pinecone/core-2020-05-10-deduplication huggingface dataset:
Structure of A*-Fibrations over One-Dimensional Seminormal Semilocal Domains

AbstractIn this paper we give a structure theorem for an A*-fibration over a one-dimensional noetherian seminormal semilocal domain and show that, in this situation, any A*-fibration whose spectrum occurs as an affine open subscheme of the spectrum of an A1-fibration (equivalently, an affine line A1) is actually A*. The structure theorem provides examples of A*-fibrations over one-dimensional noetherian seminormal semilocal domains whose spectra are not affine open subschemes of any affine line A1 over the base ring. We also construct examples of nontrivial A*-fibrations over one-dimensional noetherian non-seminormal local domains whose spectra are open subschemes of A1-fibrations over the base ring
On Simplicial Toric Varieties Which Are Set-Theoretic Complete Intersections

AbstractIn this paper we prove:&#x02022;In characteristic >0 every simplicial toric affine or projective variety with full parametrization is a set-theoretic complete intersection. This extends previous results by R. Hartshorne (1979, , 380–383) and T. T. Moh (1985, , 217–220).&#x02022;In any characteristic, every simplicial toric affine or projective variety with full parametrization is an almost set-theoretic complete intersection. This extends previous known results by M. Barile and M. Morales (1998, , 1907–1912) and A. Thoma (, to appear).&#x02022;In any characteristic, every simplicial toric affine or projective variety of codimension two is an almost set-theoretic complete intersection.Moreover the proofs are constructive and the equations we find are binomial ones
Non-isomorphic Derived-Equivalent Tubular Curves and Their Associated Tubular Algebras

AbstractLet R be the field of real numbers. We construct two tubular canonical R-algebras (in the sense of C. M. Ringel/W. Crawley–Boevey (1990, in “Topics in Algebra,” Banach Center Publ. No. 26, pp. 407–432)) which are neither isomorphic nor dual to each other, but which are tilting-equivalent. This relates to the fact that each of these algebras admits two distinct isomorphism classes of separating tubular families of standard stable tubes. The results are derived from the existence of two non-isomorphic tubular exceptional curves (in the sense of H. Lenzing (1998, in “Trends in Ring Theory” (V. Dlab et al., Eds.), CMS Conf. Proc., Vol. 22, pp. 71–97, Am. Math. Soc., Providence)) over R which are derived-equivalent, one having a commutative and the other a non-commutative function field. Furthermore, we classify all generic modules over such tubular algebras
Intersections of Matrix Algebras and Permutation Representations of PSL(n,q)

AbstractIf G is a group, H a subgroup of G, and Ω a transitive G-set we ask under what conditions one can guarantee that H has a regular orbit (=of size |H|) on Ω. Here we prove that if PSL(n,q)⊆G⊆PGL(n,q) and H is cyclic then H has a regular orbit in every non-trivial G-set (with few exceptions). This result is obtained via a mixture of group theoretical and ring theoretical methods: Let R be the ring of all n×n matrices over the finite field F and let Z be the subring of scalar matrices. We show that if A and M are proper subrings of R containing Z, and if A is commutative and semisimple, then there exists an element x∈SL(n,F) such that xAx−1∩M=Z or n=2=|F|
Character Products and Q-Polynomial Group Association Schemes

AbstractWe study a finite group having a faithful character whose square has a small number of irreducible characters as constituents. Let Irr(G) be the set of absolutely irreducible ordinary characters of a finite group G. For each φ∈Irr(G), let φ̂=φ if φ is real valued and φ̂=φ+φ̄ otherwise, where φ̄ denotes the complex conjugate of φ. Let RIrr(G)={φ̂|φ∈Irr(G)}. For χ̂∈RIrr(G), letχ̂2=b·1+a·χ̂+Ψsuch that Ψ is a character of G which does not contain χ nor the principal character 1 as a constituent. We study the case when Ψ is a scalar multiple of a sum of the characters in IRrr(G), which are in a single orbit with respect to the action of the Galois group Gal(Q̄/Q(χ̂)). Here Q̄ denotes the algebraic closure of Q in C and Q(χ̂) is the field generated by the values of χ̂. As an application, we give a classification of Q-polynomial group association schemes
Polytopes Associated to Demazure Modules of Symmetrizable Kac–Moody Algebras of Rank Two

AbstractLet ω1,ω2 be the two fundamental weights of a symmetrizable Kac–Moody algebra g of rank two (hence necessarily affine or finite), and τ an element of the Weyl group. In this paper we construct polytopes Pτ(ω1),Pτ(ω2)⊂Rl(τ) and a linear map ξ: Rl(τ)→h* such that for any dominant weight λ=k1ω1+k2ω2, we have CharEτ(λ)=eλ∑eξ(x), where the sum is over all the integral points x, of the polytope k1Pτ(ω1)+k2Pτ(ω2). Furthermore, we show that there exists a flat deformation of the Schubert variety Sτ into the toric variety defined by Pτ(ω1),Pτ(ω2)
Differential Equations and Finite Groups

AbstractThe classical solution of the Riemann–Hilbert problem attaches to a given representation of the fundamental group a regular singular linear differential equation. We present a method to compute this differential equation in the case of a representation with finite image. The approach uses Galois coverings of P1\{0,1,∞}, differential Galois theory, and a formula for the character of the Galois action of the space of holomorphic differentials. Examples are produced for the finite primitive unimodular groups of degree two and three
Ikeda–Nakayama Rings

AbstractA ring R is called a right Ikeda-Nakayama ring (right IN-ring) if the left annihilator of the intersection of any two right ideals is the sum of the two left annihilators. In this paper we show that if R is a right IN-ring and A and B are right ideals of R that are complements of each other, there exists an idempotent e in R such that A=eR and B=(1−e)R. As a consequence we show that R is right selfinjective if and only if M2(R) is a right IN-ring. It is also shown that R is a dual ring if and only if R is a left and right IN-ring and the dual of every simple right R-module is simple. Finally, we prove that R is quasi-Frobenius if and only if R is a left perfect, left and right IN-ring, extending work on both selfinjective rings and dual rings. Several examples are provided to show that our results are non-trivial extensions of the known results on the subject
Normal Subgroups in Substitution Groups of Formal Power Series

AbstractLet R be a commutative ring with 1, and let N(R)≔t+t2R[t] be the group of normalized formal power series over R under substitution. In this paper we investigate the connection between the ideal structure of R and the normal subgroup structure of N(R). In particular, we show that, if K is a finite field of characteristic not equal to two, then every proper quotient group of the so-called Nottingham group N(K) is finite. As a further application we consider the profinite completion of the group N(R). We show that, if every additive subgroup of finite index in R contains an ideal of finite index in R, then N(R̂)≅N(R̂)
Simple Lie Algebras of Special Type

AbstractLet K be a field, let A be an associative, commutative K-algebra, and let Δ be a nonzero K-vector space of commuting K-derivations of A. Then, with a rather natural definition, W(A,Δ)=A⊗KΔ=AΔ becomes a Lie algebra, a Witt type algebra. In addition, there is a map div: W(A,Δ)→A called the divergence and its kernel S=S(A,Δ) is a Lie subalgebra, a special type algebra. In this paper, we study S from a ring theoretic point of view, obtaining sufficient conditions for the Lie simplicity of [S,S]. While the main result here is somewhat cumbersome to state, it does handle a number of examples in a fairly efficient manner. Furthermore, some of the preliminary lemmas are of interest in their own right and may, in time, lead to a more satisfactory answer
Semiprime Graded Algebras of Dimension Two

AbstractSemiprime, noetherian, connected graded k-algebras R of quadratic growth are described in terms of geometric data. A typical example of such a ring is obtained as follows: Let Y be a projective variety of dimension at most one over the base field k and let E be an OY-order in a finite dimensional semisimple algebra A over K=k(Y). Then, for any automorphism τ of A that restricts to an automorphism σ of Y and any ample, invertible E-bimodule B, Van den Bergh constructs a noetherian, “twisted homogeneous coordinate ring” B=⊕H0(Y,B⊗···⊗Bτn−1). We show that R is noetherian if and only if some Veronese ring R(m) of R has the form k+I, where I is a left ideal of such a ring B and where I=B at each point p∈Y at which σ has finite order. This allows one to give detailed information about the structure of R and its modules
Hereditary Semigroup Algebras

AbstractIn “Semigroup Algebras,” Okniński posed the following question: characterize semigroup algebras that are hereditary. In this paper we describe the (prime contracted) semigroup algebras K[S] that are hereditary and Noetherian when S is either a Malcev nilpotent monoid, a cancellative monoid or a monoid extension of a finite non-null Rees matrix semigroup. Furthermore, for the class of monoids which have an ideal series with factors that are non-null Rees matrix semigroups, we obtain an upper bound for the global dimension of its contracted semigroup algebra
On Involutive Homogeneous Varieties and Representations of Weyl Algebras

AbstractThis paper is concerned with the geometry of minimal involutive homogeneous varieties in complex affine 2n-space and its application to the study of the representation theory of the nth complex Weyl algebra An. The main results are the existence of minimal involutive homogeneous varieties of any given codimension, and of An-modules that have these varieties for characteristic variety. We also determine conditions on the codimensions of An-modules M and N under which Ext1(M,N) is a finite dimensional vector space
The Prime and the Primitive Spectra of a Quantum Bruhat Cell Translate

AbstractThe prime spectra of two families of algebras, Sw, and Šw, w∈W, indexed by the Weyl group W of a semisimple finitely dimensional Lie algebra g, are studied in the spirit of Joseph. The algebras Sw have been introduced by Joseph. They are q-analogues of the algebras of regular functions on w-translates of the open Bruhat cell of a semisimple Lie group G corresponding to the Lie algebra g. We define a stratification of the spectra into components indexed by pairs (y1,y2) of elements of the Weyl group satisfying y1≤w≤y2. Each component admits a unique minimal ideal which is explicitly described. We show the inclusion relation of closures to be that induced by Bruhat order
On a-Invariant Formulas

AbstractLet G(I) be the form ring of an ideal I of positive height in a local ring A. In this work we will provide formulas for the a-invariant of G(I). Our main result will only need the assumption that A is Cohen–Macaulay and that G(I) fulfills Serre's condition (Sl) where l is the analytic spread of I. As a consequence of our formula we will prove upper bounds for the reduction exponent of I in the case that A is a regular local ring. If G(I) fulfills Serre's condition (Sl), then this upper bound is l−1. And in the case that G(I) is even Gorenstein, it is l−2
Generalized Derivations of Lie Algebras

AbstractSuppose L is a finite-dimensional Lie algebra with multiplication μ: L∧L→L. Let Δ(L) denote the set of triples (f,f′,f″), with f, f′, f″∈Hom(L,L), such that μ∘(f∧IL+IL∧f′)=f″∘μ. We consider the Lie algebra GenDer(L)={f∈Hom(L,L)|∃f′, f″: (f,f′, f″)∈Δ(L)}. Well-researched subalgebras of GenDer(L) include the derivation algebra, Der(L)={f∈Hom(L, L)|(f, f, f)∈Δ(L)}, and the centroid, C(L)={f∈Hom(L,L)|(f,0,f)∈δ(L)}. We now study the subalgebra QDer(L)={f∈Hom(L,L)|∃f′: (f,f,f′)∈Δ(L)}, and the subspace QC(L)={f∈Hom(L,L)|(f,−f,0)∈Δ(L)}. In characteristic ≠2, GenDer(L)=QDer(L)+QC(L) and we are concerned with the inclusions Der(L)⊆QDer(L) and C(L)⊆QC(L)∩QDer(L). If Z(L)=0 then C(L)=QC(L)∩QDer(L) and, under reasonable conditions on Lie algebras with toral Cartan subalgebras, we show QDer(L)=Der(L)+C(L); if L is a parabolic subalgebra of a simple Lie algebra of rank >1 in characteristic 0, then we even have GenDer(L)=ad(L)+(IL). In general QC(L) is not closed under composition or Lie bracket; however, if Z(L)=0 then QC(L) is a commutative, associative algebra, and we describe conditions that force QC(L)=C(L) or, equivalently, GenDer(L)=QDer(L). We show that, in characteristic 0, GenDer(L) preserves the radical of L, thus generalizing the classical result for Der(L). We also discuss some applications of the main results to the study of functions f∈Hom(L,L) such that f∘μ or μ∘(f∧IL) defines a Lie multiplication
The Non-abelian Specker-Group Is Free

AbstractThe paper investigates a group H which can be constructed as a subgroup of the inverse limit of the finitely generated free groups Fn by taking only those elements g∈lim←(Fn), for which there exists some bound b(g) such that none of the generators of any of the groups Fn occurs more often than b(g) times in any of the entries of at least one sequence of Fn-elements describing g. By using a similar condition, H can also be described as a subgroup of the fundamental group of the Hawaiian Earrings. Despite not having any obvious candidate for a free basis, H is proven to be free by a non-constructive basis selection method. Since H is related to lim←(Fn) the same way as the classical Specker group is related to ZN, we decided to call H the “non-abelian Specker group.
Down–Up Algebras and Ambiskew Polynomial Rings

AbstractWe show that the down–up algebras of G. Benkart (1998, in “Recent Progress in Algebra,” Contemporary Mathematics Vol. 224, Am. Math. Soc., Providence) and G. Benkart and T. Roby (1998, J. Algebra209, 305–344) lie in a certain class of iterated skew polynomial rings, called ambiskew polynomial rings, in two indeterminates x and y over a commutative ring B. In such rings, commutation of the indeterminates with elements of B involve the same endomorphism σ of B, but from different sides, that is, yb=σ(b)y and bx=xσ(b), and, for some scalar p, yx−pxy∈ B. In previous studies of ambiskew polynomial rings, σ was required to be an automorphism but, in order to cover all down–up algebras, this requirement must be dropped. The Noetherian down–up algebras are those where σ is an automorphism and, in this case, we apply existing results on ambiskew polynomial rings to determine the finite-dimensional simple modules and the prime ideals. We adapt the methods underlying these results so as to apply to the non-Noetherian down–up algebras for which they reveal a surprisingly rich structure
Right Cancellative and Left Ample Monoids: Quasivarieties and Proper Covers

AbstractThe aim of this paper is to study certain quasivarieties of left ample monoids. Left ample monoids are monoids of partial one–one mappings of sets closed under the operation α↦αα−1. The idempotents of a left ample monoid form a semilattice and have a strong influence on the structure of the monoid; however, a left ample monoid need not be inverse. Every left ample monoid has a maximum right cancellative image and a proper cover which is also left ample. The structure of proper left ample monoids is well understood. Let V be a class of right cancellative monoids. A left ample monoid has a proper cover overV if it has a proper cover with maximum right cancellative image in V. We show that if V is a quasivariety determined within right cancellative monoids by equations, then the left ample monoids having a proper cover over V form a quasivariety. We achieve our aim using the technique of graph expansions to construct proper left ample monoids from presentations of right cancellative monoids
Tight Closure and Differential Simplicity

AbstractThe behavior of the Hasse–Schmidt algebra under étale extension is used to show that the Hasse–Schmidt algebra of a smooth algebra of finite type over a field equals the ring of differential operators. These techniques show that the formation of Hasse–Schmidt derivations does not commute with localization, providing a counterexample to a question of Brown and Kuan; their conjecture is reformulated in terms of the Hasse–Schmidt algebra. These techniques also imply that a smooth domain R is differentially simple. Tight closure is used to show that the test ideal is Hasse–Schmidt stable. Indeed, differentially simple rings of prime characteristic are strongly F-regular
On the Use of the Lazard Correspondence in the Classification of p-Groups of Maximal Class

AbstractLet G be a p-group of maximal class of order pm, p an odd prime, and m≥4. In this work we reduce the construction of this group to the consideration of certain elements of HomS(R/am−2∧R/am−2,R/am−4), where R=Z[x]/(1+···+xp−1)=Z[x̄], a=(x̄−1), and S=Z[x]/(xp−1). As an application of this result we prove that if we consider in G the generators s,s1,…,sm−1 introduced by Blackburn, then the structure of G is determined by the (p−3)/2 commutators [si,si+1] for 1≤i≤(p−3)/2 and the three integers 0≤u,v,w≤p−1, where u and v are defined by the condition sp=sum−1, (ss1)p=svm−1, and w=cm−1 is the exponent of sm−1 in the commutator [s(p−1)/2,s(p+1)/2]=scpp···scm−1m−1, 0≤ci≤p−1
The Eigenvalue Distribution of a Random Unipotent Matrix in Its Representation on Lines

AbstractThe eigenvalue distribution of a uniformly chosen random finite unipotent matrix in its permutation action on lines is studied. We obtain bounds for the mean number of eigenvalues lying in a fixed arc of the unit circle and offer an approach to other asymptotics. For the case of all unipotent matrices, the proof gives a probabilistic interpretation to identities of Macdonald from symmetric function theory. For the case of upper triangular matrices over a finite field, connections between symmetric function theory and a probabilistic growth algorithm of Borodin and Kirillov emerge
Finite Semigroups Whose Varieties Have Uncountably Many Subvarieties

AbstractFor several large classes of semigroups we provide a description of all semigroups which generate varieties with uncountably many subvarieties. These include the class of all Rees quotients of free monoids, the class of finite orthodox monoids, the class of monoids of index greater than two, and the class of finite inherently not finitely based semigroups. The first example of a finite, finitely based semigroup generating a variety with uncountably many subvarieties is presented and a number of related results are obtained. All varieties found with uncountably many subvarieties contain uncountable chains of subvarieties with the same ordering as the real numbers
On Commuting and Noncommuting Complexes

AbstractIn this paper we study various simplicial complexes associated to the commutative structure of a finite group G. We define NC(G) (resp. C(G)) as the complex associated to the poset of pairwise noncommuting (resp. commuting) sets of nontrivial elements in G.We observe that NC(G) has only one positive dimensional connected component, which we call BNC(G), and we prove that BNC(G) is simply connected.Our main result is a simplicial decomposition formula for BNC(G) which follows from a result of A. Björner, M. Wachs and V. Welker, on inflated simplicial complexes (2000, A poset fiber theorem, preprint). As a corollary we obtain that if G has a nontrivial center or if G has odd order, then the homology group Hn−1(BNC(G)) is nontrivial for every n such that G has a maximal noncommuting set of order n.We discuss the duality between NC(G) and C(G) and between their p-local versions NCp(G) and Cp(G). We observe that Cp(G) is homotopy equivalent to the Quillen complexes Ap(G) and obtain some interesting results for NCp(G) using this duality.Finally, we study the family of groups where the commutative relation is transitive, and show that in this case BNC(G) is shellable. As a consequence we derive some group theoretical formulas for the orders of maximal noncommuting sets
On the Simplicity of Lie Algebras of Derivations of Commutative Algebras

AbstractLet R be a commutative algebra over a field k. We prove two related results on the simplicity of Lie algebras acting as derivations of R. If D is both a Lie subalgebra and R-submodule of DerkR such that R is D-simple and either char k≠2 or D is not cyclic as an R-module or D(R)=R, then we show that D is simple. This extends a previous result from the author (1986, J. London Math. Soc. (2)33, 33–39) so as to include characteristic 2. If Δ is a Lie subalgebra of DerkR then we show that R⊗kΔ is simple if and only if R is Δ-simple, the action of R⊗kΔ on R is faithful, and, if char k=2 and dimkΔ=1, Δ(R)=R. This generalizes the weaker of two forms of a result by D. S. Passman (1998, J. Algebra34, 682–692), where Δ is abelian. However, a stronger form, in which the action of R⊗kΔ is replaced by that of RΔ⊗kΔ, does not generalize
Finitely Determined Members of Varieties of Groups and Rings

AbstractA finitely generated algebra A in a variety V is called finitely determined in V if there exists a finite V-consistent set of equalities and inequalities in an alphabet containing the generating set of A, which, together with the identities of V, yields all relations and non-relations of A. Obviously, if the equational theory of V is recursively enumerable then any finitely determined algebra in V has solvable word problem. The known algebraic characterizations of groups and semigroups with solvable word problem imply that in the varieties of all groups and all semigroups the members with solvable word problem are finitely determined. We construct a finitely generated center-by-metabelian group with solvable word problem, which is not finitely determined in every group variety V with ZA2⊆V⊆A3. We show that every extension of a finitely generated abelian group by a finite group from a variety W is finitely determined in every variety V⊇ZA2W. However, in any abelian-by-nilpotent variety no infinite group is finitely determined; moreover, in every variety, in which all finitely presented algebras are residually finite, each finitely determined algebra is finite. In the variety of all associative linear algebras over a finitely generated field every member with solvable word problem is finitely determined. We construct an example, which shows that for the variety of all associative rings it is not true; however, in this variety each torsion-free member with solvable word problem is finitely determined
Primitive, Almost Primitive, Test, and Δ-Primitive Elements of Free Algebras with the Nielsen–Schreier Property

AbstractWe study generalized primitive elements of free algebras of finite ranks with the Nielsen–Schreier property and their automorphic orbits. A primitive element of a free algebra is an element of some free generating set of this algebra. Almost primitive elements are not primitive elements which are primitive in any proper subalgebra. Δ-primitive elements are elements whose partial derivatives generate the same one-sided ideal of the universal multiplicative envelope algebra of a free algebra as the set of free generators generate. We prove that an endomorphism preserving an automorphic orbit of a nonzero element of a free algebra of rank two is an automorphism. An algorithm to determine test elements of free algebras of rank two is described. A series of almost primitive elements is constructed and new examples of test elements are given. We prove that if the rank n of the free Lie algebra L is even, n=2m, then any Δ-primitive element of L is an automorphic image of the element w=[x1,x2]+···+[x2m−1,x2m], there are no Δ-primitive elements of L if n is odd, and the group of automorphisms of the algebra L acts transitively on the set of all Δ-primitive elements