pinecone/core-2020-05-10-deduplication huggingface dataset:
Simple Lie Algebras of Small Characteristic II. Exceptional Roots

AbstractLetLbe a finite dimensional simple Lie algebra of absolute toral rank 2 over an algebraically closed field of characteristicp>3 andTa 2-dimensional torus in the semisimplep-envelope ofL. Suppose thatLis not isomorphic to a Melikian algebra. It is proved in this paper that, for every root α∈Γ(L,T), the subalgebraK′(α) generated by ∑i∈F*pKiα(whereKiα={x∈Liα∣α([x,L−iα])=0}) acts triangulably onL. In particular, this implies that, in the terminology of R. E. Block and R. L. Wilson (1988,J. Algebra114, 115–259), all roots of Γ(L,T) are nonexceptional
The Affineq-Schur Algebra

AbstractWe introduce an analogue of theq-Schur algebra associated to Coxeter systems of typeÁn−1. We give two constructions of this algebra. The first construction realizes the algebra as a certain endomorphism algebra arising from an affine Hecke algebra of typeÁr−1, wheren≥r. This generalizes the originalq-Schur algebra as defined by Dipper and James, and the new algebra contains the ordinaryq-Schur algebra and the affine Hecke algebra as subalgebras. Using this approach we can prove a double centralizer property. The second construction realizes the affineq-Schur algebra as the faithful quotient of the action of a quantum group on the tensor power of a certain module, analogous to the construction of the ordinaryq-Schur algebra as a quotient ofU(gln)
An Asphericity Conjecture and Kaplansky Problem on Zero Divisors

AbstractSuppose a group representationH=〈A‖R〉 is aspherical,x∉A,W(A∪x) is a word in alphabet (A∪x)±1with nonzero sum of exponents onx, and the groupHnaturally embeds inG=〈A∪x‖R∪W(A∪x)〉. It is conjectured that the presentationG=〈A∪x‖R∪W(A∪x)〉 is aspherical if and only ifGis torsion free. It is proven that if this conecture is false andG=〈A∪x‖R∪W(A∪x)〉 is a counterexample, then the integral group ring Z(G) of torsion free groupGwill contain zero divisors. Some special cases when this conjecture holds are also indicated
Addendum to “Construction of (16,6)-Configurations from a Group of Order 16” by Gonzalez-Dorrego

AbstractRecently, Gonzalez-Dorrego has given a list of 9 different ways of constructing a nondegenerate (16,6) configuration from a group of order 16 (up to automorphisms and translations of the group). In this paper, using her algorithm with a minor correction, we obtain 27 such constructions, completing the list. It turns out that the Kummer configuration, which appears geometrically as 16 nodes and 16 tropes of a Jacobian Kummer surface in P3, can be obtained in 24 different ways from 12 different groups. The configuration of type (A) arises in a unique way from the abelian group C8×C2, and the one of type (B) is obtained in two ways from the groups 16Γ2b and Q8×C2
Noetherian Domains Inside a Homomorphic Image of a Completion

AbstractOver the past 60 years, important examples of Noetherian domains have been constructed using power series, homomorphic images, and intersections. In these constructions it is often crucial that the resulting domain is computable as a directed union. In this article we analyse this construction and show that the Noetherian property for the associated directed union is equivalent to a flatness condition. LetRbe a Notherian integral domain with fraction fieldL. Letxbe a nonzero nonunit ofRand letR* denote the (x)-adic completion ofR. SupposeIis an ideal ofR* with the propertyp∩R=(0) for eachp∈Ass(R*/I).Theorem. The embeddingR↪(R*/I)xis flat if and only ifA≔L∩(R*/I) is Noetherian and is realizable as a localization of a subring ofRx=R[1/x]. We present several examples where this flatness condition holds; one of these examples is a local Noetherian domain that is not universally catenary, but has geometrically regular formal fibers
Weak Cayley Table Groups

AbstractLetG,Hbe groups. We study bijectionsf:G→Hwhich preserve conjugacy classes and satisfy: for allg,g′∈Gthe elementsf(gg′) andf(g)f(g′) are conjugate. These are related to 2-characteristics of groups, objects originally studied by Frobenius and more recently by K. W. Johnsonet. al.IfG=H, then any automorphism or anti-automorphism satisfies these conditions. We call thesetrivial.We show, among other things, that any suchf:G→GforGbeing the symmetric groupSn,n>0, dihedral groupDn,n∈N∪{∞}, or the free groupFn,n≠3, is trivial
On Certain Closure Operators Defined by Families of Semiring Morphisms

AbstractGiven a continuous semiring A and a collection H of semiring morphisms mapping the elements of A into finite matrices with entries in A we define H-closed semirings. These are fully rationally closed semirings that are closed under the following operation: each morphism in H maps an element of the H-closed semiring on a finite matrix whose entries are again in this H-closed semiring.H-closed semirings coincide under certain conditions with abstract families of elements. If they contain only algebraic elements over some A′, A′⊆A, then they are characterized by Rat(A′)-algebraic systems of a specific form. The results are then applied to formal power series and formal languages. In particular, H-closed semirings are set in relation to abstract families of elements, power series, and languages. The results are strong “normal forms” for abstract families of power series and languages
Projective Representations and Relative Semisimplicity

AbstractLet R be a local (commutative) ring and let p be a prime not invertible in R. Let G be a finite group of order divisible by p. It is well known that the group ring RG admits nonprojective lattices (e.g., R itself with the trivial action). For any element α∈H2(G,R*) one can form the twisted group ring RαG. The “twisting problem” asks whether there exists a class α s.t. the corresponding twisted group ring admits only projective lattices. For fields of characteristic p, the answer is in E. Aljadeff and D. J. S. Robinson [J. Pure Appl. Algebra94 (1994), 1–15]. Here we answer this question for rings of the form Zps, s≥2. The main tools are the classification of modular representation of the Klein 4 group over Z2 and a Chouinard-like theorem [E. Aljadeff and Y. Ginosar, J. Algebra179 (1996), 599–606] for twisted group rings
On Mockor's Question

AbstractFor certain classes of Prüfer domains A, we study the completion Á,T of A with respect to the supremum topology T=sup{Tw|w∈Ω}, where Ω is the family of nontrivial valuations on the quotient field which are nonnegative on A and Tw is a topology induced by a valuation w∈Ω. It is shown that the concepts “SFT Prüfer domain” and “generalized Dedekind domain” are the same. We show that if E is the ring of entire functions, then Ê,T is a Bezout ring which is not a T̂-Prüfer ring, and if A is an SFT Prüfer domain, then Á,T is a Prüfer ring under a certain condition. We also show that under the same conditions as above, Á,T is a T̂-Prüfer ring if and only if the number of independent valuation overrings of A is finite. In particular, if A is a Dedekind domain (resp., h-local Prüfer domain), then Á,T is a T̂-Prüfer ring if and only if A has only finitely many prime ideals (resp., maximal ideals). These provide an answer to Mockor's question
Dual Pairs and Tensor Categories of Modules over Lie Algebrasgl̂∞and W1+∞

AbstractWe introduce a tensor category O+(resp. O−) of certain modules ofgl̂∞with nonnegative (resp. nonpositive) integral central charges with the usual tensor product. We also introduce a tensor category (Of,⊙) consisting of certain modules overGL(N) for allN. We show that the tensor categories O±and Ofare semisimple abelian and all equivalent to each other. We give a formula to decompose a tensor product of two modules and obtain a reciprocity law in each of these categories. We also introduce a tensor category Owof certain modules over W1+∞with nonnegative integral central charges. We show that Owis semisimple abelian and give an explicit formula to decompose a tensor product of two modules in Ow
Cyclic Extensions of Free Pro-p Groups

AbstractLet G be a pro-p group, which is a finite cyclic p-extension of a pro-p product of a free pro-p group F. We show that G can be described as the free pro-p product of the normalizers of a suitable collection of its cyclic subgroups of order p and some free pro-p group. From this we deduce that G can be realized as the fundamental group of a profinite connected graph of finite cyclic groups of bounded order. This result is thus the pro-p analogue of a special case of the well known characterization of virtually free groups as fundamental groups of graphs of groups, owing to Karrass, Pietrowski, Solitar, Cohen, and Scott. We conclude the paper by presenting a counterexample to a possible pro-p analogue of this result for noncyclic extensions of free pro-p groups, which we use in turn to construct a counterexample to pro-p versions of the Kurosh subgroup theorem
Global and Krull Dimensions of Quantum Weyl Algebras

AbstractGiven a Hecke symmetry R, A. Giaquinto and J. Zhang defined a natural quantization An(R) of the nth Weyl algebra An based on R and studied many ring theoretic properties of rings A2(Ja,b) (arising from the “Jordan” Hecke symmetry) and An(q,pi,j) (arising from the standard multiparameter Hecke symmetry). Here we compute the global and Krull dimensions in the cases that were left open; namely, we show that over any field k of characteristic zero, gldim(A2(Ja,b))=Kdim(A2(Ja,b))=3 for any a, b∈k with a≠b, and gldim(An(±1,pi,j))=Kdim(An(±1,pi,j))=n
Maximal Subgroups of GL1(D)

AbstractLet D be a division algebra of degree m over its center F. Herstein has shown that any finite normal subgroup of D*≔GL1(D) is central. Here, as a generalization of this result, it is shown that any finitely generated normal subgroup of D* is central. This also solves a problem raised by Akbari and Mahdavi-Hezavehi (Proc. Amer. Math. Soc., to appear) for finite-dimensional division algebras. The structure of maximal multiplicative subgroups of an arbitrary division ring D is then investigated. Given a maximal subgroup M of D* whose center is algebraic over F, it is proved that if M satisfies a multilinear polynomial identity over F, then [D:F]<∞
Compatible Subtriples of Jordan-Triples

AbstractThe purpose of this paper is to extend the notion of compatibility of tripotents in an anisotropic Jordan*-triple A to that of subtriples of A. For a subtriple B of A define the kernel Ker(B) and the annihilator B⊥ of B to be the subsets {a∈A: {BaB}={0}} and {a∈A: {BaA}={0}}, respectively. The Peirce spaces Bi, i=0,1,2, of B are defined in terms of the kernel and the annihilator by B0=B⊥, B1=Ker(B)∩Ker(B⊥), and B2=B. A pair (B,C) of subtriples of A is said to be compatible if A coincides with ⊕i,j=0,1,2Bi∩Cj. The subtriple B is said to be self-compatible if it is compatible with itself. A self-compatible subtriple B is complemented in that B⊕Ker(B) coincides with A, which implies that B is an inner ideal in A. It follows that a pair (u,v) of tripotents in A is compatible if and only if the ranges imP2(u),imP2(v) of the Peirce projections P2(u) and P2(v) form a compatible pair of subtriples. Moreover, for any tripotent u, imP2(u) is a self-compatible subtriple of A. For the special case of a JBW*-triple A it is shown that a subtriple I is compatible with every self-compatible subtriple J if and only if I is a complemented ideal in A. These results are then applied to abelian JBW*-triples, W*-algebras, and Hilbert spaces
On a Theorem of Shintani

AbstractLet χ be an irreducible character of Gd=GLn(Fqd) invariant under the automorphism φ of Gd induced by the field automorphism Fqd→Fqd, x↦xq, and let e be a divisor of d. By a theorem of Shintani, there exists an extension χ̃e of χ to Gd⋊〈φe〉 whose Shintani descent to Ge is, up to a sign ε, an irreducible character of Ge. It is shown in this paper that χ̃e may always be chosen such that ε=1. With this particular choice, χ̃e is the restriction of χ̃1. Our methods rely on the work of Digne and Michel on Deligne–Lusztig theory for nonconnected reductive groups
Decompositions of Difference Sets

AbstractWe characterize those symmetric designs with a Singer group G which admit a quasi-regular G-invariant partition into strongly induced symmetric subdesigns. In terms of the corresponding difference sets, the set associated with the larger design can be decomposed into a difference set describing the small designs and a suitable relative difference set. This generalizes the decomposition of the classical design with the complements of hyperplanes in PG(m−1,q) as blocks into sub-designs arising from PG(d−1,q) whenever d divides m. Parametrically, these geometrical examples provide the only known examples of the situation we are studying. But there are many nonisomorphic examples with the same parameters, namely the complements of the classical GMW designs and some generalizations. We also discuss the possibilities for obtaining new difference sets in this way and point out a connection to the recent constructions of Ionin for symmetric designs
Some Thin Pro-p-Groups

AbstractA pro-p-group is called thin if its lattice of closed normal subgroups contains no more than p+1 pairwise incomparable elements. An investigation of certain (infinite) thin pro-p-groups was made in [A. Caranti et al. Quart. J. Math. Oxford47 (1996), 279–296], by studying the graded Lie algebras associated with their lower central series. In particular, the main result of [A. Caranti et al. Quart. J. Math. Oxford47 (1996), 279–296] asserts that those graded Lie algebras are, in certain cases, uniquely determined by their quotients of low dimension. In the present paper we construct pro-p-groups which have, associated with their lower central series, the graded Lie algebras of [A. Caranti et al. Quart. J. Math. Oxford47 (1996), 279–296]
Galois Objects over Generalized Drinfeld Doubles, with an Application to uq(sl2)

AbstractRalf Günther has determined all the cleft extensions over the finite quotient Hopf algebra uq(sl2) of the quantized universal enveloping algebra of sl2 at a root of unity [R. Günther, Ph.D. thesis, Universität München, 1999]. His techniques (applications of the diamond lemma) are similar to those used by A. Masuoka [Comm. Algebra22 (1994), 4537–4559] for the two-generator Taft algebras. In the present paper we give another proof of a special case of Günther's classification, namely, the case of (cleft) Galois extensions of the base field. The idea is that uq(sl2) is the quotient of the Drinfeld double of a Taft algebra by a normal Hopf subalgebra. We use techniques that allow us to calculate all Galois objects of such a composed Hopf algebra
Maps Between Representations of Clans

AbstractWe give a description of the homomorphisms between representations of a clan (see Crawley-Boevey, J. London Math. Soc.40 (1989), 9–30), more or less as it was done by Krause (J. Algebra137 (1991), 186–194) for string-algebras. As a consequence of our findings we can establish the irreducibility of all maps which are expected to be so. This allows us, after defining a natural exact structure on the category of representations, to calculate in many situations Auslander–Reiten sequences like in the string-algebra case
Direct Sums of Self-Small Mixed Groups

AbstractThe class G of self-small mixed abelian groups of finite rank has recently been the focus of a number of investigations. It is known that, up to quasi-isomorphism, G is dual to the category of locally free torsion-free abelian groups of finite rank. We utilize the reduced mixed groups in G as building blocks of a more extensive class ∑G, the smallest class containing the reduced groups in G that is closed under taking infinite direct sums and direct summands. Our central result is the determination of a complete set of isomorphism invariants for the groups in ∑G. We supplement this broad classification theorem with an investigation of the fine structure of completely decomposable groups in ∑G
Groups Generated by Symplectic Transvections over Local Rings

AbstractLet R be a commutative local ring with the unique maximal ideal A. Let V be a free module of rank n over R. And let Spn(V) be the symplectic group on V with an alternating bilinear form f: V×V→R. We study the generation of a subgroup TR(M) of Spn(V), where M={x∈V|f(x,V)=R} and TR(M) is defined as the subgroup generated by all symplectic transvections with axis x⊥ for x∈M.Our main goal is to get a nice necessary and sufficient condition for any subset N⊆M satisfying TR(N)=TR(M), where TR(N) is the group generated by all symplectic transvections with axis x⊥ for x∈N. In particular, if f is nonsingular we have TR(M)=Spn(V), and therefore our necessary and sufficient condition gives us a criterion for an arbitrarily given N⊆M satisfying TR(N)=Spn(V).Also we shall investigate the TR(N) orbit of each x∈M, find some small sets of generators of TR(M) consisting of transvections in TR(N), and as a result solve the author's conjecture in “Generators and Relations in Groups and Geometries” (A. Barlotti et al., Eds.), pp. 47–67, Proc. NATO ASI (C)
Further Reflections on Thompson's Conjecture

AbstractLet G be a finite group and let N(G)={n∈N|G has a conjugacy class C, such that |C|=n}. Professor J. G. Thompson has conjectured that: If G is a finite group with Z(G)=1 and M a non-abelian simple group satisfying N(G)=N(M), then G≅M.We have proved previously that: If M is a sporadic simple group or a simple group having its prime graph with at least three prime graph components, then Thompson's conjecture is correct. In this paper, we shall prove:Main Theorem. Let G be a finite group with Z(G)=1 and M=G2(q) or G2(2)′, where q≥2, such that N(G)=N(M). Then G≅M
The Space of Triangles, Vanishing Theorems, and Combinatorics

AbstractWe consider compactifications of (Pn)3\∪Δij, the space of triples of distinct points in projective space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson; and a third is the triangle space of Schubert and Semple.We compute the sections of line bundles on these spaces, and show that they are equal as GL(n) representations to the generalized Schur modules associated to “bad” generalized Young diagrams with three rows (Borel–Weil theorem). On the one hand, this yields Weyl-type character and dimension formulas for the Schur modules; on the other, a combinatorial picture of the space of sections. Cohomology vanishing theorems play a key role in our analysis
On the Orbit Method and the Homomorphism of Harish-Chandra

AbstractLet g be a complex semisimple Lie algebra with adjoint group G. Let U(g) be the enveloping algebra of g. Following an idea of J. Dixmier, (Sur la méthode des orbites, in “Proceedings de la conférence: Non commutative Harmonic analysis, Marseille–Luminy, 1978,” Lecture Notes in Mathematics, Vol. 728, Springer-Verlag, New York/Berlin) we construct a bijection from the set of regular coadjoint orbits onto the set of minimal primitive ideals in U(g), without using the notion of “polarization” (see J. Dixmier, Algèbres Enveloppantes, Gauthier-Villars, Paris, 1974). This is a partial answer to the problem posed by J. Dixmier in the former reference
Quasitriangular and Differential Structures on Bicrossproduct Hopf Algebras

AbstractLet X=GM be a finite group factorisation. It is shown that the quantum double D(H) of the associated bicrossproduct Hopf algebra H=kM▷◀k(G) is itself a bicrossproduct kX▷◀k(Y) associated to a group YX, where Y=G×Mop. This provides a class of bicrossproduct Hopf algebras which are quasitriangular. We also construct a subgroup YθXθ associated to every order-reversing automorphism θ of X. The corresponding Hopf algebra kXθ▷◀k(Yθ) has the same coalgebra as H. Using related results, we classify the first order bicovariant differential calculi on H in terms of orbits in a certain quotient space of X
On the Irreducible Representations of the Specializations of the Generic Hecke Algebra of Type F4

AbstractA complete determination of the irreducible modules of specialized Hecke algebras of type F4, with respect to specializations with equal parameters, was obtained by M. Geck and K. Lux (1991, Manuscripta Math.70, 285–306). A similar determination for specializations with v=u2 and v=u4 was obtained by K. Bremke (1994, Manuscripta Math.83, 331–346). In this paper, we determine the irreducible modules for all remaining specializations other than those into fields of characteristic 2 or 3, obtaining en route decompositions of the generic irreducible modules under such specializations. We find that the decomposition matrices may be expressed in lower uni-triangular form in all these cases
Stable Norms on Complex Numbers and Quaternions

AbstractIn this paper, we study stability properties of norms on the complex numbers and on the quaternions. Our main findings are that these norms are stable if and only if they majorize the modulus function and that not all stable norms are strongly stable. Part of the paper is devoted to the standard matrix representations of the above number systems, where we show that norms on the corresponding matrix algebras are stable if and only if they are spectrally dominant. We conclude by considering proper seminorms, observing that none are stable on the complex numbers or on the quaternions
Quasialgebra Structure of the Octonions

AbstractWe show that the octonions are a twisting of the group algebra of Z2×Z2×Z2 in the quasitensor category of representations of a quasi-Hopf algebra associated to a group 3-cocycle. In particular, we show that they are quasialgebras associative up to a 3-cocycle isomorphism. We show that one may make general constructions for quasialgebras exactly along the lines of the associative theory, including quasilinear algebra, representation theory, and an automorphism quasi-Hopf algebra. We study the algebraic properties of quasialgebras of the type which includes the octonions. Further examples include the higher 2n-onion Cayley algebras and examples associated to Hadamard matrices
On Thompson's Theorem

AbstractLet p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129–134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of p′-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters χ1 and χ2 of distinct degrees such that p does not divide χ1(1)χ2(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of p′-degree are proved. Some open questions are posed
The Minimal Basis for the Centre of an Iwahori–Hecke Algebra

AbstractThis paper arose out of an attempt to generalize the Q[q,q−1]-basis for the centre of an Iwahori–Hecke algebra Hq found by Jones to Z[q,q−1] and to other types. Considering the Iwahori–Hecke algebra H over a subring Z[ξ] of Z[q1/2,q−1/2], where ξ=q1/2−q−1/2, we use a new and natural definition of positivity on H to describe the “minimal” Z[ξ]-basis for Z(H) in terms of a partial order on the positive part of H. The main result is to show that this minimal basis is the set of “primitive” minimal elements of the positive part of the centre, for any Weyl group. In addition, the primitive minimal positive central elements can be characterized as exactly those positive central elements which specialize (on setting ξ=0, the equivalent of setting q=1 in Z[q1/2,q−1/2]) to the sum of elements in a conjugacy class, and which apart from the shortest elements from that conjugacy class sum contain no other terms corresponding to shortest elements of any conjugacy class. A constructive algorithm is provided for obtaining the minimal basis. We use the results of Jones to achieve the result in type An without the need for character theory, and give the result for all Weyl groups using the character theoretical results of Geck and Rouquier. Finally we discuss the non-crystallographic cases and give some explicit examples