pinecone/core-2020-05-10-deduplication huggingface dataset:
The Spectrum of the Γ-Invariant of a Bilinear Space

AbstractTo every symmetric bilinear space (X,φ) of regular uncountable dimension κ, an invariant Γ(X,φ)∈P(κ)/F(κ) (where F(κ) is the club filter) can be assigned. We prove that in dimension ℵ2the spectrum of Γ cannot be determined inZFC. For this, on the one hand we show that underCH, Γ attains the maximal (with respect to a restriction provable inZFC) spectrum; we also show thatCHis not necessary for this result. On the other hand we show that in a variation of Mitchell's model, which is obtained by collapsing a weakly compact cardinal to ω2, the spectrum of Γ in dimension ℵ2is much thinner than the maximal one
Automorphism Subgroups of Finite Index in Algebraic Mapping Class Groups

AbstractWe give an algebraic proof of the Birman–Bers theorem—an algebraic result whose previous proofs used topology or analysis, and which says that a certain subgroup of finite index in the (algebraic) mapping class group of an oriented punctured surface is isomorphic to a certain group of automorphisms. The index 2 case gives rise to an automorphism of the group consisting of those automorphisms of a free group that stabilize the normal subgroup generated by an oriented-surface relator, and we analyze this curious automorphism
Connected Graded Gorenstein Algebras with Enough Normal Elements

AbstractWe generalize [12, 1.1 and 1.2] to the following situation.Theorem 1.Let A be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of A contains a homogeneous normal element of positive degree. Then:(1)A is Auslander–Gorenstein and Cohen–Macaulay.(2)A has a quasi-Frobenius quotient ring.(3)Every minimal prime ideal P is graded andGKdimA/P=d.(4)If, moreover, A has finite global dimension, then A is a domain and a maximal order in its quotient division ring.To prove the above we need the following result, which is a generalization of [3, 2.46(ii)].Theorem 2.Let A be a connected graded noetherian AS-Gorenstein algebra of injective dimension d. Then:(1)The last term of the minimal injective resolution of AAis isomorphic to a shift of A*.(2)For every noetherian graded A-module M,Extd(M,A)is finite dimensional over k
Drinfeld Double and Ringel–Green Theory of Hall Algebras

AbstractAs a continuation of the work of Ringel and Green on Hall algebras, the Hopf algebra structure of a (extended twisted) Hall algebra is presented here. By introducing the Ringel pairing of Hall algebras, the Drinfeld double of a Hall algebra is constructed. The reduced form of this Drinfeld double is applicable to the generic composition algebra, which gives a complete realization of quantum group. This Ringel–Green theory makes it possible to give the formulae for the computation of commutators in quantum groups in terms of Hall algebras
Valuations of Lattice-Ordered Groups

AbstractIn this paper, we introduce the concept of a valuation mapping of anl-groupGonto a distributive lattice and use such valuations to investigate the structure ofG. Then we examine the maximal immediate extensions ofGwith respect to these valuations. For the natural valuation, these are the archimedean extensions (a-extensions) first investigated by12and30. This leads to new results and new proofs of old results abouta-extensions. Then we obtain new structure theorems for Δ-extensions ofl-groups. In another paper, it will be shown that this valuation theory determinesallthe torsion classes ofl-groups that have invariant torsion radicals. This includes most of the well known and interesting torsion classes that are notl-varieties
Nonmatrix Varieties and Nil-Generated Algebras Whose Units Satisfy a Group Identity

AbstractLetR×denote the group of units of an associative algebraRover an infinite fieldF. We prove that ifRis unitarily generated by its nilpotent elements, thenR×satisfies a group identity precisely whenRsatisfies a nonmatrix polynomial identity. As an application, we examine the group algebraFGof a torsion groupGand the restricted enveloping algebrau(L) of ap-nil restricted Lie algebraL. Giambruno, Sehgal, and Valenti recently proved that if the group of units (FG)×satisfies a group identity, thenFGsatisfies a polynomial identity, thus confirming a conjecture of Brian Hartley. We show that, in fact, (FG)×satisfies a group identity if and only ifFGsatisfies a nonmatrix polynomial identity. In the case of restricted enveloping algebras, we prove thatu(L)×satisfies a group identity if and only ifu(L) satisfies the Engel condition
Varieties of Anticommutativen-ary Algebras

AbstractA fundamental problem in the theory ofn-ary algebras is to determine the correct generalization of the Jacobi identity. This paper describes some computational results on this problem using representations of the symmetric group. It is well known that over a field of characteristic 0 any variety ofn-ary algebras can be defined by multilinear identities. In the anticommutative case, it is shown that forn≤8 the[formula]-dimensionalS2n−1-module of multilinear identities in which each term involves twon-ary products (i.e., two pairs ofn-ary anticommutative brackets) decomposes as the direct sum of thendistinct simple modules labelled by thenpartitions of 2n−1 in which only 1 and 2 occur as parts. In the casesn=3 (resp.n=4), the kernel of the commutator expansion map and a generator for each of the 7 (resp. 15) nonzero submodules are determined. The paper concludes with some conjectures forn≥5
On the Trace of Graded Automorphisms

AbstractLetA=⊕d≥0Adbe a connected algebra with a graded algebra endomorphism σ. The trace of σ is defined to be Tr(σ,t)=∑d≥0tr(σ|Ad)td. We prove that Tr(σ,t) is a rational function ifAis either finitely generated commutative or right noetherian with finite global dimension or regular. A version of Molien's theorem follows in these three cases. IfAis a regular algebra or a Frobenius algebra we prove a reciprocity for the trace. We also partially generalize a theorem of Watanabe on the Gorenstein property to the noncommutative case
A Proof of Hozo's Conjecture on the Homology of a Class of Nilpotent Lie Algebras

AbstractLetPbe a rooted tree with rootrand vertex set {1,2,…,n}. ConsiderPto be a poset by sayingu≤vifuis on the path fromrtov. LetZ[P] be the span of all matriceszuvsuch thatu<Pv, wherezuvis then×nmatrix with a 1 in theu,ventry and 0's elsewhere. Note thatZ[P] is a nilpotent Lie subalgebra of sℓn(C). I. Hozo (Electron. J. Combin., to appear) studied the Laplacian of the Koszul complex for computing the Lie algebra homology ofZ[P]. He showed that the tree structure ofPforces significant simplifications in the Laplacian. He went on to conjecture that the eigenvalues of the Laplacian are indexed by certain labellings ofPby partitions. He gave a simple method for determining the eigenvalue indexed by a labelling. In this paper we prove Hozo's conjecture. As a consequence, we deduce that all eigenvalues of the Laplacian are integers
Free Resolutions and Change of Rings

AbstractProjective resolutions of modules over a ringRare constructed starting from appropriate projective resolutions over a ringQmapping toR. It is shown that such resolutions may be chosen to be minimal in codimension ≤2, but not in codimension 3. This is used to obtain minimal resolutions for essentially all modules over local (or graded) ringsRwith codimension ≤2. Explicit resolutions are given for cyclic modules over multigraded rings, and necessary and sufficient conditions are obtained for their minimality
Symplectic Groups, Symplectic Spreads, Codes, and Unimodular Lattices

AbstractIt is known that the symplectic groupSp2n(p) has two (complex conjugate) irreducible representations of degree (pn+1)/2 realized overQ(−p), provided thatp≡3mod4. In the paper we give an explicit construction of an odd unimodularSp2n(p)·2-invariant lattice Δ(p,n) in dimensionpn+1 for anypn≡3mod4. Such a lattice has been constructed by R. Bacher and B. B. Venkov in the casepn=27. A second main result says that these lattices are essentially unique. We show that forn≥3 the minimum of Δ(p,n) is at least (p+1)/2 and at mostp(n−1)/2. The interrelation between these lattices, symplectic spreads of Fp2n, and self-dual codes over Fpis also investigated. In particular, using new results of U. Dempwolff and L. Bader, W. M. Kantor, and G. Lunardon, we come to three extremal self-dual ternary codes of length 28
Invariant Subspaces of the Ring of Functions on a Vector Space over a Finite Field

AbstractIfFqis the finite field of characteristicpand orderq=ps, let F(q) be the category whose objects are functors from finite dimensionalFq-vector spaces toFq-vector spaces, and with morphisms the natural transformations between such functors.A fundamental object in F(q) is the injectiveIFqdefined byIFq(V)=FqV*=S*(V)/(xq−x).We determine the lattice of subobjects ofIFq. It is the distributive lattice associated to a certain combinatorially defined poset I(p,s) whoseqconnected components are all infinite (with one trivial exception). An analysis of I(p,s) reveals that every proper subobject of an indecomposable summand ofIFqis finite. ThusIFqis Artinian.FilteringIFqand I(p,s) in various ways yields various finite posets, and we recover the main results of papers by Doty, Kovács, and Krop on the structure ofS*(V)/(xq) overFq, andS*(V) overF̄p
K-Theory of Endomorphism Rings and of Rings of Invariants

AbstractThis paper gives the following description ofK0of the endomorphism ring of a finitely generated projective module.Theorem.Let T be a ring and P a finitely generated,projective T-module. Let I be the trace ideal of P. Then K0(EndPT)is isomorphic to a subgroup of K0(T,I).If,further,the natural map K1(T)→K1(T/I)is surjective then K0(EndPT)is isomorphic to the subgroup of K0(T)generated by the direct summands of Pn,for n∈N.As a corollary we can determineK0of the ring of invariants for many free linear actions. In particular, the following result is proved.Theorem.Let V be a fixed-point-free linear representation of a finite group G over a field k of characteristic zero and let S(V)be the symmetric algebra of V.Let K be any finite-dimensional k-vector space. Then K0(S(V)G⊗kS(K))=〈[S(V)G⊗kS(K)]〉.Similar results are given for suitable noncommutative versions ofS(V)
Double Quantum Groups and Iwasawa Decomposition

AbstractThe double quantum groups Cq[D(G)]=Cq[G]⋈Cq[G] are the Hopf algebras underlying the complex quantum groups of which the simplest example is the quantum Lorentz group. They are nonstandard quantizations of the double groupG×G. We construct a corresponding quantized universal enveloping algebraUq(d(g)) and prove that the pairing between Cq[D(G)] andUq(d(g)) is nondegenerate. We analyze the representation theory of these Cq[D(G)], give a detailed version of the Iwasawa decomposition proved by Podles and Woronowicz for the quantum Lorentz group, and show that Cq[D(G)] is noetherian. Finally we outline how to construct more general nonstandard quantum groups using quantum double groups and their generalizations
PI-Algebras Generated by Nilpotent Elements of Bounded Index

AbstractLetRbe an associative algebra over a fieldFof positive characteristicp. We address the following problem: ifRis generated by nilpotent elements with bounded index, under what conditions can we conclude thatRitself is nil of bounded index? We prove that wheneverRsatisfies the Engel condition andp>0, orRis Lie soluble andp>2, then the conclusion holds. Along the way we prove that ifRsatisfies the Engel condition, then there exists a positive integernsuch that the mapx↦xpnis additive. The converse also holds whenFis infinite orRis generated by nilpotent elements
On Finite and Locally Finite Subgroups of Free Burnside Groups of Large Even Exponents

AbstractThe following basic results on infinite locally finite subgroups of a freem-generator Burnside groupB(m,n) of even exponentn, wherem>1 andn≥248,nis divisible by 29, are obtained: A clear complete description of all infinite groups that are embeddable inB(m,n) as (maximal) locally finite subgroups is given. Any infinite locally finite subgroup L ofB(m,n) is contained in a unique maximal locally finite subgroup, while any finite 2-subgroup ofB(m,n) is contained in continuously many pairwise nonisomorphic maximal locally finite subgroups. In addition, L is locally conjugate to a maximal locally finite subgroup ofB(m,n). To prove these and other results, centralizers of subgroups inB(m,n) are investigated. For example, it is proven that the centralizer of a finite 2-subgroup ofB(m,n) contains a subgroup isomorphic to a free Burnside groupB(∞,n) of countably infinite rank and exponentn; the centralizer of a finite non-2-subgroup ofB(m,n) or the centralizer of a nonlocally finite subgroup ofB(m,n) is always finite; the centralizer of a subgroup S is infinite if and only if S is a locally finite 2-group
Some Homotopy Equivalences for Sporadic Geometries

AbstractA previous work [RSY90] established the projectivity of the reduced Lefschetz modules of certain sporadic group geometries, and the present paper continues that work in a wider context.Recent developments in sporadic-group cohomology include some applications of [RSY90], which in turn suggested treatment of a broader class of geometries. Recurring similarities in the proofs also led to a more unified treatment—establishing the stronger result of homotopy equivalence of thep-local geometry with the usual elementary poset Ap(G). One equivalence method proceeds by means of a new “closed set” in a standard technique of Quillen. It was further observed that the larger list of simple groups now treated essentially coincides with those of characteristicp-type, suggesting another equivalence method via the poset Bp(G) of radical (or stubborn)p-subgroups. In particular, one finds that these sporadic groups satisfy an analogue of the Borel–Tits theorem—that normalizers ofp-groups lie in simplex stabilizers. Still further intriguing coincidences remain to be explained
Internal Categories and Groupoids in Congruence Modular Varieties

AbstractIt is well known that there exist several simple descriptions of the internal categories and groupoids in various classical algebraic categories such as groups, rings, Lie algebras, etc. One of those descriptions is extended to arbitrary congruence modular varieties: we describe all internal categories satisfying a certain commutator condition which always holds for internal groupoids, and for all internal categories in Mal'tsev (=congruence permutable) varieties. The results of this paper show a deep connection between internal category theory and commutator theory in universal algebra; the so-called Kiss difference operation also plays a central role
On Spinor Vertex Operator Algebras and Their Modules

AbstractIn this paper, we first present a generator theorem and introduce an asymmetric tensor of two colored vertex operator superalgebras with the same grading and supermap. Then we give axioms which characterize a class of vertex operator superalgebras constructed from Clifford algebras. In particular, we are able to give a shorter argument for the Jacobi identity than the existing proofs by using elements analogous to coherent states and our generator theorem. Using the new tensor and the same techniques, we classify twisted irreducible modules of these algebras in a constructive way
The Universal Nonabelian Representation of the Petersen Type Geometry Related toJ4

AbstractLet G be a geometry with three points on each line. The universal representation groupR(G) of G is generated by a set of involutions, one for each point, subject to the relations that for every line the product of the three generators corresponding to the points on the line is the identity. LetJ4be the fourth sporadic simple group of Janko and let G(J4) be the Petersen type geometry ofJ4whose points and lines are subgroups of order 2 and 22inJ4with normalizers containing Sylow 2-subgroups; the incidence relation is given by inclusion. We show thatR(G(J4)) is exactlyJ4
Short Presentations for Finite Groups

AbstractWe conjecture that every finite groupGhas a short presentation (in terms of generators and relations) in the sense that the totallengthof the relations is (log|G|)O(1).We show that it suffices to prove this conjecture for simple groups.Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name ofG, assuming in the case of Lie type simple groups overGF(pm) that an irreducible polynomialfof degreemoverGF(p) and a primitive root ofGF(pm) are given.We verify this (stronger) conjecture for all finite simple groups except for the three families of rank 1 twisted groups: we do not handle the unitary groupsPSU(3,q)=2A2(q), the Suzuki groupsSz(q)=2B2(q), and the Ree groupsR(q)=2G2(q). In particular,all finite groups G without composition factors of these types have presentations of length O((log|G|)3).For groups of Lie type (normal or twisted) of rank≥2, we use a reduced version of the Curtis–Steinberg–Tits presentation
The Continued Fraction Algorithm and Regulator for Quadratic Function Fields of Characteristic 2

AbstractThe continued fraction expansion and infrastructure for quadratic congruence function fields of odd characteristic have been well studied. Recently, these ideas have even been used to produce cryptosystems. Much less is known concerning the continued fraction expansion and infrastructure for quadratic function fields of even characteristic. We will explore these ideas, and show that the situation is very similar to the odd characteristic case. This exploration will result in a method for computing the regulator for quadratic function fields of characteristic 2
Computation of First Cohomology Groups of Finite Covers

AbstractWe give several applications of standard methods of group cohomology to some problems arising in model theory concerning finite covers. We prove a conjecture of the author that forG-finite, ℵ0-categorical structures the kernels of minimal superlinked finite covers have bounded rank. We show that the cohomology groups associated to finite covers of certain structures (amongst them, the primitive, countable, totally categorical structures) have to be finite. From this we deduce that the finite covers of these structures are determined up to finitely many possibilities by their kernels
On Groups Admitting a Disconnected Common Divisor Graph

AbstractLetGbe an arbitrary group with a subgroupA. Each double cosetAgAis a union of right cosetsAu. The cardinality of the set {Au∣u∈G,Au⊆AgA} is called asubdegreeof (A,G) and is denoted by [AgA:A]. Thus for each double cosetAgAwe have a corresponding subdegree. An equivalent definition of the subdegree concept is given in [2]. IfAis not normal inGand all the subdegrees of (A,G) are finite, we attach to (A,G) thecommon divisor graphΓ: its vertices are the nonunit subdegrees of (A,G), and two different subdegrees are joined by an edge iff they arenotcoprime. It is proved in [2] that Γ has at most two connected components. We prove that if Γ is disconnected andAsatisfies a certain “regularity” property (a property which holds whenAor [G:A] is finite, and is called in this paper stability), thenGhas a nice structure. To be more precise, letDdenote the subdegree set of (A,G) and letD1be the set of all the subdegrees in the connected component of Γ containing min(D−{1}). Then we prove (Theorem A) that the setH=⋃[AgA:A]∈D1∪{1}AgAis a subgroup ofGandNG(A)<H<G. Some interesting properties of the subgroupHare described in Theorem B. Theorems D and E describe some properties of the subdegrees in the disconnected case
Complexity and Varieties for Infinite Groups, I

AbstractThis two-part paper generalizes the usual notion of complexity and varieties for modules over the group algebra of a finite group, to a large class of infinite groups. The context is modules of typeFP∞ for groups in Kropholler's classLHF. One of the main results is that the category of such modules is generated in a suitable sense by modules induced from finite elementary abelian subgroups. This implies that an element of complete cohomology of such a module is nilpotent if and only if its restriction to every finite elementary abelian subgroup is nilpotent. It also implies that the complexity of a module of typeFP∞ is finite, and that the variety is supported on some finite collection of finite elementary abelian subgroups. An example is given which shows that the complexity does not determine the rate of growth of the number of generators in a projective resolution, in the way it does for finite groups
Bounds for Codimensions of Fitting Ideals

AbstractWe find upper bounds for codimensions of Fitting ideals of a module (over a regular ring), whose symmetric algebra is equidimensional. These bounds are stronger than the classical Eagon–Northcott bounds. Together with lower bounds of Simis and Vasconcelos, they show that the Fitting ideals of a module whose symmetric algebra has an irreducible spectrum can appear only in a finite number of codimensions. Our proof is based on earlier results concerning minimal primes of tensor powers of a symmetric algebra and uses one of the dimension formulæ of Huneke and Rossi
Dimension Estimates for Representable Equivalences of Module Categories

AbstractLetRbe a ring,Pbe a ∗-module, andS=EndP. Inspired by the tilting theory, we investigate relations between the global dimensions gldimRand gldimS, and between the structure of the Grothendieck groupsK0(mod-R) andK0(mod-S). We prove that gldimS≤gldimR+D, whereD=1, e.g., forPalmost tilting, but the symmetry fails: for eachn<ω, there are almost tilting modules with gldimR−gldimS>n. IfRis right artinian andSright noetherian, then there is an isomorphismK0(mod-S)≅K0(mod-R̄), whereR̄=R/AnnR(P). We also prove that ifRandR′ are right artinian rings such that there is a torsion theory counter equivalence between Mod-Rand Mod-R′, thenK0(mod-R)≅K0(mod-R′)
A Characterization of Zero-Dimensional Annihilators of Tableaus

AbstractIn the theory of linear recursive sequences over a fieldKthere is a natural action of the polynomial ringK[x] on sequences (the end-off left shift), and every ideal ofK[x] is the annihilator with respect to this action of some sequences. This is a direct consequence of the fact thatK[x] is a principal ideal domain. In dimensionn≥2 the analogue of a sequence is atableau, and there is an analogous action ofK[x1,x2,…,xn] on tableaus. However, a given ideal ofK[X]=K[x1,x2,…,xn] is not necessarily the annihilator of some tableau. We will give an example in Section 2. Our main result is to characterize an important subset of those ideals which are annihilators. More specifically, in the theory of linear recursive sequences, an idealIofK[x] and a finite set of initial conditions (the initial fill) completely determine a sequence annihilated byI. This is a consequence of the fact thatK[x]/Iis a finite-dimensionalK-vector space. In order for a tableau to be completely determined by an idealIand a finite set of initial conditions, we must have thatK[X]/Iis a finite-dimensionalK-vector space, i.e., the only primes containingIare maximal, i.e.,Iis a 0-dimensional ideal. In particular, ifmis maximal andIism-primary thenIis 0-dimensional. Using methods of primary decomposition, we will be able to reduce to this case. Following a suggestion of B. Sturmfels, we show that a 0-dimensional ideal is the annihilator of some tableau if and only if it isGorenstein
On Antipodes and Integrals in Hopf Algebras over Rings and the Quantum Yang-Baxter Equation

AbstractThe authors showed previously (2) that every Frobenius algebra over a commutative ring defines a solution of the quantum Yang-Baxter equation. Applying this result to Hopf algebras over commutative rings which are finitely generated and projective as modules, we obtain an explicit formula for this solution. It turns out that this solution can be expressed in terms of the integral and antipode. We use this solution to characterize separable Hopf algebras over rings. Some results on the order of the antipode are also obtained