pinecone/core-2020-05-10-deduplication huggingface dataset:
On the Construction of the Finite Simple Groups with a Given Centralizer of a 2-Central Involution

AbstractLet H be a finite group having center Z(H) of even order. By the classical Brauer–Fowler theorem there can be only finitely many non-isomorphic simple groups G which contain a 2-central involution t for which CG(t)≅H. In this article we give a deterministic algorithm constructing from the given group H all the finitely many simple groups G having an irreducible p-modular representation M over some finite field F of odd characteristic p>0 with multiplicity-free semisimple restriction M|H to H, if H satisfies certain natural conditions. As an application we obtain a uniform construction method for all the sporadic simple groups G not isomorphic to the smallest Mathieu group M11. Furthermore, it provides a permutation representation, and the character table of G
Degenerate Affine Hecke Algebras and Centralizer Construction for the Symmetric Groups

AbstractIn our recent papers the centralizer construction was applied to the series of classical Lie algebras to produce the quantum algebras called (twisted) Yangians. Here we extend this construction to the series of the symmetric groups S(n). We study the “stable” properties of the centralizers of S(n−m) in the group algebra C[S(n)] as n→∞ with m fixed. We construct a limit centralizer algebra A and describe its algebraic structure. The algebra A turns out to be closely related with the degenerate affine Hecke algebras. We also show that the so-called tame representations of S(∞) yield a class of natural A-modules
Classification of Left-Covariant Differential Calculi on the Quantum Group SLq(2)

AbstractFor transcendental values of q the quantum tangent spaces of all left-covariant first order differential calculi of dimension less than four on the quantum group SLq (2) are given. All such differential calculi Γ are determined and investigated for which the left-invariant differential one-forms ω(u12), ω(u21), and ω(u11−u22) generate Γ as a bimodule and the universal higher order differential calculus has the same dimension as in the classical case. Important properties (cohomology spaces, ∗-structures, braidings, generalized Lie brackets) of these calculi are examined as well
A Pseudo-Analyzer Approach to Formal Group Laws Not of Operad Type

AbstractFormal group schemes, associated to affine group schemes or Lie groups by completion, can be described by classical formal group laws. More generally, cogroup objects in categories of complete algebras (e.g., associative) are described by group laws for operads or analyzers. M. Lazard has introduced analyzers to study formal group laws and group law chunks (truncated formal power series). A main example of a type of generalized formal group laws not given by an operad or analyzer are group laws corresponding to noncommutative complete Hopf algebras. To cover this case and other types of group laws, pseudo-analyzers are introduced. We point out differences to the (quadratic) operad case; e.g., there is no classification of group laws by Koszul duality. On the other hand we show how pseudo-analyzer cohomology can be used to describe extension of group law chunks
Extension of Ideal-Theoretic Properties of a Domain to Submodules of Its Quotient Field

AbstractWe examine when multiplicative properties of ideals extend to submodules of the quotient field of an integral domain. An integral domain R is stable if each non-zero ideal of R is invertible as an ideal over its ring of endomorphisms. We show that an integral domain R is stable if and only if an analogue of this invertibility property extends to submodules of the quotient field of R. By contrast, the class of integral domains for which every non-zero ideal is locally free over its ring of endomorphisms is shown to properly contain the class of domains R for which each submodule of the quotient field is locally free over its ring of endomorphisms, and we give complete characterizations of both classes of domains
On the Growth of Subalgebras in Lie p-Algebras

AbstractLet L be a finitely generated Lie p-algebra over a finite field F. Then the number, an(L), of p-subalgebras of finite codimension n in L is finite. We say that L has PSG (polynomial p-subalgebras growth) if the growth of an(L) is bounded above by some polynomial in ∣F∣n. We show that if L has PSG then the lower central series of L stabilises after a finite number of steps. On the other hand, if L is nilpotent then L has PSG. We deduce the following group-theoretic result. Let G be a group and let Gp̂ denote a pro-p completion of G. Then the associated Lie p-algebra Lp(G) of G has PSG if and only if Gp̂ is a p-adic analytic Lie group
On Algebraically Exact Categories and Essential Localizations of Varieties

AbstractAlgebraically exact categories have been introduced in J. Adámek, F. W. Lawvere, and J. Rosický (to appear), as an equational hull of the 2-category VAR of all varieties of finitary algebras. We will show that algebraically exact categories with a regular generator are precisely the essential localizations of varieties and that, in this case, algebraic exactness is equivalent to (1) exactness, (2) commutativity of filtered colimits with finite limits, (3) distributivity of filtered colimits over arbitrary products, and (4) product-stability of regular epimorphisms. This can be viewed as a nonadditive generalization of the classical Roos Theorem characterizing essential localizations of categories of modules. Analogously, precontinuous categories, introduced in J. Adámek, F. W. Lawrence, and J. Rosický (to appear) as an equational hull of the 2-category LFP (of locally finitely presentable categories), are characterized by the above properties (2) and (3). Essential localizations of locally finitely presentable categories and presheaf categories are fully described
Embedding Problems over Abelian Groups and an Application to Elliptic Curves

AbstractConditions for the solvability of certain embedding problems can be given in terms of the existence of elements with certain norm properties. A classical example, due to Witt (1936, Crelle's J.174, 237–245), is that of embedding a Klein extension into a dihedral extension. In “Construction de p-extensions Galoisiennes d'un corps de caractéristique différente de p” (1987, J. Algebra109, 508–535), Massy finds this type of condition for central p-extensions of an abelian group of exponent p. In this paper we generalize the results of Massy to the case of central p-extensions of any abelian group. In the last section we discuss the reason for our interest in these problems: they appear in the theory of elliptic Q-curves if one is interested in computing representatives with especially good arithmetic properties in the isogeny class of a given curve
The Regularity of a Toric Variety

AbstractWe give a method for computing the degrees of the minimal syzygies of a toric variety by means of combinatorial techniques. Indeed, we complete the explicit description of the minimal free resolution of the associated semigroup algebra, using the simplicial representation of Koszul homology which appeared in A. Campillo and C. Marijuán (1991, Sém. Théor. Nombres Bordeaux3, 249–260). As an application, we obtain an algorithm for computing the Castelnuovo–Mumford regularity of a projective toric variety. This regularity is explicitly bounded by means of the semigroup generators which parametrize the variety
Absolute Irreducibility of Polynomials via Newton Polytopes

AbstractA multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein criterion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field, but keeping a certain collection of them nonzero
Quasi-permutation Representations of the Group GL2(q)

AbstractA square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rationals and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G), and r(G) are calculated for the group G=GL2(q)
The Category PME(G,V) and Generalized Covering Groups

AbstractThe properties of covering and universality between the central extensions and the structure of a covering group of perfect groups have been generalized by S. Kayvanfar and M. R. R. Moghaddam (1997, Indag. Math. N.S.8(4), 537–542) to the variety of groups defined by a set of outer commutator words. In this paper we generalize the above results to any variety of groups. Then we introduce the category PME(G,V) and, using the above generalization, show that if G is V-perfect, then there exists a universal object in this category and its structure will be determined. Finally it is shown that any two V-covering groups of a V-perfect group are isomorphic and the structure of the unique generalized covering group of an arbitrary V-perfect group is introduced
Categorical Aspects of Generating Functions (I): Exponential Formulas and Krull–Schmidt Categories

AbstractIn this paper, we study formal power series with exponents in a category. For example, the generating function of a category E with finite hom sets is defined by E(t)=∑tX/|Aut(X)|, where the summation is taken over all isomorphism classes of objects of E. We can use such power series to enumerate the number of E-structures along a faithful functor (Theorem 4.6). Our theory is closely related to the theory of species (Joyal, 1981). A species can be identified with a faithful functor from a groupoid to the category of finite sets (Theorem 3.6). We use mainly the concept of faithful functors with finite fibers instead of species, so that we can separate the roles categories and functors play. For example, the exponential formula E(t)=exp(Con(E)(t)) means the unique coproduct decomposition property (Theorem 5.8). In the final section, we give some applications of our theory to rather classical enumerations
The Module of Logarithmic p-Forms of a Locally Free Arrangement

AbstractFor an essential, central hyperplane arrangement A⊆V≃kn+1 we show that Ω1(A) (the module of logarithmic one forms with poles along A) gives rise to a locally free sheaf on Pn if and only if, for all X∈LA with rank X<dimV, the module Ω1(AX) is free. Motivated by a result of L. Solomon and H. Terao (1987, Adv. Math.64, 305–325), we give a formula for the Chern polynomial of a bundle E on Pn in terms of the Hilbert series of ⊕m∈ZH0(Pn,∧iE(m)). As a corollary, we prove that if the sheaf associated to Ω1(A) is locally free, then π(A,t) is essentially the Chern polynomial. If Ω1(A) has projective dimension one and is locally free, we give a minimal free resolution for Ωp and show that ΛpΩ1(A)≃Ωp(A), generalizing results of L. Rose and H. Terao (1991, J. Algebra136, 376–400) on generic arrangements
A Decomposition Theory for Cyclotomic Modules under the Complete Point of View

AbstractIn 1986, D. Blessenohl and K. Johnsen (1986, J. Algebra103, 141–159) proved that for any finite extension E/F of Galois fields there exists a complete normal basis generator w of E/F, which means that w simultaneously generates a normal basis for E over every intermediate field of E/F. In a recent monograph by the author (1997, “Finite Fields: Normal Bases and Completely Free Elements,” Kluwer Academic, Boston) a theory is developed which allows the study of module structures of Galois fields as extensions with respect to various subfields and which led to an exploration of the structure of complete normal basis generators as well as explicit and algorithmic constructions of these objects. In the present paper we continue the development of that theory by providing various structural results: the Complete Decomposition Theorem, the Complete Product Theorem, a Theorem on Simultaneous Generators, and a Uniqueness Theorem
Speciality of Lie–Jordan Algebras

AbstractThe class of so-called Lie–Jordan algebras, which have one binary (Lie) operation [x,y] and one ternary (Jordan) operation {x,y,z}, that satisfy the identitiesx,y=−y,x,x,y,z=z,y,x,x,y,z=x,y,z−y,x,z,x,y,z,t=x,t,y,z+x,y,t,z+x,y,z,t,x,y,z,t,v=x,t,v,y,z−x,y,v,t,z+x,y,z,t,v.is introduced. It is proved that any such algebra is special, that is, isomorphic to a subalgebra of a Lie–Jordan algebra of the type A±, obtained from an associative algebra A via the operations [x,y]=xy−yx, {x,y,z}=xyz+zyx. As an application, we prove the conjecture about associativity of a certain loop constructed by A. Grishkov
On the Congruence Extension Property for Semigroups: Preservation under Homomorphic Images

AbstractA congruence ρT on a subsemigroup T of S extends to the semigroup S, if there exists a congruence ρ on S such that ρ∣T=ρT. A semigroup S has the congruence extension property (CEP) if each congruence on each subsemigroup extends to S. Previously, it was shown that a semigroup S has CEP if and only if it satisfies the so-called CEP conditions, a construction for semigroups with CEP was given, and we characterized congruences on this kind of semigroup. In the 1970s, the question whether or not CEP for semigroups is preserved in homomorphic images was asked. We give a positive answer for the problem in this paper
On TI and TI Defect Blocks

AbstractWe show that every trivial intersection block of a finite group (as introduced by J. L. Alperin and M. Broué (1979, Ann. of Math.110, 143–157)) has trivial intersection (TI) defect groups but that the converse is not true in general. We then present some conditions equivalent to B being a TI block, generalizing the idea of a k-generated p-core to B-subgroups. In particular we give further weight to Olsson's assertion that TI blocks are a better generalization of groups with TI Sylow p-subgroups than are TI defect blocks. Finally we describe the rôle of the generalized k-generated p-core in the control of fusion of subpairs
On Herstein's Lie Map Conjectures, II

AbstractThe theory of functional identities is used to study derivations of Lie algebras arising from associative algebras. Definitive results are obtained modulo algebras of “low dimension.” In particular, Lie derivations of [K,K]/([K,K]∩Z), where K is the Lie algebra of skew elements of a prime algebra with involution and Z is its center, are described. This solves the last remaining open problem of Herstein on Lie derivations. For a simple algebra with involution the Lie algebra of all derivations of [K,K]/([K,K]∩Z) is thoroughly analyzed. Maps that act as derivations on arbitrary fixed polynomials are also discussed, and in particular a solution is given for Herstein's question concerning maps of K which act like a derivation on xm, m being a fixed odd integer
Cross Product Bialgebras Part II

AbstractThis is the central article of a series of three papers on cross product bialgebras. We present a universal theory of bialgebra factorizations (or cross product bialgebras) with cocycles and dual cocycles. We also provide an equivalent (co-)modular (co-)cyclic formulation. All known examples as for instance bi- or smash, doublecross and bicross product bialgebras as well as double biproduct bialgebras and bicrossed or cocycle bicross product bialgebras are now united within a single theory. Furthermore our construction yields various novel types of cross product bialgebras
Functorial Representation Theorems for MVΔ Algebras with Additional Operators

AbstractWe prove functorial representation theorems for MVΔ algebras, and for varieties obtained from MVΔ algebras by the adding of additional operators corresponding to natural operations in the real interval [0,1], namely PMVΔ algebras, obtained by the adding of product, and ŁΠ algebras, obtained by the adding of product and of its residuum. Our first result is that the category of MVΔ algebras is equivalent to that of lattice ordered abelian groups with strong unit and with some kind of characteristic function. Our second result is that the category of PMVΔ algebras is equivalent to a category of commutative f-rings with strong unit, again with a suitable characteristic function, whose members (modulo a forgetful functor) are isomorphic to subdirect products of linearly ordered domains of integrity. Our third result (in our opinion, the most interesting) is that the category of ŁΠ algebras is equivalent to a category whose members are regular commutative f-rings with unit, equipped with an ideal with suitable properties
Packing and Covering Groups with Subgroups

AbstractWe study the problem of covering or packing a finite group with subgroups of a specified order and obtain bounds on the size of such covers and packings. Our main results provide characterizations of the elementary abelian groups by the existence of large packings or small covers, respectively. Hence large packings and small covers can be thought of as geometric objects: they correspond to large partial t-spreads and small t-covers of a suitable projective space PG(d,p) for some prime p. We shall also exhibit some series of examples which show that our bounds are reasonable
Generalized Weyl Algebras Are Tensor Krull Minimal

AbstractFormulae for calculating the Krull dimension of noetherian rings obtained by the authors and their collaborators are used to calculate Krull dimension for certain classes of algebras. An F-algebra T is said to be tensor Krull minimal (TKM) with respect to a class of F-algebras Ω if K(T⊗B)=K(T)+K(B), for each B∈Ω. We show that generalized Weyl algebras over affine commutative F-algebras, where F is an uncountable algebraically closed field, are TKM with respect to the class of countably generated left noetherian F-algebras. This simplifies the task of calculating many Krull dimensions. In addition, we develop an improved formula for the Krull dimension of a skew Laurent extension D[x,x−1;σ], where D is a polynomial algebra over an algebraically closed field, and σ is an affine automorphism. Finally, we calculate the Krull dimension of the noetherian down–up algebras introduced by Benkart
Reduction in the Rationality Problem for Multiplicative Invariant Fields

AbstractFor a faithful ZG lattice A and a field K on which the group G acts by field automorphisms, let R be the normal subgroup generated by the elements of G which act trivially on K and act as reflections on A. We prove that the rationality of the multiplicative invariant field K(A)G over K(AR)G is equivalent to the rationality of K(A)ΩG over K (AR)ΩG where ΩG is a particular subgroup of G such that G /R≅ΩG. We then use this reduction result to prove that K(A)G is rational over K where G is the automorphism group of a crystallographic root system Ψ, G acts trivially on K and A is any lattice on the space QΨ
Exponent Reduction for Projective Schur Algebras

AbstractIn this paper it is proved that the “exponent reduction property” holds for all projective Schur algebras. This was proved in an earlier paper of the authors for a special class, the “radical abelian algebras.” The precise statement is as follows: let A be a projective Schur algebra over a field k and let k(μ) denote the maximal cyclotomic extension of k. If m is the exponent of A⊗kk(μ), then k contains a primitive mth root of unity. One corollary of this result is a negative answer to the question of whether or not the projective Schur group PS(k) is always equal to Br(L/k), where L is the composite of the maximal cyclotomic extension of k and the maximal Kummer extension of k. A second consequence is a proof of the “Brauer–Witt analogue” in characteristic p: if char(k)=p≠0, then every projective Schur algebra over k is Brauer equivalent to a radical abelian algebra
Direct-Sum Decompositions over Local Rings

AbstractLet (R,m) be a local ring (commutative and Noetherian). If R is complete (or, more generally, Henselian), one has the Krull–Schmidt uniqueness theorem for direct sums of indecomposable finitely generated R-modules. By passing to the m-adic completion R̂, we can get a measure of how badly the Krull–Schmidt theorem can fail for a more general local ring. We assign to each finitely generated R-module M a full submonoid Λ(M) of Nn, where n is the number of distinct indecomposable direct summands of R̂⊗RM. This monoid is naturally isomorphic to the monoid +(M) of isomorphism classes of modules that are direct summands of direct sums of finitely many copies of M. The main theorem of this paper states that every full submonoid of Nn arises in this fashion. Moreover, the local ring R realizing a given full submonoid can always be taken to be a two-dimensional unique factorization domain. The theorem has two non-commutative consequences: (1) a new proof of a recent theorem of Facchini and Herbera characterizing the monoid of isomorphism classes of finitely generated projective right modules over a non-commutative semilocal ring, and (2) a characterization of the monoids +(N), where N is an Artinian right module over an arbitrary ring
Semigroup Algebras and Noetherian Maximal Orders

AbstractIn this paper we describe when a monoid algebra K[S] is a noetherian PI domain which is a maximal order. Our work relies on the study of the height one primes of K[S] and of the minimal primes of the monoid S and leads to a characterization purely in terms of S. It turns out that the primes P intersecting S plays a crucial role, and therefore we reduce the problem to certain “local” monoids SP, that is, monoids with only one minimal prime. However, we illustrate by examples that such monoids and their algebras are much more complicated than the discrete valuation rings arising in the commutative case. Our work is based on the results of Brown concerning the group ring case and of Anderson and Chouinard on commutative monoid rings. We also rely on the description of noetherian monoid algebras obtained earlier by the authors
Maximal Cohen–Macaulay Modules and Gorenstein Algebras

AbstractLet B be a graded Cohen–Macaulay quotient of a Gorenstein ring, R. It is known that sections of the dual of the canonical module, KB, can be used to construct Gorenstein quotients of R. The purpose of this paper is to place this method of construction into a broader context. If M is a maximal Cohen–Macaulay B-module whose sheafified top exterior power is a twist of K̃B and if M satisfies certain additional homological conditions then regular sections of M* can again be used to construct Gorenstein quotients of R. On Cohen–Macaulay quotients, the normal module, the first Koszul homology module and several other associated modules all have sheafified top exterior power equal to a twist of K̃B. If additional restrictions are placed on the Cohen–Macaulay quotients then these modules will satisfy the required additional homological conditions. This places the canonical module within a broad family of easily manipulated maximal Cohen–Macaulay modules whose sections can be used to construct Gorenstein quotients of R
The Brauer Group of Irreducible Coalgebras

AbstractIn this paper we solve a previously formulated conjecture (B. Torrecillas, F. Van Oystaeyen, and Y. H. Zhang, J. Algebra177 (1995), 568). That is, for a cocommutative irreducible coalgebra C, the homomorphism (−)*:Br(C)→Br(C*) is injective. The proof uses Morita–Takeuchi theory and the linear topology of all closed cofinite left ideals in C*. As an inmediate consequence, Br(C) is a torsion group. Some cases where the map (−)* is an isomorphism are studied. It is also deduded from the main result that the inclusion of the coradical C0 into C induces a monomorphism i∗:Br(C)→Br(C0). New examples of Brauer groups of cocommutative coalgebras may be given using this fact