pinecone/core-2020-05-10-deduplication huggingface dataset:
Triangular Decomposition of Tame Non-Simply Laced Composition Algebras

AbstractLet A be a finite dimensional hereditary algebra over a finite field, and let C(A) be the composition algebra of A. Let P (resp. I) be the subalgebra of C(A) generated by the preprojective (resp. preinjective) A-modules, and let T be the subalgebra generated by rd, where rd=∑[M]∈H(A),[M] runs over the isomorphism classes of the regular A-modules with dimension vector d, and H(A) is the Ringel–Hall algebra of A. Then we prove that C(A)=P·T·I, if A is one of the types B̃n, C̃n, BC̃n, BD̃n, CD̃n, F̃41, F̃42, G̃21, G̃22, or Ã11
On Torsion-by-Nilpotent Groups

AbstractLet C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C are torsion-by-nilpotent, then all soluble groups of C are torsion-by-nilpotent. From that, we deduce the following consequence, similar to a well-known result of P. Hall (1958, Illinois J. Math.2, 787–801): if H is a normal subgroup of a group G such that H and G/H′ are (locally finite)-by-nilpotent, then G is (locally finite)-by-nilpotent. We give an example showing that this last statement is false when “(locally finite)-by-nilpotent” is replaced with “torsion-by-nilpotent.
The Bethe Equation at q=0, the Möbius Inversion Formula, and Weight Multiplicities. II. The Xn Case

AbstractWe study a family of power series characterized by a system of recursion relations (Q-system) with a certain convergence property. We show that the coefficients of the series are expressed by the numbers which formally count the off-diagonal solutions of the Uq(X(1)n) Bethe equation at q=0. The series are conjectured to be the Xn-characters of a certain family of irreducible finite-dimensional Uq(X(1)n)-modules which we call the KR (Kirillov–Reshetikhin) modules. Under the above conjecture, these coefficients give a formula of the weight multiplicities of the tensor products of the KR modules, which is also interpreted as the formal completeness of the XXZ-type Bethe vectors
Glauberman Correspondence of p-Blocks of Finite Groups

AbstractLet G and A be finite groups with coprime orders. Suppose that A is solvable and that it acts on G by automorphisms. Let C=CG(A). By Irr(G) and IrrA(G) we denote the set of all irreducible and all A-invariant irreducible characters of G, respectively. Let D≤C be a fixed p-subgroup of G for a prime p. Using the Glauberman correspondenceπG,A:IrrAG→IrrC,A. Watanabe (J. Algebra216 (1999), 548–565) recently established a Glauberman correspondence between A-invariant p-blocks B of G with defect group D and p-blocks B1 of C with defect group D. Let Br(B) and Br(B1) be the Brauer correspondents of B and B1 in NG(D) and NC(D), respectively. The main result of this article asserts that the block algebras Br(B) and Br(B1) are Morita equivalent. Furthermore, if G is p-solvable and D is abelian, then the block algebras B and B1 are Morita equivalent
Local PI Theory of Jordan Systems II

AbstractWe pursue the study, initiated in a previous paper, of Jordan systems having nonzero local algebras that satisfy a polynomial identity. We define the extended centroid of a nondegenerate Jordan system, the corresponding central extension, which we call the extended central closure, and prove a Jordan analogue of Martindale's theorem on prime algebras having a generalized identity: If J is a nondegenerate Jordan system with nonzero PI-elements, then the extended central closure of J has nonzero socle, equal to its PI ideal
Hilbert Functions and Level Algebras

AbstractThe authors consider certain quotients A=R/I of the polynomial ring R=K[x1,…,xr] over an arbitrary field K. They first determine upper and lower bounds on the Hilbert functions of any algebra having the form A=R/V, where V̄ is the largest ideal of R agreeing in degrees at least j with the ideal (V) generated by a vector subspace V⊂Rj of degree j forms: these bounds extend the Macaulay bounds (Theorem 1.4). A level Artinian quotient of A=R/I has socle Soc(A)=(0:M),M=(x1,…,xr) in a single degree j. The authors next determine the extremal Hilbert functions Hmax(t,j,r) and Hmin(t,j,r) that occur for level graded Artin algebra quotients of R having socle degree j and “type” t=dimKSoc(A), and they describe the extremal strata (Theorem 1.8).They next give a natural upper bound for the Hilbert function of level algebra quotients of the coordinate ring OZ of a punctual subscheme Z⊂Pn,n=r−1, in terms of the Hilbert function HZ and j,t. This bound was known and known to be sharp in the case Z is locally Gorenstein and t=1. Finally, they show that there are no level algebras of Hilbert function T=(1,3,4,5,6,…,2) (Proposition 2.7). Given that there are smooth punctual subschemes Z of P2 with Hilbert function HZ=(1,3,4,5,6,6,…), this result shows that the sharpness of the natural upper bound in the case where Z is Gorenstein and t=1 does not extend to level algebra quotients of type 2, even when Z is smooth.The extremality results for the Hilbert functions of level algebras, and some counterexamples, use the extremality theorem of F. H. S. Macaulay [1927, Proc. London Math. Soc.26, 531–555] and G. Gotzmann's [1978, Math. Z.158, 61–70] results on the Hilbert scheme
Generalized Pure Modules

AbstractFor right modules M<N over a ring R, consider any system of equations in M of the form ∑{xirij∣i∈I}=dj∈M,j∈J, where rij∈R. The usual definition of M as pure in N is that for any such a finite system, if the system is solvable in the bigger module N, then it is already solvable in M. Here the above ordinary concept of purity will be generalized by allowing I and J to be of possibly infinite cardinalities |I|<μ and |J|<ℵ for fixed cardinals μ and ℵ. In this way, generalized (μ<,ℵ<)-pure and absolutely pure concepts are defined in terms of μ and ℵ and studied. Here the number ℵ of relations of a module is simultaneously studied with the more familiar number μ of generators
Automorphic Orbits in Free Non-associative Algebras

AbstractWe consider finitely generated free non-associative algebras, free commutative non-associative algebras, and free anti-commutative non-associative algebras. We study orbits of elements of these algebras under the action of automorphism groups. Using free differential calculus we obtain matrix criteria for a system of elements to have given rank (or to be primitive). It gives us a possibility to construct fast algorithm to recognize primitive systems of elements. We show that if an endomorphism of a free algebra preserves the automorphic orbit of a nonzero element, then it is an automorphism of this algebra. In particular, endomorphisms preserving primitivity of elements are automorphisms
The Combinatorial Laplacian of the Tutte Complex

AbstractLet M be an ordered matroid and C••(M) be an exterior algebra over its underlying set E, graded by both corank and nullity. Then C•0(M) is the simplicial chain complex of IN(M), the simplicial complex whose simplices are indexed by the independent sets of the matroid. Dually, C0•(M) is the cochain complex of IN(M*). We give a combinatorial description of a basis of eigenvectors for the combinatorial Laplacian of a family of boundary maps on the double complex, extending work by W. Kook, V. Reiner, and D. Stanton [2000, J. Amer. Math. Soc.13, 129–148] on IN(M). The eigenvalues are enumerated by a weighted version of the Tutte polynomial, using an identity of G. Etienne and M. Las Vergnas [1998, Discrete Math.179, 111–119]. As an application, we prove a duality theorem for the cohomology of Orlik–Solomon algebras
An Independence Result on Cotorsion Theories over Valuation Domains

AbstractIt is shown that, over suitable valuation domains R with field of quotients Q, the cotorsion theory GK generated by K=Q/R coincides with the cotorsion theory C∂ cogenerated by the Fuchs' divisible module ∂, provided that Gödel's Axiom of Constructibility V=L is assumed. On the other hand, assuming Martin's Axiom and the negation of the Continuum Hypothesis, it is proved that the cotorsion theory GK is strictly smaller than C∂ by exhibiting a strongly (ℵ1−K)-free divisible module M of projective dimension 2 such that Ext1R(M,K)=0. Applications to Whitehead modules are derived
One-Sided Noncommutative Gröbner Bases with Applications to Computing Green's Relations

AbstractStandard noncommutative Gröbner basis procedures are used for computing ideals of free noncommutative polynomial rings over fields. This paper describes Gröbner basis procedures for one-sided ideals in finitely presented noncommutative algebras over fields. The polynomials defining a K-algebra A as a quotient of a free K-algebra are combined with the polynomials defining a one-sided ideal I of A, by using a tagging notation whose essential effect is to forbid left multiplication. Standard noncommutative Gröbner basis techniques can then be applied to the mixed set of polynomials, thus calculating A/I whilst working in a free structure, avoiding the complication of computing in A. The paper concludes by showing how the results can be applied to completable presentations of semigroups and so enable calculations of Green's relations
Pseudocomplemented Semilattices, Boolean Algebras, and Compatible Products

AbstractPseudocomplemented semilattices are studied here from an algebraic point of view, stressing the pivotal role played by the pseudocomplements and the relationship between pseudocomplemented semilattices and Boolean algebras. Following the pattern of semiprime ring theory, a notion of Goldie dimension is introduced for complete pseudocomplemented lattices and calculated in terms of maximal uniform elements if they exist in abundance. Products in lattices with 0-element are studied and questions about the existence and uniqueness of compatible products in pseudocomplemented lattices, as well as about the abundance of prime elements in lattices with a compatible product, are discussed. Finally, a Yood decomposition theorem for topological rings is extended to complete pseudocomplemented lattices
Finite Presentability of Bruck–Reilly Extensions of Groups

AbstractThe purpose of this paper is to consider finite generation and finite presentability of a Bruck–Reilly extension S=BR(G,θ) of a group G with respect to an endomorphism θ. It is proved that S is finitely generated if and only if G can be generated by a set of the form ⋃∞i=0Aθi, where A⊆G is finite. The main result states that S is finitely presented if and only if G can be defined by a presentation of the form 〈A∣R〉 where A is finite and R is of the form ⋃∞i=0R̄θi for some finite set of relations R̄. Finally, it is proved that S is finitely presented as an inverse monoid if and only if it is finitely presented as an ordinary monoid
Applications of the Brauer Complex: Card Shuffling, Permutation Statistics, and Dynamical Systems

AbstractBy algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini using the affine Weyl group. Formulas for Cellini's measure in type A are found. This leads to new models of card shuffling and has interesting combinatorial and number-theoretic consequences. An analysis of type C gives another solution to a problem of Rogers in dynamical systems: the enumeration of unimodal permutations by cycle structure. The proof uses the factorization theory of palindromic polynomials over finite fields. Contact is made with symmetric function theory
Generalized Verma Modules Induced from sl(2,C) and Associated Verma Modules

AbstractWith each generalized Verma module induced from a “well-embedded” parabolic subalgebra of a Lie algebra with triangular decomposition, we associate a Verma module over the same algebra in a natural way. In the case when the semisimple part of the Levi factor of the parabolic subalgebra is isomorphic to sl(2,C) and the generalized Verma module is induced from an infinite-dimensional simple module, we prove that the associated Verma module is simple if and only if the original generalized Verma module is simple
On Formations of Finite Groups with the Wielandt Property for Residuals

AbstractGiven two subgroups U,V of a finite group which are subnormal subgroups of their join 〈U,V〉 and a formation F, in general it is not true that 〈U,V〉F=〈UF,VF〉. A formation is said to have the Wielandt property if this equality holds universally. A formation with the Wielandt property must be a Fitting class. Wielandt proved that the most usual Fitting formations (e.g., nilpotent groups and π-groups) have the Wielandt property. At present, neither a general satisfactory result on the universal validity of the Wielandt property nor a counterexample is known. In this paper a criterion for a Fitting formation to have the Wielandt property is given. As an application, it is proved that many of the known Fitting formations have the Wielandt property
A Stable Range for Dimensions of Homogeneous O(n)-Invariant Polynomials on the n×n Matrices

AbstractThe complex orthogonal group O(n) acts on the n×n matrices, Mn, by restricting the adjoint action of GL(n,C). This action provides us with an action on the ring of complex valued polynomial functions on the n×n matrices, P(Mn). The polynomials of degree d, denoted Pd(Mn), form a finite dimensional representation of O(n) and provide a graded module structure on P(Mn) as well as the subring of invariant polynomials, P(Mn)O(n). For 0≤d≤n, it is shown that dim Pd(Mn)O(n) is equal to the coefficient of qd in Π∞k=1(1/(1−qk))ck, where ck is the number of k vertex cyclic graphs with directed edges counted up to dihedral symmetry. The above formula provides a combinatorial interpretation of an initial segment of the Hilbert series for this ring
Envelopes and Covers by Modules of Finite Injective and Projective Dimensions

AbstractIn this paper, we study the existence of L⊥-envelopes, L-envelopes, D⊥-envelopes, D-covers, and L-covers where L and D denote the classes of modules of injective and projective dimension less than or equal to a natural number n, respectively. We prove that over any ring R, special D⊥-preenvelopes and special D-precovers always exist. If the ring is noetherian, the same holds for L⊥- envelopes, and for D⊥-envelopes and d-covers when the ring is perfect. When inj.dimR≤n then L-covers exist, and if R is such that a given class of homomorphisms is closed under well ordered direct limits then L-envelopes exist
Double Quantum Groups

AbstractThe construction of the Drinfeld double D(H) of a finite dimensional Hopf algebra H was one of the first examples of a quasitriangular Hopf algebra whose category of modules MD(H) is braided. The braided category of Yetter–Drinfeld modules YDHH is the analogue for infinite dimensional Hopf algebras. It uses a strong dependence between the H-module and the H-comodule structures. We generalize this construction to the category MCA(ψ) of entwined modules, that is, A-modules and C-comodules over Hopf algebras A and C where the structures are only related by an entwining map ψ: C⊗A→A⊗C. We show that MCA(ψ) is braided iff there is an r-map r:C⊗C→A⊗A satisfying suitable axioms that generalize the axioms of an R-matrix. For finite dimensional C there is a quasitriangular Hopf algebra structure on Hom(C,A), the quantum group double, generalizing the construction of the Drinfeld double
Involutions, Classical Groups, and Buildings

AbstractIn [Invent. Math.58 (1980), 201–210], Curtis et al. construct a variation of the Tits building. The Curtis–Lehrer–Tits building L(G,k) of a connected reductive k-group G has the important feature that it is a functor from the category of reductive groups defined over a field k and monomorphisms to the category of topological spaces and inclusions. An important consequence derived by Curtis et al. from the functorial nature of the Curtis–Lehrer–Tits building L(G,k) is that if s is a semisimple element of the group G(k) of k-rational points, and G′ is the connected component group of the centralizer of s, then the fixed point set L(G,k)s of s in L(G,k) is the Curtis–Lehrer–Tits building L(G′,k). We generalize this result to arbitrary involutions of Autk(G), and we also prove an analogue in the context of affine buildings
Curves of Degree r+2 in Pr: Cohomological, Geometric, and Homological Aspects

AbstractLet C⊆Prk be a non-degenerate projective curve of degree r+2, where r≥3. By means of the Hartshorne–Rao module of C we distinguish 4 different possible cases (and an exceptional case which only appears if r=3). In any case C is obtained either by means of an embedding of an arbitrary smooth curve C0 of genus 2, or by projecting an elliptic normal curve from a point or by projecting a rational normal curve from a line. Finally, the possible minimal free resolutions of the homogeneous coordinate ring of C are studied
Modules whose Injective Endomorphisms Are Essential

AbstractAn R-module M is called weakly co-Hopfian if any injective endomorphism of M is essential. The class of weakly co-Hopfian modules lies properly between the class of co-Hopfian and the class of Dedekind finite modules. Several equivalent conditions are given for a module to be weakly co-Hopfian. Being co-Hopfian, weakly co-Hopfian, or Dedekind finite are all equivalent conditions on quasi-injective modules. Some other properties of weakly co-Hopfian modules are also obtained. The ring R is said to be right strong stably finite if all the finitely generated free right R-modules are weakly co-Hopfian. We shall characterize such rings and show that they are stably finite and satisfy the right strong rank condition. Examples show that stably finite rings and rings with the right strong rank conditions need not be strong stably finite. Both weakly co-Hopfian and right strong stably finite are Morita invariants, although the right and left strong stably finite are different properties. The class of commutative rings and the class of rings with finite right uniform dimension are proper subclasses of the class of right strong stably finite rings. We shall also investigate conditions that are relevant to weakly co-Hopfian modules. Equivalent statements are found on a ring to have all its finitely generated right modules weakly co-Hopfian
Bijective Relative Gabriel Correspondence over Rings with Torsion Theoretic Krull Dimension

AbstractA series of results by Asensio and Torrecillas (1992, Comm. Algebra20, 847–866), Gordon and Robson (1973, “Krull Dimension,” Memoirs of the American Mathematical Society, Vol. 133), Kim and Krause (1999, Comm. Algebra27, 3339–3351), and Năstăsescu (1981, Comm. Algebra9, 1395–1426) give information about rings that have Krull dimension or are noetherian relative to a torsion theory. The aim of this paper is to extend these results to rings R having relative Krull dimension with respect to a hereditary torsion theory τ on Mod-R such that any τ-torsion-free right R-module M has nonempty assassinator. Since any ideal invariant hereditary torsion theory has this property in view of a recent result by the authors (2000, J. Algebra229, 498–513), these results apply in particular to the commutative case
Closed Subbifunctors of the Extension Functor

AbstractGiven a subbifunctor F of Ext1(,), one can ask if one can generalize the construction of the derived category to obtain a relative derived category, where one localizes with respect to F-acyclic sequences. We show that this is possible if and only if F is closed. We also show that for artin algebras the closed subbifunctors correspond to Serre subcategories of a category of finitely presented functors that vanish on projectives, and we use this to find new examples of closed subbifunctors. Using relatively derived categories, we give a relative version of Happel's result on derived equivalences induced by tilting, and we show in an example how this can be used to find ordinary derived equivalences
Linear Groups Definable in o-Minimal Structures

AbstractWe study subgroups G of GL(n,R) definable in o-minimal expansions M=(R,+,·,…) of a real closed field R. We prove several results such as: (a) G can be defined using just the field structure on R together with, if necessary, power functions, or an exponential function definable in M. (b) If G has no infinite, normal, definable abelian subgroup, then G is semialgebraic. We also characterize the definably simple groups definable in o-minimal structures as those groups elementarily equivalent to simple Lie groups, and we give a proof of the Kneser–Tits conjecture for real closed fields
Hochschild (Co)Homology of Differential Operator Rings

AbstractWe show that the Hochschild homology of a differential operator k-algebra E=A#fU(g) is the homology of a deformation of the Chevalley–Eilenberg complex of g with coefficients in (M⊗[formula],b*). Moreover, when A is smooth and k is a characteristic zero field, we obtain a type of Hochschild–Kostant–Rosenberg theorem for these algebras. When A=k our complex reduces to the one obtained by C. Kassel (1988, Invent. Math. 91, 221–251) for the homology of filtrated algebras whose associated graded algebras are symmetric algebras. In the last section we give similar results for the cohomology
Lusztig's Canonical Quotient and Generalized Duality

AbstractWe give a new characterization of Lusztig's canonical quotient, a finite group attached to each special nilpotent orbit of a complex semisimple Lie algebra. This group plays an important role in the classification of unipotent representations of finite groups of Lie type. We also define a duality map. To each pair of a nilpotent orbit and a conjugacy class in its fundamental group, the map assigns a nilpotent orbit in the Langlands dual Lie algebra. This map is surjective and is related to a map introduced by Lusztig (and studied by Spaltenstein). When the conjugacy class is trivial, our duality map is just the one studied by Spaltenstein and by Barbasch and Vogan which has an image of the set of special nilpotent orbits
Sheets and Topology of Primitive Spectra for Semisimple Lie Algebras

AbstractLet g be a semisimple Lie algebra. Consider the set X of primitive ideals of the enveloping algebra U(g), given the Jacobson topology. A basic open problem is to describe X as a countable union of algebraic varieties V with strata V as large as possible. Towards this aim the notion of a sheet in X is introduced here in analogy with the notion of sheet in the adjoint orbit space g/G. The sheets in X are classified and given a purely topological description. Moreover Goldie-rank is constant and maximal on a dense open subset of a sheet. For every positive integer n it is shown that there are only finitely many sheets in X with maximal Goldie-rank n. For n=1, which corresponds to completely prime ideals, an extension of the Dixmier-map (from some adjoint orbits to primitive ideals) is introduced with the aim of showing that the sheets with n=1 are homeomorphic to sheets in g/G and hence explicitly described as algebraic varieties. This goal is partly though not fully achieved. The proofs of the above results require a detailed knowledge of Goldie-rank polynomials and even a new difficult result concerning the support of their restrictions to the walls. An extension of a theorem of Soergel on the topology of X is also required, and for this a new approach is given, which is both simpler and more general
The Picard–Vessiot Antiderivative Closure

AbstractF is a differential field of characteristic zero with algebraically closed field of constants C. A Picard–Vessiot antiderivative closure of F is a differential field extension E⊃F which is a union of Picard–Vessiot extensions of F, each obtained by iterated adjunction of antiderivatives, and such that every such Picard–Vessiot extension of F has an isomorphic copy in E. The group G of differential automorphisms of E over F is shown to be prounipotent. When C is the complex numbers and F=C(t) the rational functions in one variable, G is shown to be free prounipotent