pinecone/core-2020-05-10-deduplication huggingface dataset:
On Barnes' Multiple Zeta and Gamma Functions

AbstractWe show how various known results concerning the Barnes multiple zeta and gamma functions can be obtained as specializations of simple features shared by a quite extensive class of functions. The pertinent functions involve Laplace transforms, and their asymptotics is obtained by exploiting this. We also demonstrate how Barnes' multiple zeta and gamma functions fit into a recently developed theory of minimal solutions to first order analytic difference equations. Both of these new approaches to the Barnes functions give rise to novel integral representations
The Noether Bound in Invariant Theory of Finite Groups

AbstractLet R be a commutative ring, V a finitely generated free R-module and G⩽GLR(V) a finite group acting naturally on the graded symmetric algebra A=Sym(V). Let β(AG) denote the minimal number m, such that the ring AG of invariants can be generated by finitely many elements of degree at most m. Furthermore, let H◁G be a normal subgroup such that the index |G:H| is invertible in R. In this paper we prove the inequality β(AG)⩽β(AH)·|G:H|.For H=1 and |G| invertible in R we obtain Noether's bound β(AG)⩽|G|, which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order, |G|!, is invertible in R
Invariant Modules and the Reduction of Nonlinear Partial Differential Equations to Dynamical Systems

AbstractWe completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations and provides a nonlinear counterpart to quasi-exactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented
Strength of Convergence in Duals of C*-Algebras and Nilpotent Lie Groups

AbstractBy using trace formulae, the recent concept of upper multiplicity for an irreducible representation of a C*-algebra is linked to the earlier notion of strength of convergence in the dual of a nilpotent Lie group G. In particular, it is shown that if π∈Ĝ has finite upper multiplicity then this integer is the greatest strength with which a sequence in Ĝ can converge to π. Upper multiplicities are calculated for all irreducible representations of the groups in the threadlike generalization of the Heisenberg group. The values are computed by combining new C*-theoretic results with detailed analysis of the convergence of coadjoint orbits and they show that every positive integer occurs for this class of groups
New Upper Bounds for Finite Bh Sequences

17 páginas.Let Fh(N) be the maximum number of elements that can be selected from the set\ud
{1, ..., N} such that all the sums a1+ ....+ ah , a1<=...<=ah are different. We introduce new combinatorial and analytic ideas to prove new upper bounds for Fh(N).\ud
In particular we prove\ud
F3(N)<= (4/\ud
1+16/(pi+2)4 . N) 1/3 + o(N1/3),\ud
F4(N)<= (8/\ud
1+16/(pi+2)4 .N) 1/4 + o(N1/4).\ud
Besides, our techniques have an independent interest for further research in additive\ud
number theory.Peer reviewe
The Subquasivariety Lattice of a Discriminator Variety

AbstractLet V be a discriminator variety such that the class B={A∈V: A is simple and has no trivial subalgebra} is closed under ultraproducts. This property holds, for example, if V is locally finite or if the language is finite. Let v(V) and q(V) denote the lattice of subvarieties and subquasivarieties of V, respectively. We prove that q(V) is modular iff q(V) is distributive iff v(V) satisfies a certain condition where the case in which the language has a constant symbol is “v(V) is a chain or q(V)=v(V).” We give an isomorphism between q(V) and a lattice constructed in terms of v(V). Via this isomorphism we characterize the completely meet irreducible (prime) elements of q(V) in terms of the completely meet irreducible elements of v(V). We conclude the paper with applications to the varieties of Boolean algebras, relatively complemented distributive lattices, Łukasiewicz algebras, Post algebras, complementary semigroups of rankk (xn≈x)-rings, R5 lattices (P-algebras, B-algebras), and monadic algebras
Hall–Littlewood Vertex Operators and Generalized Kostka Polynomials

AbstractA family of vertex operators that generalizes those given by Jing for the Hall–Littlewood symmetric functions is presented. These operators produce symmetric functions related to the Poincaré polynomials referred to as generalized Kostka polynomials in the same way that Jing's operator produces symmetric functions related to Kostka–Foulkes polynomials. These operators are then used to derive commutation relations and new relations involving the generalized Kostka coefficients. Such relations may be interpreted as identities in the (GL(n)×C)-equivariant K-theory of the nullcone
Graded Multiplicities in the Exterior Algebra

AbstractThis paper deals with the graded multiplicities of the “smallest” irreducible representations of a simple Lie algebra in its exterior algebra. An explicit formula for the graded multiplicity of the adjoint representation in terms of the Weyl group exponents was conjectured by A. Joseph; a proof of this conjecture, based on the properties of Macdonald polynomials, is given in the present paper. The same method allows us to calculate the multiplicity of the simple module with highest weight equal to the short dominant root
Curvature Invariant for Hilbert Modules over Free Semigroup Algebras

AbstractWe introduce a notion of relative curvature (resp. Euler characteristic) for finite rank contractive Hilbert modules over CF+n, the complex free semigroup algebra generated by the free semigroup F+n on n generators. Asymptotic formulas and basic properties for both the curvature and the Euler characteristic are established. In particular, it is shown that the standard relative curvature invariant (resp. Euler characteristic) of a Hilbert module H is a nonnegative number less than or equal to the rank of H, and it depends only on the properties of the completely positive map φT(X)≔∑ni=1TiXT*i, where [T1, …, Tn] is the row contraction of (not necessarily commuting) operators uniquely determined by the CF+n-module structure of H. Moreover, we prove that for every t⩾0 there is a Hilbert module H such that curv(H)=χ(H)=t. The module structure defined by the left creation operators on the full Fock space F2(Hn) on n generators occupies the position of the rank-one free module in the algebraic theory. We obtain a complete description of the closed submodules (resp. quotients) of the free Hilbert module F2(Hn) and calculate their curvature invariant. It is shown that the curvature is a complete invariant for the finite rank submodules of the free Hilbert module F2(Hn)⊗K, where K is a finite dimensional Hilbert space. A noncommutative version of the Gauss–Bonnet–Chern theorem from Riemannian geometry is obtained for graded Hilbert modules over CF+n. In particular, it is proved that the curvature and Euler characteristic coincide for certain classes of pure Hilbert modules. Our investigation is based on noncommutative Poisson transforms, noncommutative dilation theory, and harmonic analysis on Fock spaces
Semi-Infinite Cohomology and Hecke Algebras

AbstractThis paper provides a homological algebraic foundation for generalizations of classical Hecke algebras introduced in (1999, A. Sevostyanov, Comm. Math. Phys.204, 137). These new Hecke algebras are associated to triples of the form (A, A0, ε), where A is an associative algebra over a field k containing subalgebra A0 with augmentation ε:A0→k. These algebras are connected with cohomology of associative algebras in the sense that for every left A-module V and right A-module W the Hecke algebra associated to triple (A, A0, ε) naturally acts in the A0-cohomology and A0-homology spaces of V and W, respectively. We also introduce the semi-infinite cohomology functor for associative algebras and define modifications of Hecke algebras acting in semi-infinite cohomology spaces. We call these algebras semi-infinite Hecke algebras. As an example we realize the W-algebra Wk(g) associated to a complex semisimple Lie algebra g as a semi-infinite Hecke algebra. Using this realization we explicitly calculate the algebra Wk(g) avoiding the bosonization technique used in (1992, B. Feigin and E. Frenkel, Internat. J. Mod. Phys. A7, Suppl. 1A, 197–215)
Conjugacy Relation on Coxeter Elements

AbstractLet (W, S, Γ) be an irreducible finitely presented Coxeter system. The present paper is mainly concerned with a conjugacy relation on Coxeter elements in the case where Γ contains just one circle, in particular when Γ is itself a circle. In the cases where Γ is either a three multiple circle or a circle with three nodes, we show that the ss-equivalence relation on Coxeter elements of W is the same as the W-conjugacy relation. An explicit formula is given for the characteristic polynomial of a Coxeter element in the natural reflection representation of W when Γ is a circle. We also give the answers to some questions raised by Coleman and extend some results of Geck and Pfeiffer concerning conjugacy relation in Coxeter groups
Monoidal Uniqueness of Stable Homotopy Theory

AbstractWe show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, W-spaces, orthogonal spectra, simplicial functors) with symmetric spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence
Theory of Finite Pseudoalgebras

Conformal algebras, recently introduced by Kac, encode an axiomatic
description of the singular part of the operator product expansion in conformal
field theory. The objective of this paper is to develop the theory of
``multi-dimensional'' analogues of conformal algebras. They are defined as Lie
algebras in a certain ``pseudotensor'' category instead of the category of
vector spaces. A pseudotensor category (as introduced by Lambek, and by
Beilinson and Drinfeld) is a category equipped with ``polylinear maps'' and a
way to compose them. This allows for the definition of Lie algebras,
representations, cohomology, etc. An instance of such a category can be
constructed starting from any cocommutative (or more generally,
quasitriangular) Hopf algebra $H$. The Lie algebras in this category are called
Lie $H$-pseudoalgebras.
  The main result of this paper is the classification of all simple and all
semisimple Lie $H$-pseudoalgebras which are finitely generated as $H$-modules.
We also start developing the representation theory of Lie pseudoalgebras; in
particular, we prove analogues of the Lie, Engel, and Cartan-Jacobson Theorems.
We show that the cohomology theory of Lie pseudoalgebras describes extensions
and deformations and is closely related to Gelfand-Fuchs cohomology. Lie
pseudoalgebras are closely related to solutions of the classical Yang-Baxter
equation, to differential Lie algebras (introduced by Ritt), and to Hamiltonian
formalism in the theory of nonlinear evolution equations. As an application of
our results, we derive a classification of simple and semisimple linear Poisson
brackets in any finite number of indeterminates.Comment: 102 pages, 7 figures, AMS late
Endomorphism Algebras of Jacobians

AbstractVery little is known about the existence of curves, and families of curves, whose Jacobians are acted on by large rings of endomorphisms. In this paper, we show the existence of curves X with an injection K↪Hom(Jac(X), Jac(X))⊗ZQ, where K is a subfield of even index at most 10 in a primitive cyclotomic field Q(ζp), or a subfield of index 2 in Q(ζpq) or Q(ζpα). This result generalizes previous work of Brumer, Mestre, and Tautz, Top, and Verberkmoes. Our curves are constructed as branched covers of the projective line, and the endomorphisms arise as quotients of double coset algebras of the Galois groups of these coverings. In certain cases, we show the existence of continuous families of Jacobians admitting the desired endomorphism algebras. At the end, we raise some questions about upper bounds for endomorphism algebras of the type described here and ask about a “non-Galois version of Hurwitz's theorem.
The Picard Group of Equivariant Stable Homotopy Theory

AbstractLet G be a compact Lie group. We describe the Picard group Pic(HoGS) of invertible objects in the stable homotopy category of G-spectra in terms of a suitable class of homotopy representations of G. Combining this with results of tom Dieck and Petrie, which we reprove, we deduce an exact sequence that gives an essentially algebraic description of Pic(HoGS) in terms of the Picard group of the Burnside ring of G. The deduction is based on an embedding of the Picard group of the endomorphism ring of the unit object of any stable homotopy category C in the Picard group of C
Symplectic Manifolds and Isomonodromic Deformations

AbstractWe study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint (generalising the Atiyah–Bott approach). This enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of Jimbo, Miwa and Ueno, thereby putting the existing results for the six Painlevé equations and Schlesinger's equations into a uniform framework
The Space of Composants of an Indecomposable Continuum

AbstractThe family of all composants of an indecomposable continuum is studied. We investigate the equivalence relation induced on an indecomposable continuum by its partition into composants. We show that up to Borel bireducibility such an equivalence relation can be of only two types: E0 and E1, the E0 type being “simple” and the E1 type being “complicated.” As a consequence of this, we show that each indecomposable continuum carries a Borel probability measure which assigns 0 to each composant and 0 or 1 to each Borel set which is the union of a family of composants. In particular, it follows that there is no Borel transversal for the family of all composants. This solves an old problem in the theory of continua. We prove that all hereditarily indecomposable continua are of the complicated type, that is, they fall into the E1 type. We analyze the properties of being of type E1 or of type E0. We show, using effective descriptive set theory, that the first of these properties is analytic and so the second one is coanalytic. We construct examples of continua of both types; in fact, we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic, respectively, hence non-Borel, so they do not admit simple topological characterizations. We also use continua from this family to show that an indecomposable continuum may be of type E1 only because of the behavior of composants on a small subset of the continuum. This, in particular, shows that certain natural approaches to solving Kuratowski's problem on generic ergodicity of the composant equivalence relation will not work. We finish with some open problems
Hopf Algebra Actions on Strongly Separable Extensions of Depth Two

AbstractWe bring together ideas in analysis on Hopf *-algebra actions on II1 subfactors of finite Jones index [9, 24] and algebraic characterizations of Frobenius, Galois and cleft Hopf extensions [3, 13, 14] to prove a non-commutative algebraic analogue of the classical theorem: a finite degree field extension is Galois iff it is separable and normal. Suppose N↪M is a separable Frobenius extension of k-algebras with trivial centralizer CM(N) and split as N-bimodules. Let M1≔End(MN) and M2≔End(M1)M be the endomorphism algebras in the Jones tower N↪M↪M1 ↪M2. We place depth 2 conditions on its second centralizers A≔CM1(N) and B≔CM2(M). We prove that A and B are semisimple Hopf algebras dual to one another, that M1 is a smash product of M and A, and that M is a B-Galois extension of N
On Some Universal Algebras Associated to the Category of Lie Bialgebras

AbstractIn our previous work (math/0008128), we studied the set Quant(K) of all universal quantization functors of Lie bialgebras over a field K of characteristic zero, compatible with the operations of taking duals and doubles. We showed that Quant(K) is canonically isomorphic to a product G0(K)×ш(K), where G0(K) is a universal group and ш(K) is a quotient set of a set B(K) of families of Lie polynomials by the action of a group G(K). We prove here that G0(K) is equal to the multiplicative group 1+ℏK[[ℏ]]. So Quant(K) is “as close as it can be” to ш(K). We also prove that the only universal derivations of Lie bialgebras are multiples of the composition of the bracket with the cobracket. Finally, we prove that the stabilizer of any element of B(K) is reduced to the 1-parameter subgroup generated by the corresponding “square of the antipode.
The Infimum, Supremum, and Geodesic Length of a Braid Conjugacy Class

AbstractAlgorithmic solutions to the conjugacy problem in the braid groups Bn,n =2, 3, 4, … were given in earlier work. This note concerns the computation of two integer class invariants, known as “inf” and “sup.” A key issue in both algorithms is the number m of times one must “cycle” (resp. “decycle”) in order to either increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the class. Our main result is to prove that m is bounded above by ((n2−n)/2)−1 in the situation stated by E. A. Elrifai and H. R. Morton (1994, Quart. J. Math. Oxford45, 479–497) and by n−2 in the situation stated by authors (1998, Adv. Math.139, 322–353). It follows immediately that the computation of inf and sup is polynomial in both word length and braid index, in both algorithms. The integers inf and sup determine (but are not determined by) the shortest geodesic length for elements in a conjugacy class, and so we also obtain a polynomial-time algorithm for computing this length
Littlewood–Paley Theory and the T(1) Theorem with Non-doubling Measures

AbstractLet μ be a Radon measure on Rd which may be non-doubling. The only condition that μ must satisfy is μ(B(x, r))⩽Crn, for all x∈Rd, r>0, and for some fixed 0<n⩽d. In this paper, Littlewood–Paley theory for functions in Lp(μ) is developed. One of the main difficulties to be solved is the construction of “reasonable” approximations of the identity in order to obtain a Calderón type reproducing formula. Moreover, it is shown that the T(1) theorem for n-dimensional Calderón–Zygmund operators, without doubling assumptions, can be proved using the Littlewood–Paley type decomposition that is obtained for functions in L2(μ), as in the classical case of homogeneous spaces.
Hopf Bimodules, Coquasibialgebras, and an Exact Sequence of Kac

AbstractBased on the ideas of Tannaka–Kreın reconstruction, we present a categorical construction that assigns to any cleft Hopf algebra inclusion K⊂H a coquasibialgebra having K* as a Hopf subalgebra. As a special case, the construction gives an intrinsic connection between the bismash product K#Q and the double cross- product Q⋈K* constructed from the same combinatorial data. A cocommutative coquasibialgebra is the same as a cocommutative bialgebra equipped with a Sweedler three-cocycle. Thus our construction assigns to every bicrossproduct (or Hopf algebra extension) of a commutative and a cocommutative factor a corresponding cocommutative double crossproduct equipped with a Sweedler three-cocycle. Based on this observation we use the construction to prove generalizations of Kac's exact sequence for the group of Hopf algebra extensions of a group algebra by a dual group algebra
The Algebra of Quasi-Symmetric Functions Is Free over the Integers

AbstractLet Z denote the Leibniz–Hopf algebra, which also turns up as the Solomon descent algebra and the algebra of noncommutative symmetric functions. As an algebra Z=Z〈Z1, Z2,…〉, the free associative algebra over the integers in countably many indeterminates. The coalgebra structure is given by μ(Zn)=∑ni=0Zi⊗Zn−i, Z0=1. Let M be the graded dual of Z. This is the algebra of quasi-symmetric functions. The Ditters conjecture says that this algebra is a free commutative algebra over the integers. In this paper the Ditters conjecture is proved
A Global Theory of Algebras of Generalized Functions

AbstractWe present a geometric approach to defining an algebra G(M) (the Colombeau algebra) of generalized functions on a smooth manifold M containing the space D′(M) of distributions on M. Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of G(M). G(M) is a differential algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of D′(M) into G(M) that renders C∞(M) a faithful subalgebra of G(M). Finally, it is shown that this embedding commutes with Lie derivatives. Thus G(M ) retains all the distinguishing properties of the local theory in a global context
On Harmonic Elements for Semi-simple Lie Algebras

AbstractLet g be a semi-simple complex Lie algebra and g=n−⊕h⊕n its triangular decomposition. Let U(g), resp. Uq(g), be its enveloping algebra, resp. its quantized enveloping algebra. This article gives a quantum approach to the combinatorics of (classical) harmonic elements and Kostant's generalized exponents for g. A quantum analogue of the space of harmonic elements has been given by A. Joseph and G. Letzter (1994, Amer. J. Math.116, 127–177). On the one hand, we give specialization results concerning harmonic elements, central elements of Uq(g), and the decomposition of Joseph and Letzter (cited above). For g=sln+1, we describe the specialization of quantum harmonic space in the N-filtered algebra U(sln+1) as the materialization of a theorem of A. Lascoux et al. (1995, Lett. Math. Phys.35, 359–374). This enables us to study a Joseph–Letzter decomposition in the algebra U(sln+1). On the other hand, we prove that highest weight harmonic elements can be calculated in terms of the dual of Lusztig's canonical basis. In the simply laced case, we parametrize a basis of n-invariants of minimal primitive quotients by the set C0 of integral points of a convex cone
The Validity of the Multifractal Formalism: Results and Examples

AbstractBy obtaining a new sufficient condition for a valid multifractal formalism, we improve in this paper a result developed by L. Olsen (1995, Adv. Math.116, 82–196). In particular, we describe a large class of measures satisfying the multifractal formalism and for which the construction of Gibbs measures is not possible. Some of these measures are not unidimensional but have a nontrivial multifractal spectrum, giving a negative answer to a question asked by S. J. Taylor (1995, J. Fourier Anal. Appl., special issue). We also describe a necessary condition of validity for the formalism which is very close to the sufficient one. This necessary condition allows us to describe a measure μ for which the multifractal packing dimension function Bμ(q) is a nontrivial real analytic function but the multifractal formalism is nowhere satisfied. This example gives also a solution to a problem posed by Taylor (cited above)