pinecone/core-2020-05-10-deduplication huggingface dataset:
The Generalized Restricted Representations of Graded Lie Algebras of Cartan Type

AbstractLetFbe an algebraically closed field of characteristicp>2. In this paper, the concepts of generalized restricted Lie algebras and their generalized restricted representations overFare introduced. Any graded Lie algebra of Cartan typeL=X(m:n)(2),X=W,S,H, orK, is a generalized restricted Lie algebra associated with a standard basisEofLandsa positive integer ≥max1≤i≤m{ni}. The irreducible generalized restricted representations ofLare determined, which are in a one-to-one correspondence withpsdweight functions; heredis the dimension of the standard torus inL1+δX,Kεm,[0]. The descriptions of the blocks and a Brauer-type reciprocity are obtained
Constructing Local Generic Formal Fibers

AbstractLet (T,M) be a complete local (Noetherian) unique factorization domain of dimension at least two such that the cardinality of the residue fieldT/Mis at least the cardinality of the real numbers. Suppose (0)=p0⊂p1⊂···⊂pn≠Mis a chain of distinct prime ideals ofTsuch thatpnintersected with the prime subring ofTis the zero ideal. We construct a chain of local unique factorization domainsAn⊂An−1⊂···⊂A1⊂A0=Tsuch that the completion of eachAiisTand for everyithe generic formal fiber ofAiis local (this means that the ringT⊗AiKiis a local ring whereKiis the quotient field ofAi) withpi⊗AiKiits maximal ideal
Archimedean Superrigidity of SolvableS-Arithmetic Groups

AbstractLet G be a connected, solvable linear algebraic group over a number fieldK, letSbe a finite set of places ofKthat contains all the infinite places, and let O(S) be the ring ofS-integers ofK. We define a certain closed subgroup[formula]of GS=∏v∈SGKvthat contains GO(S), and prove that GO(S)is a superrigid lattice in[formula], by which we mean that finite-dimensional representations α:GO(S)→GLn(R) more or less extend to representations of[formula]. The subgroup[formula]may be a proper subgroup of GSfor only two reasons. First, it is well known that GO(S)is not a lattice in GSif G has nontrivialK-characters, so one passes to a certain subgroup GS(1). Second, GO(S)may fail to be Zariski dense in GS(1)in an appropriate sense; in this sense, the subgroup[formula]is the Zariski closure of GO(S)in GS(1). Furthermore, we note that a superrigidity theorem for many nonsolvableS-arithmetic groups can be proved by combining our main theorem with the Margulis Superrigidity Theorem
Transposition Hypergroups: Noncommutative Join Spaces

AbstractThe study of various kinds of algebraic hypergroups is unified in the theory of transposition hypergroups. The hyperoperation in any hypergroup has two inverses given bya/b={x|a∈xb} andb/a={x|a∈bx}. A transposition hypergroup is a hypergroup where transposition,b/a∩c/d≠∅ impliesad∩bc≠∅, holds. The algebra of transposition hypergroups is developed. Closed subhypergroupsNthat are reflexive,a/N=N/a, have distinguished structural significance. The quotient space of a transposition hypergroup modulo a reflexive closed subhypergroup forms a transposition hypergroup that is a polygroup. Then generalizations of the isomorphism theorems and the Jordan–Hölder theorem of group theory are obtained
Slenderness, Completions, and Duality for Primary Abelian Groups

AbstractIfAis a fixed abelian group with endomorphism ringE, then for any groupG, letG*=Hom(G,A) and for anyE-moduleM, letM*=HomE(M,A). The evaluation map σG:G→G** is defined in the usual way andGisA-reflexive if σGis an isomorphism. This is strongly related to the question of whetherAis slender as anE-module, and we discuss thep-groups for which this holds. In some important cases,G** can be viewed as the completion ofGin a linear topology. It is known that ifA=⊕,nZpn, andGis ap-group of non-measurable cardinality, thenG** can be identified with the completion ofGin the ⊕c-topology, and we provide a generalization of this result. We also show that for any groupNof non-measurable cardinality there is a groupGsuch thatG**/σG(G)≅N
The Resolution of the Ideal of 2×2 Minors of a 2×nMatrix of Linear Forms

AbstractLetkbe an algebraically closed field and letS=k[x1,…,xm]. LetMbe a 2×nmatrix of linear forms ofSand letI2(M) denote the ideal generated by the determinants of the 2×2 minors ofM. We study in this paper the minimal finite free resolution ofS/I2(M) as anS-module.Mcorresponds to a certain 2-dimensional vector spaceLofm×nmatrices, that is, to a matrix pencil. The Kronecker–Weierstrass theory of such matrix pencils provides a normal form forL, and we characterize the resolution ofS/I2(M) in terms of this normal form. In particular, if the general element ofLis injective, we explicitly construct the minimal resolution ofS/I2(M) by repeated application of the horseshoe lemma. For anyM, we express the regularity ofS/I2(M) as a function of the invariants ofL
Parametric Decomposition of Monomial Ideals, II

AbstractLetx1,…,xdbe anR-sequence in a commutative ringRand letIbe a monomial idea (soIis generated by elements of the formx1e1···xded, where eacheiis a nonnegative integer). The main results of this paper: (a) establish a practical formula which computes the monomial length ofIwhen Rad(I)=Rad((x1,…,xd)R); (b) determine necessary and sufficient conditions for the intersection of finitely many monomial ideals to again be a monomial ideal; (c) show that ifC, the set of all monomial ideals inRthat containI, is closed under finite intersections, then each idealJinChas a unique decomposition as an irredundant finite intersection of ideals of the form (xτ(1)a1,…,xτ(h)ah)R, where τ is a permutation of {1,…,d},h∈{1,…,d}, anda1,…,ahare positive integers; and, (d) give additional results for certain form rings and Rees rings ofR, related to the unique parametric decomposition theorem
Quasi-Tilting Modules and Counter Equivalences

AbstractGiven two ringsRandS, we study the category equivalences T⇄Y, where T is a torsion class ofR-modules and Y is a torsion-free class ofS-modules. These equivalences correspond to quasi-tilting triples (R,V,S), whereRVSis a bimodule which has, “locally,” a tilting behavior. Comparing this setting with tilting bimodules and, more generally, with the torsion theory counter equivalences introduced by Colby and Fuller, we prove a local version of the Tilting Theorem for quasi-tilting triples. A whole section is devoted to examples in case of algebras over a field
The Endomorphism Ring of a Localized Coherent Functor

AbstractLetCbe a commutative artinian ring and Λ an artinC-algebra. The category of coherent additive functorsA: mod-Λ→Ab on the finitely presented right Λ-modules will be denoted by Ab(Λ). This category is equivalent to the free abelian category over the ring Λ. If S0⊆Ab(Λ) is the Serre subcategory of the finite length objects of Ab(Λ) andA∈Ab(Λ), it is proved that the endomorphism ring EndAb(Λ)/S0AS0of the localized objectAS0is a locally artinC-algebra. This is used to show that the Krull-Gabriel dimension of the category Ab(Λ) cannot equal 1. In particular, this holds for finite rings
Spectral Sequences for the Classification of Extensions of Hopf Algebras

AbstractWe construct spectral sequences which provide a way to compute the cohomology theory that classifies extensions of graded connected Hopf algebras over a commutative ring as described by William M. Singer. Specifically, for (A,B) an abelian matched pair of graded connectedR-Hopf algebras, we construct a pair of spectral sequences relatingH*(B,A) to Ext*,*B(R,Cotor*,*A(R,R)). To illustrate these spectral sequences, we examine the special case ofBa monogenic graded connected Hopf algebra and also analyze an extension of Hopf algebras given by James P. Lin
Two Theorems about Equationally Noetherian Groups

AbstractAn algebraic set over a groupGis the set of all solutions of some system {f(x1,…,xn)=1∣f∈G∗〈x1,…,xn〉} of equations overG. A groupGis equationally noetherian if every algebraic set overGis the set of all solutions of a finite subsystem of the given one. We prove that a virtually equationally noetherian group is equationally noetherian and that the quotient of an equationally noetherian group by a normal subgroup which is a finite union of algebraic sets is again equationally noetherian. On the other hand, the wreath productW=U≀Tof a non-abelian groupUand an infinite groupTis not equationally noetherian
Simple Connectedness of the 3-Local Geometry of the Monster

AbstractWe consider the 3-local geometry M of the Monster groupMintroduced as a locally dual polar space of the group Ω8−(3) and independently in the context of minimalp-local parabolic geometries for sporadic simple groups. More recently the geometry appeared implicitly within theZ3-orbifold construction of the Moon-shine moduleV♮. In this paper we prove the simple connectedness of M. This result makes unnecessary the refereeing to the classification of finite simple groups in theZ3-orbifold construction ofV♮and realizes an important step in the classification of the flag-transitivec-extensions of the classical dual polar spaces. We make use of the simple connectedness results for the 2-local geometry ofMand for a subgeometry in M which is the 3-local geometry of the Fischer groupM(24)
Exceptional Polynomials of Affine Type

AbstractLetKbe a finite field of characteristicp. A polynomialfwith coefficients inKis said to beexceptionalif it induces a permutation on infinitely many finite extensions ofK. Lettbe a transcendental, andK̄be an algebraic closure ofK. The exceptional polynomials known to date are constructed from certain twists of polynomialsg, such that the splitting field ofg(X)−toverK̄is rational. On the other hand we know that except forp=2 or 3, either the degree of an exceptional polynomialfis a power of the characteristicp, orfis a Dickson polynomial. The case degf=phas been settled completely. We give a new series of exceptional polynomials of degreepmwithmeven which does not follow the above mentioned construction principle. We show under certain additional hypotheses that the associated monodromy groups of exceptional polynomials will be severely restricted. In particular, we determine precisely what the geometric monodromy groups of exceptional polynomials of degreep2will be
On the Automorphism Group of the One-Rooted Binary Tree

AbstractLetAbe the automorphism group of the one-rooted regular binary treeT2andGthe subgroup ofAconsisting of those automorphisms admitting a “finite description” in their action onT2. LetNA(G) be the normaliser ofGinA, let Aut(G) be the group of automorphisms ofG, and let EndA(G) be the semigroup of endomorphisms ofGinduced by conjugation by elements ofA. ThenGis the infinite iterated wreath product (…≀C2)≀C2, andAis the topological limit ofG. We study in some detail the structure ofG, Aut(G), and EndA(G). In particular, we proveNA(G) is isomorphic to Aut(G), contains a copy ofAitself, and is a proper subgroup of EndA(G). Furthermore we discuss connections with automata and introduce the notion of functionally recursive automorphisms ofT2
Fixed Point Free Action with Some Regular Orbits

AbstractLet the finite groupAbe acting on a finite (solvable) groupGand suppose that no non-trivial element ofGis fixed under the action of all the elements ofA. Assume furthermore that (|A|,|G|)=1. A long-standing conjecture is that then the Fitting height ofGis bounded by the length of the longest chain of subgroups ofA. In3, this was proved in the case where for every proper subgroupBofAand everyB-invariant elementary abelian sectionSofG, there exists somev∈Ssuch thatCB(v)=CB(S) (we say thatBhas a regular orbit onS). In the present paper we establish the conjecture assuming only that some of these sections have regular orbits
Almeida's Generalized Variety Problem

AbstractLet Com denote the variety of all commutative semigroups, let Niℓ denote the generalized variety of all nil semigroups, and let N denote the generalized variety of all nilpotent semigroups. For any class W of semigroups let L(W) denote the lattice of all varieties contained in W, and let G(W) denote the lattice of all generalized varieties contained in W. Almeida has shown that the map ϕ: L(Niℓ∩Com)∪{Niℓ∩Com}→G(N∩Com) given by Wϕ=W∩N is an isomorphism, and asked whether the extension of this map to L(Niℓ)∪{Niℓ} is also an isomorphism. In this article a negative answer is given: two varieties U,V∈L(Niℓ) are defined and used to show that the map is not injective. In the process, congruence classes of the fully invariant congruences on the free semigroup on any countable setXwith |X|≥3 which correspond to U and V are described which are denumerable and contain words of unbounded length
Groups without Faithful Transitive Permutation Representations of Small Degree

AbstractA subgroupHof a groupGiscore-freeifHcontains no non-trivial normal subgroup ofG, or equivalently the transitive permutation representation ofGon the cosets ofHis faithful. We study the obstacles to a group having large core-free subgroups. We call a subgroupDa “dedekind” subgroup ofGif all subgroups ofDare normal inG. Our main result is the following: If a finite groupGhas no core-free subgroups of order greater thank, thenGhas two dedekind subgroupsD1andD2such that every subgroup inGof order greater thanf(k) has non-trivial intersection with eitherD1orD2(wherefis a fixed function independent ofG). Examples show that the dedekind subgroups need not have index bounded by a function ofk, and the result would not be true with one dedekind subgroup instead of two. We exhibit various related properties ofp-groups and infinite locally finite groups without large core-free subgroups, including the following: IfGis a locally finite group with no infinite core-free subgroup, then every infinite subgroup ofGcontains a non-trivial cyclic normal subgroup ofG. We also exhibit asymptotic bounds for some related problems, including the following: If a groupGhas a solvable subgroup of indexn, thenGhas a solvable normal subgroup of index at mostnc(for some absolute constantc). IfGis a transitive permutation group of degreenwith cyclic point-stabilizer subgroup, then |G|<n7
Left Cells in the Affine Weyl Group of TypeF̃4

AbstractBy applying an algorithm designed before, we complete the description for all the left cells of the affine Weyl groupWaof typeF̃4by finding a representative set of its left cells together with all its left cell graphs (or with all the associated essential graphs) in each of its two-sided cells. The generalized τ-invariants of left cells ofWaare exhibited graphically. A group-theoretical interpretation is given on the numbers of left cells ofWain some two-sided cells. Thus so far the left cells of all the affine Weyl groups of ranks less than or equal to 4 have been known explicitly. Some techniques are developed in applying the algorithm. As a consequence, we complete the verification of a conjecture concerning the characterization of left cells of Weyl groups and affine Weyl groups
Stability of Schur Functors

AbstractSymplectic (resp. orthogonal) Schur functors are defined on a vector spaceVwith a non-degenerate skew-symmetric (resp. symmetric) bilinear form. Each of these functors determines an irreducible representation of sp(V) (resp. so(V)). We prove that if dimVis sufficiently large, then the splitting into irreducible components of the composition of a symplectic (orthogonal) Schur functor and an ordinary Schur functor applied toVis independent of dimV. We also prove the stability of the branching for the symplectic (orthogonal) Schur functors
Simple Involutive Quantales

AbstractInvolutive quantales were introduced in [7] as complete lattices equipped with a multiplication and an involution. Such structures are well known from the calculus of relations: the set Rel(X) of binary relations on any setXforms an involutive quantale. However, the motivating example of an involutive quantale is the spectrum MaxAof a non-commutativeC*-algebraA, where MaxAis the involutive quantale consisting of all closed linear subspaces ofA. In [8] it was indeed shown thatAcan be reconstructed from the involutive quantale MaxA, demonstrating that involutive quantales do provide a convenient algebraic invariant of non-commutativeC*-algebras.The aim of this paper is to study involutive quantales from the algebraic point of view. Our main result is the characterization of simple involutive quantales, that is, involutive quantales on which the only surjections are isomorphisms or constant morphisms. We also show that this class contains all quantales Q(S) of sup-preserving endomorphisms on a complete latticeShaving a duality, a fortiori, all Hilbert quantales as defined in [7]. A special case of this is Rel(X), which can be seen to be Q(2X).Our investigation of the class of simple involutive quantales is also motivated byC*-algebras. It is proved in [8] that for anyC*-algebraA, MaxAhas enough homomorphisms into the Hilbert quantales Q(P(H)), where P(H) is the complete lattice of closed linear subspaces of a Hilbert spaceH. In this spirit, viewing involutive quantales as duals of generalized (non-commutative) topological spaces, simple involutive quantales correspond to spaces having only trivial subspaces, that is, to points. We begin a study of those involutive quantales which have enough homomorphisms into simple involutive quantales, noting their “spatial” character. Such a study has algebraic antecedents in the early work [4] and the more recent work [1] which studied certain lattices (noncomplete involutive quantales) having enough homomorphisms into simple involutive quantales of the form Rel(X)
On the Variety of Automorphisms of the Affine Plane

AbstractThe main subject of our study is GA2,n, the variety of automophisms of the affine plane of degree bounded by a positive integern. After detailing some definitions and notations in Section 1, we give in Section 2 an algorithm to decide whether an endomorphism of the affine plane over an integral domain is a tame automorphism. Then, by applying this algorithm to the Nagata automorphism, we recover easily its known results. In Section 6, we compute the number of irreducible components of GA2,nwhenn≤9 and we show that GA2,nis reducible whenn≥4. Our proofs are based on a precise decomposition theorem for automorphisms given in Section 3 and a characterization of length one automorphisms given in Section 5. Finally, in Section 7, we give some details on the casen=4
Hilbert–Kunz Functions of Cubic Curves and Surfaces

AbstractWe determine the Hilbert–Kunz function of plane elliptic curves in odd characteristic, as well as over arbitrary fields the generalized Hilbert–Kunz functions of nodal cubic curves. Together with results of K. Pardue and P. Monsky, this completes the list of Hilbert–Kunz functions of plane cubics. Combining these results with the calculation of the (generalized) Hilbert–Kunz function of Cayley's cubic surface, it follows that in each degree and over any field of positive characteristic there are curves resp. surfaces taking on the minimally possible Hilbert–Kunz multiplicity
Tor and Torsion on a Complete Intersection

AbstractLet (R,m) be a complete intersection, that is, a ring whose m-adic completion is the quotient of a regular local ring by a regular sequence. SupposeMandNare finitely generatedR-modules. We give a necessary condition for the vanishing of TorRi(M,N) for alli⪢0 in terms of the intersection of certain affine algebraic sets associated toMandN. We apply this condition to the study of torsion in tensor products. For example, we show that ifRis a domain andMis anR-module of infinite projective dimension then there exist infinitely manynfor which the tensor product ofMwith one of itsnth syzygy modules has torsion.We also give a sufficient condition for the vanishing TorRi(M,N) for alli⪢0 in terms of the ability to leftMandNto “disjoint” complete intersections of smaller codimension. We use this condition to construct tensor products of non-free modules which are maximal Cohen–Macaulay
Lie Stack Deformations

AbstractEnveloping algebras of Lie stacks give irreducible Hopf algebra deformations ofU(g) which are neither commutative nor cocommutative. In this paper we present and study a large class of examples of Lie stacks. In particular, we show that the PBW-bases of these Hopf algebras do not have to be finite in general. Further, we construct a non-cocommutative Hopf structure onU(g) (usually with antipode of infinite order) whenever g has a codimension one Lie ideal h such that the quotient has the h-weight of an eigenvector of ⋀2h
A Cohomological Transfer Map for Profinite Groups

AbstractLetGbe a group,AaG-module, andHa subgroup ofG. The standard cohomological transfer map fromH*(H,A) toH*(G,A) is defined in the case thatHis of finite index inGand is given explicitly in each dimension by a formula involving a sum over a set of representives forH\G. In this paper, we obtain a new transfer in the case thatGis a profinite group,Ais an abelian protorsion group on whichGacts continuously,His a closed subgroup ofG, and the cohomology is continuous. We do this by developing a theory of integration for continuous functions from a compact space to a projective limit of discrete modules and replacing the finite sum in the formula for the standard transfer with an integral. As an application of the new transfer, we prove a profinite version of the well-known result that forAabelian andGfinite, an extension0→A→E→βG→1splits if, for every prime numberp, there exists a homomorphism γpfrom ap-Sylow subgroupSpofGtoEsuch that β∘γpis the identify onSp. Of particular importance in our proof is the fact that the composition of the restriction map fromH*(G,A) toH*(H,A) and the transfer which we introduce is equal to multiplication by [G:H], where the index in this case is a suitably defined element in  Z
Embedding a Quantum Nonsingular Quadric in a Quantum P3

AbstractA definition of regularity has been given for non-commutative graded algebras and results of Artin, Schelter, Tate, and Van den Bergh classify the regular algebras of global dimension three that are generated by degree one elements. Our purpose is to classify a certain class of quadratic regular algebras of global dimension four. LetSbe a twisted homogeneous coordinate ring of a nonsingular quadricQ⊂P3. Our interest is in algebrasRsuch that there is an embedding ProjS↪ProjR. In this paper, we classify all the quadratic regular algebrasRof global dimension four which have the same Hilbert series as that of the polynomial ring on four variables, and which map ontoSvia a graded degree zero homomorphism. Our approach makes use of the point modules ofRand their associated geometric data. We classify the algebrasRaccording to their “point scheme”Pand corresponding automorphism σ∈Aut(P); those algebrasRwhich are determined by (P,σ) belong to at most a five-parameter family, but those which are not determined by (P,σ) belong to at most a four-parameter family. In the first case,Pis either P3or consists ofQtogether with a lineL, while in the second caseP=Q. It is also proved that under certain sufficient conditions, the zero locus of the defining relations of a quadratic regular algebra of global dimension four is the graph of an automorphism
Polynomial Automorphisms and Gröbner Reductions

AbstractLetPn=K[x1,…,xn] be the polynomial algebra over a fieldKof characteristic 0. We show that applying an automorphism to a given polynomialp∈Pnis mimicked by Gröbner transformations of a basis of the ideal ofPngenerated by partial derivatives of this polynomial. In the case ofP2, this yields a miraculously simple algorithm for deciding whether or not a given polynomial fromP2is part of a basis. Another application is an algorithm which, given a polynomialp∈P2that is part of a basis, finds a sequence of elementary automorphisms that reducesptox1. We also speculate on how our method may be used for constructing a possible counterexample to the Jacobian conjecture in higher dimensions