for a valuation ν on K[x]. We consider the set Ψα of key polynomials for ν of degree α. We set p to be the exponent characteristic of ν. Our first main result (Theorem 1.1) is that if F is a limit key polynomial for Ψα, then the degree of F is prα for some r∈N. Moreover, in Theorem 1.2, we show that there exist Q∈Ψα and F a limit

In this paper, we prove that the inductive blockwise Alperin weight condition holds for 2-blocks with abelian defect groups, which are not inertial and have a Klein four hyperfocal subgroup.

Let G be a finite group. We prove that if the set of degrees of characters in the principal p-block of G has size at most 2 then G is p-solvable, and G/Op′(G) has a metabelian normal Sylow p-subgroup. The general question of proving that if an arbitrary p-block has two degrees then their defect groups are metabelian remains open.

Herschend-Liu-Nakaoka introduced the notion of n-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of n-exact categories in the sense of Jasso and (n+2)-angulated in the sense of Geiss-Keller-Oppermann. Let C be an n-exangulated category with enough projectives and enough injectives

We discuss the existence of zero coefficients in the powers of the determinant polynomial of order n. D. G. Glynn proved that the coefficients of the mth power of the determinant polynomial are all nonzero, if m=p−1 with a prime p. We show that the converse also holds, if n≥3. The proof is quite elementary.

For every Lie group G, we compute the maximal n such that an n-fold product of nonabelian free groups embeds into G.

A classical theorem of Veldkamp describes the center of an enveloping algebra of a Lie algebra of a semi-simple algebraic group in characteristic p. We generalize this result to a class of Lie algebras with a property that they arise as the reduction modulo p≫0 from an algebraic Lie algebra g, such that g has no nontrivial semi-invariants in Sym(g) and Sym(g)g is a polynomial algebra. As an application