pinecone/core-2020-05-10-deduplication huggingface dataset:
Balanced and Cobalanced Butler Groups

AbstractThe class of Butler groups—pure subgroups of finite-rank torsion-free completely decomposable groups—has been widely studied by abelian group theorists. Here we present several classification results for groups in two chains of special subclasses of the Butler groups, the K(n)- and co-K(n)-groups. The classes K(n) and co-K(n),n≥1, are defined by balanced and cobalanced exact sequences of Butler groups, respectively. We give direct sum decompositions of certain pure subgroups of K(n)-groups and certain torsion-free quotients of co-K(n)-groups. The direct sum decompositions characterize the groups in those classes and extend some well-known results for Butler groups
Clifford Regular Domains

AbstractThe class semigroup of a commutative integral domain R is the semigroup S(R) of the isomorphism classes of the nonzero ideals of R with operation induced by multiplication. A domain R is said to be Clifford regular if S(R) is a Clifford semigroup, i.e. S(R) is the disjoint union of the subgroups associated to the idempotent elements. In this paper we characterize the noetherian and the integrally closed Clifford regular domains and find some properties of an arbitrary Clifford regular domain
Localizing Systems, Module Systems, and Semistar Operations

AbstractWe present the concept of module systems for cancellative monoid. This concept is a common generalization of the notion of an ideal system (as presented by F. Halter-Koch (“Ideal Systems,” Dekker, New York, 1997)) and the notion of a semistar operation (as introduced by A. Okabe and R. Matsuda (Math. J. Toyama Univ.17 (1994), 1–21)). It allows a new insight into the connection between semistar operations and localizing systems (as developed in by M. Fontana and J. A. Huckaba (in “Commutative Rings in a Non-Noetherian Setting” (S. T. Chapman and S. Glanz, Eds.), Kluwer Academic, Dordrecht/Norwell, MA, 2000)), a general theory of flatness (including results of M. Fontana (in “Advances in Commutative Ring Theory” (D. E. Dobbs et al., Eds.), pp. 271–306, Dekker, New York, 1999) and S. Gabelli (in “Advances in Commutative Ring Theory” (D. E. Dobbs et al., Eds.), pp. 391–409, Dekker, New York, 1999)) and a new presentation of the theory of generalized integral closures
Generalizing the Hilton–Mislin Genus Group

AbstractFor any group H, let χ(H) be the set of all isomorphism classes of groups K such that K×Z≃H×Z. For a finitely generated group H having finite commutator subgroup [H,H], we define a group structure on χ(H) in terms of embeddings of K into H, for groups K of which the isomorphism classes belong to χ(H). If H is nilpotent, then the group we obtain coincides with the genus group G(H) defined by Hilton and Mislin. We obtain some new results on Hilton–Mislin genus groups as well as generalizations of known results
Forms of Coalgebras and Hopf Algebras

AbstractWe study forms of coalgebras and Hopf algebras (i.e., coalgebras and Hopf algebras which are isomorphic after a suitable extension of the base field). We classify all forms of grouplike coalgebras according to the structure of their simple subcoalgebras. For Hopf algebras, given a W*-Galois field extension K⊆L for W a finite-dimensional semisimple Hopf algebra and a K-Hopf algebra H, we show that all L-forms of H are invariant rings [L⊗H]W under appropriate actions of W on L⊗H. We apply this result to enveloping algebras, duals of finite-dimensional Hopf algebras, and adjoint actions of finite-dimensional semisimple cocommutative Hopf algebras
Categorical Quotients

AbstractEvery equivalence relation R on an algebraic variety U defines a class PR of all morphisms constant on equivalence classes of R and determines its categorical closure R̄ defined on U by xR̄y if and only if φ(x)=φ(y), for every φ∈PR. It is proved (Theorem A) that the equivalence classes of R̄ coincide with fibers of a morphism ψ∈PR. In the family of all morphisms with this property we may determine a subfamily of morphisms, called final pseudoquotients, which contains categorical quotients, if a categorical quotient exists, and, in the general case, is a substitute of such quotients
Division Algebras with PSL(2,q)-Galois Maximal Subfields

AbstractIf G is a finite group and k is a field, then G is k-admissible if there exists a G-Galois extension L/k such that L is a maximal subfield of a k-division algebra. We prove that PSL(2,7) is k-admissible for any number field which either fails to contain −1 or which has two primes lying over the dyadic prime. In addition, PSL(2,11) is shown to be admissible over Q or any number field k with at least two extensions of the dyadic prime. Indeed, there exist infinitely many linearly disjoint admissible extensions for these groups
Powers of the Fundamental Ideal in the Witt Ring

AbstractLet K be a field of characteristic different from 2. In the algebraic theory of quadratic forms, one studies the Witt ring W(K) of equivalence classes of non-degenerate quadratic forms. The Witt ring has a filtration given by the powers In(K) of the fundamental ideal I(K) of even-dimensional forms. The ideal In(K) is generated by the set Pn(K) of n-fold Pfister forms, 〈〈a1,…,an〉〉≔⊗ni=1〈1,−ai〉, where ai∈K*=K\;{0}. Many questions about this filtration and its quotients have arisen in the study of W(K). The spectacular work of Voevodsky, together with his collaborative work with Orlov and Vishik, allows one to answer many old questions. The purpose of this paper is to indicate some of these solutions
Local and Subquotient Inheritance of Simplicity in Jordan Systems

AbstractIn this paper we prove that the local algebras of a simple Jordan pair are simple. Jordan pairs all of which local algebras are simple are also studied, showing that they have a nonzero simple heart, which is described in terms of powers of the original pair. Similar results are given for Jordan triple systems and algebras. Finally, we characterize the inner ideals of a simple pair which determine simple subquotients, answering the question posed by O. Loos and E. Neher (1994, J. Algebra166, 255–295)
SCr Modules over Local Rings

AbstractWe show that the following two conditions, for each integer r≥1, are equivalent for a finitely generated module M over a complete Noetherian local ring (R,m):•(a) For each integer ≥1, there is a submodule ⊂m such that / is embeddable in , where denotes the injective hull of the residue field /m.•(b) Either ⊂, or else (/m,)=< and there is no prime ideal p such that /p=1 and (/p) is embeddable in .This is an extension of a result of M. Hochster (1977, Trans. Amer. Math. Soc.231, 463–488), which is the case r=1. We show other results related to these conditions
Asymptotic Cyclic Expansion and Bridge Groups of Formal Proofs

AbstractFormal proofs, even simple ones, may hide an unexpected intricate combinatorics. We define a new combinatorial invariant, the bridge group of a proof, which encodes the cyclic structure of proofs in the sequent calculus. We compute the bridge groups of two infinite families of proofs and identify them with the Baumslag–Solitar and Gersten groups. We observe that the distortion of cyclic subgroups in these groups equals the asymptotic growth of the procedure of elimination of lemmas from the proofs
Koszulity for Nonquadratic Algebras

AbstractIt is known that a Koszul algebra is defined as being a quadratic algebra with a “pure” resolution of the ground field. In this paper, we extend Koszulity to algebras whose relations are homogeneous of degree s>2. A cubic Artin–Schelter regular algebra has motivated our work. Generalized Koszulity is connected to lattice distributivity and to confluence. A generalized symmetric algebra is proved to be generalized Koszul, and the bimodule version of the generalized Koszul resolution is used for investigating its Hochschild homology
Decomposition Classes for Representations of Tame Quivers

AbstractFor the variety of representations of a tame quiver, we describe a finite stratification into smooth subvarieties that are invariant under the group of base changes and that admit smooth rational geometric quotients. A stratum consists of all representations that differ only in the parameters of the homogeneous indecomposable direct summands. For the special case of the loop our definition coincides with the well-known bundles (V. I. Arnold, Uspekhi Mat. Nauk26 (1971), 101–114) or decomposition classes of matrices (W. Borho and H. Kraft, Comment. Math. Helv.54 (1979), 61–104)
SK1-like Functors for Division Algebras

AbstractWe investigate the group valued functor G(D)=D*/F*D′ where D is a division algebra with center F and D′ the commutator subgroup of D*. We show that G has the most important functorial properties of the reduced Whitehead group SK1. We then establish a fundamental connection between this group, its residue version, and relative value group when D is a Henselian division algebra. The structure of G(D) turns out to carry significant information about the arithmetic of D. Along these lines, we employ G(D) to compute the group SK1(D). As an application, we obtain theorems of reduced K-theory which require heavy machinery, as simple examples of our method
Symmetric Powers of Modular Representations for Groups with a Sylow Subgroup of Prime Order

AbstractLet V be a representation of a finite group G over a field of characteristic p. If p does not divide the group order, then Molien's formula gives the Hilbert series of the invariant ring. In this paper we find a replacement of Molien's formula which works in the case that |G| is divisible by p but not by p2. We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our methods can be applied to determine the depth of the invariant ring without computing any invariants. This leads to a proof of a conjecture of the second author on certain invariants of GL2(p)
Nilpotent pairs, dual pairs, and sheets

Recently, V.Ginzburg introduced the notion of a principal nilpotent pair (=
pn-pair) in a semisimple Lie algebra {\frak g}. It is a double counterpart of
the notion of a regular nilpotent element. A pair (e_1,e_2) of commuting
nilpotent elements is called a pn-pair, if the dimension of their simultaneous
centralizer is equal to the rank of {\frak g} and some bi-homogeneity condition
is satisfied. Ginzburg proved that many familiar results of the `ordinary'
theory have analogues for pn-pairs. The aim of this article is to develop the
theory of nilpotent pairs a bit further and to present some applications of it
to dual pairs and sheets. It is shown that a large portion of Ginzburg's theory
can be extended to the pairs whose simultaneous centraliser is of dimension
rk{\frak g}+1. Such pairs are called almost pn-pairs. It is worth noting that
the very existence of almost pn-pairs is a purely "double" phenomenon, because
the dimension of "ordinary" orbits is always even. We prove that to any
principal or almost nilpotent pair one naturally associates a dual pair.
Moreover, this dual pair is reductive if and only if e_1 and e_2 can be
included in commuting sl_2-triples. We also study sheets containing members of
pn-pairs. Some cases are described, where these sheets are smooth and admit a
section.Comment: 27 pages, LaTeX 2.0
The q-Analogue of the Alternating Group and Its Representations

AbstractWe define the new algebra. This algebra has a parameter q. The defining relations of this algebra at q=1 coincide with the basic relations of the alternating group. We also give the new subalgebra of the Hecke algebra of type A which is isomorphic to this algebra. This algebra is free of rank half that of the Hecke algebra. Hence this algebra is regarded as a q-analogue of the alternating group.All the isomorphism classes of the irreducible representations of this algebra and the q-analogue of the branching rule between the symmetric group and the alternating group are obtained
The Exponential Homomorphism for the Second Syntomic Cohomology

AbstractLet OK be a complete discrete valuation ring with perfect residue field F of characteristic p≥3 and M a free filtered Dieudonné module. The second relative syntomic cohomology H2((OK,F),S(M,2)) with coefficients in M is an object related to arithmetic geometry, such as the albanese kernel. In this article, we study H2((OK,F),S(M,2)) by constructing the exponential homomorphism. We determine the structure of H2((OK,F),S(M,2)), under the assumption that M is of Hodge–Witt type and that the absolute ramification index of OK is prime to p
Algebraic Approaches to Periodic Arithmetical Maps

AbstractA residue class a+nZ with weight λ is denoted by 〈λ,a,n〉. For a finite system A={〈λs,as,ns〉}ks=1 of such triples, the periodic map wA(x)=∑ns|x−asλs is called the covering map of A. Some interesting identities for those A with a fixed covering map have been known; in this paper we mainly determine all those functions f:Ω→C such that ∑ks=1λsf(as+nsZ) depends only on wA where Ω denotes the family of all residue classes. We also study algebraic structures related to such maps f, and periods of arithmetical functions ψ(x)=∑ks=1λse2πiasx/ns and ω(x)=|{1≤s≤k:(x+as,ns)=1}|
Double Centralizer Properties, Dominant Dimension, and Tilting Modules

AbstractDouble centralizer properties play a central role in many parts of algebraic Lie theory. Soergel's double centralizer theorem relates the principal block of the Bernstein–Gelfand–Gelfand category O of a semisimple complex Lie algebra with the coinvariant algebra (i.e., the cohomology algebra of the corresponding flag manifold). Schur–Weyl duality relates the representation theories of general linear and symmetric groups in defining characteristic, or (via the quantized version) in nondefining characteristic. In this paper we exhibit algebraic structures behind these double centralizer properties. We show that the finite dimensional algebras relevant in this context have dominant dimension at least two with respect to some projective–injective or tilting modules. General arguments which combine methods from ring theory (QF-3 rings and dominant dimension) with tools from representation theory (approximations, tilting modules) then imply the validity of these double centralizer properties as well as new ones. In contrast to the traditional proofs (e.g., by the fundamental theorems of invariant theory) no computations are necessary
Vertex Algebra Approach to Fusion Rules for N=2 Superconformal Minimal Models

AbstractLet Lcm be the vertex operator superalgebra associated to the unitary vacuum module for the N=2 superconformal algebra with the central charge cm=3mm+2,m∈N. Then the unitary N=2-modules give all irreducible modules for the vertex operator superalgebra Lcm. In this paper, we determine all fusion rules for Lcm-modules from the vertex algebra point of view. These fusion rules coincide with the fusion rules obtained by M. Wakimoto (Fusion rules for N=2 superconformal modules, hep-th/9807144) using a modified Verlinde formula
Quasigroup Homogeneous Spaces and Linear Representations

AbstractUsing pseudoinverses of incidence matrices of finite quasigroups in partitions induced by left multiplications of subquasigroups, a quasigroup homogeneous space is defined as a set of Markov chain actions indexed by the quasigroup. A certain non-unital ring is afforded a linear representation by a quasigroup homogeneous space. If the quasigroup is a group, the linear representation is a factor in the usual linear representation of the group algebra afforded by the group homogeneous space. In the general case, the structure of the non-unital ring is analyzed in terms of the permutation action of the multiplication group of the quasigroup. The linear representation corestricts to the natural projection of the non-unital ring onto the quotient by its Jacobson radical
Basis of the Identities of the Matrix Algebra of Order Two over a Field of Characteristic p≠2

AbstractIn this paper we prove that the polynomial identities of the matrix algebra of order 2 over an infinite field of characteristic p≠2 admit a finite basis. We exhibit a finite basis consisting of four identities, and in “almost” all cases for p we describe a minimal basis consisting of two identities. The only possibilities for p where we do not exhibit minimal bases of these identities are p=3 and p=5. We show that when p=3 one needs at least three identities, and we conjecture a minimal basis in this case. In the course of the proof we construct an explicit basis of the vector space of the central commutator polynomials modulo the ideal of the identities of the matrix algebra of order two
On the Normalizer Problem

AbstractIn this paper the normalizer problem of an integral group ring of an arbitrary group G is investigated. It is shown that any element of the normalizer NU1(G) of G in the group of normalized units U1(ZG) is determined by a finite normal subgroup. This reduction to finite normal subgroups implies that the normalizer property holds for many classes of (infinite) groups, such as groups without non-trivial 2-torsion, torsion groups with a normal Sylow 2-subgroup, and locally nilpotent groups. Further it is shown that the commutator of NU1(G) equals G′ and NU1(G)/G is finitely generated if the torsion subgroup of the finite conjugacy group of G is finite
Amenable Orders Associated with Inverse Transversals

AbstractIf S is a regular semigroup with an inverse transversal S°={x°;x∈S} then an order ≤ on S is said to be amenable with respect to S° if•(1) ≤ is compatible with the multiplication of ;•(2) on the idempotents, ≤ coincides with the natural order ≤;•(3) ≤⇒°≤°,°≤°.This notion is in fact independent of the choice of inverse transversal. Here we consider the case where S is locally inverse (equivalently, where S° is a quasi-ideal). We give a complete description of all amenable orders on S and characterise the natural order ≤n as the smallest of these. We also establish a bijection from the set of amenable orders definable on S to the set of McAlister cones of S°, whence every amenable order on S° extends to a unique amenable order on S
Finite Soluble Groups with Permutable Subnormal Subgroups

AbstractA finite group G is said to be a PST-group if every subnormal subgroup of G permutes with every Sylow subgroup of G. We shall discuss the normal structure of soluble PST-groups, mainly defining a local version of this concept. A deep study of the local structure turns out to be crucial for obtaining information about the global property. Moreover, a new approach to soluble PT-groups, i.e., soluble groups in which permutability is a transitive relation, follows naturally from our vision of PST-groups. Our techniques and results provide a unified point of view for T-groups, PT-groups, and PST-groups in the soluble universe, showing that the difference between these classes is quite simply their Sylow structure
On Quasi-thin Association Schemes with Odd Number of Points

AbstractLet (X,R) be an association scheme in the sense of P.-H. Zieschang (1996, “An Algebraic Approach to Association Schemes,” Lecture Notes in Mathematics, Vol. 1628, Springer, New York/Berlin), where X is a finite set and R is a partition of X×X. We say that (X,R) is quasi-thin if each element of R has a valency of at most two. In this paper we focus on quasi-thin association schemes with an odd number of points and obtain that (X,R) has a regular automorphism group when nOθ(R) is square-free
The Method of Computing Formal Moduli

AbstractIn this paper we consider a method given earlier for computing the formal moduli of modules. We start with a short resume of the method, then we explain how this method can be implemented using a standard basis computation. We list the formal moduli with versal families for all torsionfree rank 1 modules over the curve singularity E6. Finally we discuss the degeneracy properties of these modules, proving that the normalization deforms to all the other torsionfree rank 1 modules. These examples are computed by implementing the algorithm in the Singular language. The program source is not included in this paper for obvious reasons, but can be found in previous work by the author (1997, Implementing the algorithm for computing formal moduli, Preprint Series, Institute of Mathematics, University of Oslo)
Stable Limits of Log Surfaces and Cohen–Macaulay Singularities

AbstractGiven a family of surfaces of general type over a smooth curve, one can apply semistable reduction and the minimal model program to obtain a stable reduction. This is the basis for a geometric compactification for moduli spaces of surfaces of general type, due to Kollár, Shepherd-Barron, and Alexeev. However, this approach hinges on the fact that the resulting stable limit has relatively mild singularities; in particular, it should be Cohen–Macaulay. Unfortunately, the standard formalism does not guarantee that stable limits of families of log surfaces are Cohen–Macaulay. Here we prove that this is the case
BGP-Reflection Functors and Lusztig's Symmetries: A Ringel–Hall Algebra Approach to Quantum Groups

AbstractAccording to the canonical isomorphisms between the Ringel–Hall algebras (composition algebras) and the quantum groups, we deduce Lusztig's symmetries T″i,1,i∈I, by applying the Bernstein–Gelfand–Ponomarev reflection functors to the Drinfeld doubles of Ringel–Hall algebras. The fundamental properties of T″i,1 including the following can be obtained conceptually. (1) T″i,1,i∈I induce automorphisms of the quantum groups Uq(g) and on the integrable modules. (2) T″i,1,i∈I satisfy the braid group relations. This extends and completes the results of B. Sevenhant and M. Van den Bergh (1999, J. Algebra221, 135–160)
The Buchsbaum–Rim Polynomial of a Module

AbstractFor a Noetherian local ring (R,m), the Hilbert functions of m-primary ideals provide considerable information about constructions and singularities coded by the ideal. For a module E which is the analogue of such ideals (E is a proper submodule of a free module so that the quotient has finite length), Buchsbaum and Rim defined a similar function and introduced the notion of Buchsbaum–Rim multiplicity br(E) of the module. Unlike in the ideal case, the information about these functions is still scant. When R is Cohen–Macaulay of dimension d>1 and E has rank r, we develop some of their properties sufficiently enough to establish the following bound for the reduction number of the module E,r(E)≤(d+r−1)·(br(E)−2)+1. The same techniques lead to bounds on the number of generators of the integral closure of the powers of certain ideals and modules
Normal Bases over GF(q)

AbstractFor q a power of a prime p, it is known that if m is a power of p or m itself is a prime different from p having q as one of its primitive roots, then the roots of any irreducible polynomial of degree m and of non-zero trace are linearly independent over GF(q). As a consequence the roots of such an mth degree polynomial form a basis of GF(qm) over GF(q). Such a basis is called a normal basis over GF(q) and the polynomial is called normal over GF(q). Normal bases over finite fields have proved very useful for fast arithmetic computations with potential applications to coding theory and to cryptography. In this paper, we prove that for mth degree irreducible polynomials the above two conditions are indeed necessary and sufficient conditions for the equivalence between the properties of having a non-zero trace and being normal over GF(q)
Examples of QF Rings without Nakayama Automorphism and H-Rings without Self-Duality

AbstractJ. Kado and K. Oshiro (J. Algebra211 (1999), 384–408) proved the equivalence of (A) the existence of a Nakayama isomorphism for every basic left H-ring (Harada ring) and (B) the existence of a Nakayama automorphism for every basic QF ring. In this paper, by using examples of J. Kraemer (“Characterizations of the Existence of (Quasi-) Self-Duality for Complete Tensor Rings,” Algebra Berichte 56, Verlag Reinhard Fischer, Munich, 1987), we give examples of QF rings without Nakayama automorphism and H-rings without self-duality