pinecone/core-2020-05-10-deduplication huggingface dataset:
Peirce Inner Ideals in Jordan*-Triples

AbstractA subspaceJof an anisotropic Jordan*-tripleAis said to be aninner idealif the subspace {JAJ} is contained inJ. An inner idealJinAis said to becomplementedifAis equal to the sum ofJand thekernelKer(J) ofJ, defined to be the subspace ofAconsisting of elementsainAfor which {JaJ} is equal to {0}. TheannihilatorJ⊥of an inner idealJinAis the inner ideal consisting of elementsainAsuch that {JaA} is equal to {0}. When bothJandJ⊥are complemented,Acan be decomposed into the direct sum ofJ, Ker(J)∩Ker(J⊥) andJ⊥. Modulo six of the generalized Peirce relations this decomposition is a grading ofAof Peirce type. Since an inner ideal in a JBW*-triple is complemented if and only if it is weak*-closed, the result described above applies to all weak*-closed inner idealsJin a JBW*-tripleA. Furthermore, it can be shown that in this case all except five of the generalized Peirce relations hold, and an example is given of a weak*-closed inner ideal in a JBW*-triple for which all five fail to hold, thereby showing that the result is the best possible. It is also shown that the condition that a weak*-closed inner ideal in a JBW*-tripleAleads to a grading ofAwhich is of Peirce type is equivalent to several other conditions, all of a topological, rather than algebraic, nature. These results are applied to W*-algebras, spin triples, and the bi-Cayley triple
Rational Functions with a Polynomial Iterate

AbstractLet φ(z)∈C(z) be a rational function of degreeD≥2. A well known theorem in dynamical systems says that if some iterate φn(z)=φ∘φ∘···∘φ(z) is a polynomial, then already φ2(z) is a polynomial. More generally, this is true for φ(z)∈K(z) for any fieldKprovided that φ is separable (i.e., φ′(z)≠0). However, it need not be true for inseparable φ. In this note we give a complete description, in all characteristics and with no separability assumptions, of the rational functions which have a polynomial iterate
On Generalised Free Products of Conjugacy Separable Groups

AbstractIn this paper we prove that a free product of conjugacy separable groupsAandB, amalgamating a cyclic subgroup, is conjugacy separable ifAandBare subgroup separable, cyclic conjugacy separable, 2-free, and residuallyp-finite, for all prime numbersp. The following result is an example of the applications we obtain as consequences of our main theorem. LetAandBeach be a free product of surface groups, amalgamating a maximal cyclic subgroup. Then a free product ofAandB, amalgamating a cyclic subgroup, is conjugacy separable
Finite-Rank Free Metabelian Subgroups of Finitely Presented Metabelian Groups

AbstractA sufficient condition is obtained for the residual torsion-free nilpotence of certain finitely presented metabelian groups that arise from a matrix representation developed by Magnus (1939,Ann. of Math.40, 764–768) for metabelian groups. Using this condition and a construction due to Baumslag (1973,J. Austral. Math. Soc.16, 98–110), we prove that a free metabelian group of finite rank can be embedded in a finitely presented metabelian group that is also residually torsion-free nilpotent
Universally Coefficient Domains and Their Relation to the Cancellation Problem for Rings

AbstractWe call an integral domainDauniversally coefficient domainif for any domainRwithD⊆R[x1,…,xn] thenD⊆R. It is true that every universally coefficient domain is strongly invariant but not conversely. We show that ifKis a field of characteristic zero andD=S−1K[x,y] thenDis a universally coefficient domain if and only if[formula], wherepis such that[formula]for some[formula]and where[formula]is the algebraic closure ofK. We then prove that any localization ofK[x,y], whereKis algebraically closed of characteristic zero, is a universally coefficient domain if and only if it is strongly invariant, giving necessary and sufficient conditions for localizations ofK[x,y] to be strongly invariant. This, in turn, shows that every localization ofK[x,y],Kalgebraically closed of characteristic zero, is invariant and strongly invariant if it is not a polynomial ring. We also discuss generalizations tonvariables and overrings, and we give examples of classes of polynomialsfsuch thatK[x1,…,xn,1/f] is a universally coefficient domain
Discrete Quantum Groups

AbstractLetGbe any discrete group. Consider the algebraAof all complex functions with finite support onGwith pointwise operations. The multiplication onGinduces a comultiplication Δ onAby (Δf)(p,q)=f(pq) wheneverf∈Aandp,q∈G. IfGis finite, one can identify the algebra of complex functions onG×GwithA⊗Aso that Δ:A→A⊗A. Then (A,Δ) is a Hopf algebra. IfGis infinite, we still have Δ(f)(g⊗1) and Δ(f)(1⊗g) inA⊗Afor allfandg. In this case (A,Δ) is a multiplier Hopf algebra. In fact, it is a multiplier Hopf *-algebra whenAis given the natural involution defined by[formula]for allf∈Aandp∈G. In this paper we call a multiplier Hopf *-algebra (A,Δ)adiscrete quantum group if the underlying *-algebraAis a direct sum of full matrix algebras. We study these discrete quantum groups and we give a simple proof of the existence and uniqueness of a left and a right invariant Haar measure
Segre Product of Artin–Schelter Regular Algebras of Dimension 2 and Embeddings in Quantum P3's

AbstractThe Segre embedding of P1×P1as a smooth quadricQin P3corresponds to the surjection of the four-dimensional polynomial ring onto the Segre productSof two copies of the homogeneous coordinate ring of P1. We study Segre products of noncommutative algebras. If in particularAandBare two copies of a quantum P1thenS=⊕i(Ai⊗kBi) is a twisted homogeneous coordinate ring of the quadricQ. The main result of this paper is the classification of all embeddings of the Segre product of two quantum planes into so-called quantum P3's. These are (the Proj of) Artin–Schelter regular algebrasRof global dimension four with the Hilbert series of a commutative polynomial ring and which map ontoS. IfRis not a twist of a polynomial ring, then the point scheme ofReither is the union of the quadricQwith a line or is only the quadricQ. In the first case,Ris a central extension of a three-dimensional Artin–Schelter regular algebra and a twist of an algebra mapping onto the (commutative) homogeneous coordinate ring ofQ; in the second case, such an algebraRis the first known example of a four-dimensional Artin–Schelter regular algebra which is not determined by its point scheme
Balanced Butler Groups

AbstractThe class of Butler groups, pure subgroups of finite rank completely decomposable groups, has been studied extensively by abelian group theorists in recent years. Classification by numerical invariants up to quasi-isomorphism and even isomorphism has been achieved for special subclasses. Here we highlight a new class in which to extend and expand classification results, the balanced Butler groups or K(1)-groups. These are the pure balanced subgroups of finite rank completely decomposable groups. A strictly decreasing chain of classes of Butler groups, introduced by Kravchenko, is obtained by defining the K(n)-groups (n≥2) to be those balanced subgroups of a completely decomposable group for which the quotient is a K(n−1)-group. We establish an internal characterization of K(n)-groups, give a method for constructing examples, and derive decomposition results
A Generalization of Borcherds Algebra and Denominator Formula

AbstractWe study generalized Lie superalgebras (an extension of Kac's generalized Lie superalgebras) and show that they are transformed forms ofL-graded Lie superalgebras for some abelian groupL. We then introduce a generalized Lie superalgebra version of the generalized Kac–Moody algebra (Borcherds algebra). Since it is a transformed Borcherds superalgebra, it has several properties similar to those of Borcherds superalgebra. For example, it is defined by similar relations and has a similar denominator formula and character formulas
Koszul Modules and Gorenstein Algebras

AbstractI first define Koszul modules, which are a generalization to arbitrary rank of complete intersections. After a study of some of their properties, it is proved that Gorenstein algebras of codimension one or two over a local or graded CM ring are Koszul modules, thus generalizing a well known statement for rank one modules. The general techniques used to describe Koszul modules are then used to obtain a structure theorem for Gorenstein algebras in codimension one and two, over a local or graded CM ring
Finite Group Factorizations and Braiding

AbstractWe compute the quantum double, braiding, and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebraHassociated to the factorization of a finite groupXinto two subgroups. The representations of the quantum double are described by a notion of bicrossed bimodules, generalising the cross modules of Whitehead. We also show that basis-preserving self-duality structures for the bicrossproduct Hopf algebras are in one–one correspondence with factor-reversing group isomorphisms. The example Z6Z6is given in detail. We show further that the quantum doubleD(H) is the twisting ofD(X) by a nontrivial quantum cocycle
A Classification of Multiplicity Free Actions

AbstractThe action of a complex algebraic groupGon an affine varietyVis said to be multiplicity free if each irreducible representation ofGoccurs at most once in the ring C[V] of regular functions onV. This paper concerns the classical setting whereVis a vector space andGacts linearly onV. The irreducible linear multiplicity free actions have been classified by V. Kac. The focus here is on nonirreducible linear multiplicity free actions. We exhibit many such actions which do not decompose as direct products of irreducible actions. Our main results provide a classification for all linear multiplicity free actions on a vector space
Binary Codes and Vertex Operator (Super)Algebras

AbstractWe study a vertex operator algebra whose Virasoro element is a sum of pairwise orthogonal rational conformal vectors with central charge 1/2. The most important example is the moonshine moduleV♮. In particular, we construct a series of vertex operator algebras whose full automorphism groups are finite. Namely, we construct vertex operator algebrasMD=∑∞i=0(MD)ifrom even linear binary codeD⊆Zn2and prove that if the minimum weight ofDis greater than 2 then (MD)1=0 and the full automorphism group ofMDis finite. From the viewpoint of finite group theory, the construction of a vertex operator algebra has one advantage. We can expect not only automorphisms ofD, but also another one. Indeed, ifDcontains a [8, 4, 4] Hamming subcodeC, thenCdefines a nontrivial involutive automorphism ofMD, which is not induced from the automorphism group ofD. In particular, we have a finite group extension of Aut(D)(Zn2/D⊥)
Gorenstein Flat Covers of Modules over Gorenstein Rings

AbstractA left and right Noetherian ringRis called Gorenstein if bothRRandRRhave finite injective dimensions. These rings were studied by Bass for the commutative case and Iwanaga for the noncommutative case. In this paper, we define Gorenstein flat modules over a Gorenstein ring. These modules are a generalization of flat modules. After discussing the properties of modules over a Gorenstein ring, we prove that every module over a Gorenstein ring has a Gorenstein flat cover or a minimal right approximation by a Gorenstein flat module in Auslander and Reiten's terminology
Strong Morita Equivalence and a Generalisation of the Rees Theorem

AbstractIn our earlier work we associated a natural category to a semigroup with local units and proved semigroup analogues of the celebrated Morita theorems for rings. In this article we use the notion of a Morita context to define an equivalence relation on a far wider class of semigroups. We give a semigroup analogue of Morita I and show that Morita equivalent semigroups can be constructed with ease. Our construction is based on the classical Rees theorem and a generalisation of this theorem by Hotzel. We then use properties of equivalent categories to deduce properties of this construction. Finally, we give a new generalisation of the Rees theorem and applications of this generalisation
Leibniz Representations of Lie Algebras

AbstractALeibniz representationof the Lie algebra g is a vector spaceMequipped with two actions (left and right) [–,–]: g⊗M→Mand [–,–]:M⊗g→Mwhich satisfy the relations[formula]when one of the variables is inMand the two others are in g. In this paper we show that the categoryL(g) of finite dimensional Leibniz representations of a finite dimensional semi-simple Lie algebra is not semi-simple, but thatL(g) has global dimension 2. We give an explicit description of the extensions of simple objects and we obtain the description of the quiver ofL(g) (in the sense of Gabriel). It turns out thatL(g) is tame only for g=sl2. We give the complete list of indecomposable objects in that case
Subgroups of HolQ8as Galois Groups

AbstractIn this paper, we consider nonsplit Galois theoretical embedding problems with cyclic kernel of prime orderp, in the case where the ground field has characteristic ≠p. It is shown that such an embedding problem can always be reduced to another embedding problem, in which the ground field contains the primitivepth roots of unity, and the group extension is central. The reduction is effective, in the sense that a solution to the reduced embedding problem induces a solution to the original embedding problem and that all solutions to the original embedding problem are induced in this way from solutions to the reduced embedding problem. The simplest case of this reduction is then used to give criteria for the realisability of four subgroups of the holomorph HolQ8, whereQ8is the quaternion group of order 8, including the holomorph itself
The Bernstein Problem for Type (n−2,2)

AbstractIn a recent paper we solved the Bernstein problem in the nonregular nonexceptional case for the type (3,2). The aim of this paper it is to describe explicitly all simplicial stochastic nonexceptional nonregular nonnuclear Bernstein algebras of type (n−2,2). According to the Lyubich conjecture every nuclear Bernstein algebra with stochastic realization is regular. Consequently, this paper would completely solve the Bernstein problem for the type (n−2,2) if the Lyubich conjecture is proved to be true
Hall Polynomials for Symplectic Groups, II

AbstractThis is the second paper on Hall polynomials for symplectic groups. The definition is analogous to that of Hall polynomials for general linear groups. In both papers we compute the number of all totally isotropic subspacesWof type μ in a vector space with symplectic geometryVof type λ denotedgλμ(see Section 0.1 for definitions). Let the dimensions ofVandWbe 2mandm, respectively. We represent the basis ofVof type λ=rd11rd22…rdsswith a diagram consisting ofsblocks. The entries of this diagram represent elements of a basis ofVconsisting ofmhyperbolic pairs (see Definition 0.1.1). Because of the existence of S–S basis (see 0.2 for definition) we always have an obvious totally isotropic subspace. Our method is to “deform” this initial subspace into all others by adding and discarding basis elements from the current basis. We must keep track of the type of subspace we thus obtain and the geometric structure (that is, the positions of hyperbolic pairs in the diagram) of the entire space, ensuring isotropy at each step. In our first paper [10] we obtain the result for the case when λ=rd, corresponding to a diagram with one block. In this case we can choose our basis so that each type ofWcorresponds to one geometric structure ofVonly. This makes our task and notation simpler and we obtain a closed formula forgλμ(see Theorem 0.5.5). Unfortunately it is not so in the case of λ=rd11rd22…rdss,s≠1, where two subspaces may have the same type but induce different geometric structures inV(see the example in Section 1). This case is studied below. The Hall polynomials for general linear groups have been computed by T. Klein [4] (later, I. G. Macdonald [5] and F. M. Maley [6]). Our method differs from that of either author substantially. Many of the results of [10] are used in this paper. A quick review of those results is given in the first section of this paper
Left-Invariant Affine Structures on Reductive Lie Groups

AbstractWe describe left-invariant affine structures (that is, left-invariant flat torsion-free affine connections ∇) on reductive linear Lie groupsG. They correspond bijectively to LSA-structures on the Lie algebra g ofG. Here LSA stands for left-symmetric algebra. If g has trivial or one-dimensional center z then the affine representation α=λ⊕1 of g, induced by any LSA-structure gλon g isradiant, i.e., the radiance obstructioncα∈H1(g,gλ) vanishes. If dimz=1 we prove that g=s⊕z, where s is split simple, admits LSA-structures if and only if s is of typeAl, that is, g=gln.Here we have the associative LSA-structure given by ordinary matrix multiplication corresponding to the bi-invariant affine structure on GL(n), which was believed to be essentially the only possible LSA-structure on gln. We exhibit interesting LSA-structures different from the associative one. They arise as certain deformations of the matrix algebra. Then we classify all LSA-structures on glnusing a result of Baues. Forn=2 we compute all structures explicitly over the complex numbers
Transitive Permutation Groups of Minimal Movement

AbstractFor a permutation groupGon a setS, themovementofGis defined as the maximum cardinality of subsetsTofSfor which there exists an elementx∈Gsuch thatTis disjoint from its translateTx(that is, when such subsets have bounded cardinality). It was shown by the second author that, ifGhas bounded movementmand ifGhas no fixed points inS, thenSis finite, and |S| is bounded above by a function ofm. In particular, ifGis transitive, then |S|≤3m. This paper completes the proof of a conjecture of Gardiner and Praeger that the only transitive groups on a set of size 3mwhich have movementmare transitive permutation groups of exponent 3 (whenmis a power of 3), the symmetric groupS3in its natural representation on a set of three points, and the alternating groupsA4andA5, in their transitive representations on six points
Reflective Integral Lattices

AbstractA latticeLwith a positive definite quadratic form is called reflective if the unique largest subgroup generated by reflections of the orthogonal group O(L) has no fixed vector. Equivalently, the “root system” R(L) has maximal rank. The root system of a lattice is defined in Section 1; the roots are not necessarily of length 1 or 2. In Section 2, the structure of reflective lattices is worked out. They are described and classified by pairs (R,L), where R is a “scaled root system” and the “code” L is a subgroup of the “reduced discriminant group” T(R). The crucial point is that T(R) only depends on the combinatorial equivalence class of the root system R. In Section 3, we give a precise description of the full root system of a reflective lattice if one starts with a sub-root-system of combinatorial typenA1ormA2. In Section 4, our techniques are applied to a complete and explicit description of all reflective lattices in dimensions ≤6
On a Special Case of Carrell's Conjecture

AbstractIn this paper we study a conjecture of J. B. Carrell on the rationality of a compact Kähler manifold admitting a holomorphic vector field with isolated zeroes. The conjecture, formulated in terms of C+-actions, says that if C+is acting on a nonsingular projective varietyXwith exactly one fixed point, thenXis rational. We prove this is true under the additional assumption that in the tangent space at the fixed point there is only one fixed direction. To prove this result we embedXas a fibre in a familyX* equipped with a suitable C*-action. Then we use a C*-equivariant projection onto the tangent space toX* at the sink
Nondegenerate Jordan Algebras Satisfying Local Goldie Conditions

AbstractA structure theorem is given for nondegenerate Jordan algebrasJsatisfying the ascending chain condition on annihilators of a single element and such thatJcontains no infinite direct sum of inner deals inside the inner ideal generated by each elementx∈J. As a consequence of this theorem and of the main results of a previous paper by the authors (J. Algebra174(1995), 1024–1048), it is obtained that such Jordan algebrasJare precisely the local orders in nondegenerate Jordan algebras satisfying dcc on principal inner ideals and without non-artinian quadratic ideals, which extends to local orders the Zel'manov theorem for Goldie Jordan algebras
Rings of Generalized and Stable Invariants of Pseudoreflections and Pseudoreflection Groups

AbstractLet ϱ:G↪GL(n,F) be a representation of a finite groupGover the field F and F[V] the space of polynomial functions onV=Fn. We associate toGan idealJ∞(G)⊂F[V] called the ideal of stable invariants ofG(or more precisely ϱ). If S⊂GL(n,F) is a set of pseudoreflections we associate to S an idealI(S)⊂F[V] called the ideal of generalized invariants of S. WhenGis a pseudoreflection group we investigateI(S) for various choices of S⊂Gand the relation betweenJ∞(G) andI(S). To a representation ϱ:G↪GL(n,F) of a finite group, respectively to a set S⊂GL(n,F) of pseudoreflections, we also associate a ring[formula], respectively[formula]. We show that[formula]is always a polynomial algebra over F, and whenever ϱ(G) is generated by semisimple pseudoreflections S that[formula]
Virasoro Vertex Operator Algebras, the (Nonmeromorphic) Operator Product Expansion and the Tensor Product Theory

AbstractA theory of tensor products of modules for a vertex operator algebra is being developed by Lepowsky and the author. To use this theory, one first has to verify that the vertex operator algebra satisfies certain conditions. We show in the present paper that for any vertex operator algebra containing a vertex operator subalgebra isomorphic to a tensor product algebra of minimal Virasoro vertex operator algebras (vertex operator algebras associated to minimal models), the tensor product theory can be applied. In particular, intertwining operators for such a vertex operator algebra satisfy the (nonmeromorphic) commutativity (locality) and the (nonmeromorphic) associativity (operator product expansion). Combined with a result announced by Lepowsky and the author in 1994, the results of the present paper also show that the category of modules for such a vertex operator algebra has a natural structure of a braided tensor category. In particular, for any pairp,qof relatively prime positive integers larger than 1, the category of minimal modules of central charge 1−6[(p−q)2/pq] for the Virasoro algebra has a natural structure of a braided tensor category
Monoids of Intervals of Ordered Abelian Groups

AbstractFor any partially ordered abelian groupG, we relate the structure of the ordered monoid Λ(G) ofintervalsofG(i.e., nonempty, upward directed lower subsets ofG), to various properties ofG, as for example interpolation properties, or topological properties of the state space whenGhas an order-unit. This allows us to solve a problem by K. R. Goodearl by proving that even in most natural cases, multiplier groups of dimension groups often fail to be interpolation groups. Furthermore, the study of monoids of intervals in the totally ordered case yields a characterization of Hahn powers of the real line by afirst-order sentenceon the positive interval monoid
Kostant's Conjecture and the Geometry of Cartan Subalgebras in the Rank Two and Exceptional Cases

AbstractThis paper deals with certain aspects of a conjecture made by B. Kostant in 1983 relating the Coxeter number to the occurrence of the simple finite groupsL(2,q) in simple complex Lie groups. In particular, we examine how the conjecture gives rise to certain presentations of the Lie algebra as a sum of Cartan subalgebras for the rank two and exceptional cases. The presentations studied are those with invariant properties with respect to Kostant and Kac elements. The techniques also connect certain presentations of theq+1 dimensional representations ofL(2,q) to certain properties of Jacobi sums