pinecone/core-2020-05-10-deduplication huggingface dataset:
Real Even Symmetric Ternary Forms

AbstractLet Sen,m denote the set of all real symmetric forms of degree m=2d. Let PSen,m and ΣSen,m denote the cones of positive semidefinite (psd) and sum of squares (sos) elements of Sen,m, respectively. For m=2 or 4, these cones coincide. For m=6, they do not, and were analyzed in Even Symmetric Sextics, by M. D. Choi, T. Y. Lam, and B. Reznick (1987, Math. Z.195, pp. 559–580).We present an easily checked, necessary and sufficient condition for an even symmetric n-ary octic to be in PSen,8 and for an even symmetric ternary decic to be in PSe3,10; we also show that there is no corresponding condition for even symmetric ternary forms of degree greater than 10. We proceed to discuss the extremal elements of the cones PSe3,8 and PSe3,10. This leads to the question: how many of these extremal forms have sos representations? We prove that PSe3,8=ΣSe3,8 and demonstrate that PSe3,10\ΣSe3,10 is nonempty, providing many new examples of psd forms which are not sos.We give a graphic representation of ternary forms which also indicates whether or not an element of Se3,8 or Se3,10 is psd. We interpret elements of PSe3,m as inequalities; in particular, we give all symmetric polynomial inequalities of degree ≤5 satisfied by the sides of a triangle
Rank One Lattice Type Vertex Operator Algebras and Their Automorphism Groups II. E-Series

AbstractLet L be the A1 root lattice and let G be a finite subgroup of Au(V), where V=VL is the associated lattice VOA (in this case, Aut(V)≅PSL(2,C)). The fixed point sub-VOA, VG, was studied previously by the authors, who found a set of generators and determined the automorphism group when G is cyclic (from the “A-series”) or dihedral (from the “D-series”). In the present article, we obtain analogous results for the remaining possibilities for G, that it belong to the “E-series”: G≅Alt4, Alt5, or Sym4. For such L and G, the above VGL may be rational VOAs
Clifford Correspondence for Finite Dimensional Hopf Algebras

AbstractLet B#σH be a crossed product algebra over an algebraically closed field, with H a finite dimensional Hopf algebra. We give an explicit equivalence between the category of finite dimensional B#σH-modules whose restriction to B is a direct sum of copies of a stable irreducible B-module, and the category of modules for a twisted product of H with the field. This describes all finite dimensional irreducible B#σH-modules containing a stable irreducible B-submodule, and thus generalizes the classical stable Clifford correspondence for groups. In case H is cocommutative, we extend this correspondence to the nonstable case
Hecke Algebras and Semisimplicity of Monoid Algebras

AbstractWe begin by constructing Hecke algebras for arbitrary finite regular monoids M. We then show that the semisimplicity of the complex monoid algebra CM is equivalent to the semisimplicity of the associated Hecke algebras and a condition on induced group characters. We apply these results to finite Lie-type monoids M. We show that the monoid algebra FM over a field F is semisimple if and only if the characteristic of F does not divide the order of the unit group G. This is accomplished by developing formulas for the unities of CJ, J a J-class of M. The unity is explicitly given when G is a simply connected Chevalley group and J is associated with a Borel subgroup of G
Noetherian Semigroup Algebras

AbstractIt is shown that a semigroup S is finitely generated whenever the semigroup algebra K[S] is right Noetherian and has finite Gelfand–Kirillov dimension or S is a Malcev nilpotent semigroup. If, furthermore, S is a submonoid of a finitely generated nilpotent-by-finite group G, then K[S] is right Noetherian if and only if K[S] is left Noetherian, or equivalently S satisfies the ascending chain condition on right (left) ideals. The latter condition is completely described in terms of the structure of S: in case G is a nilpotent group the quotient group H of S contains a normal subgroup F such that H/F is abelian-by-finite and F⊆S. Finally, also prime Goldie contracted semigroup algebras are described
A Munn Type Representation for a Class of E-Semiadequate Semigroups

AbstractMunn's construction of a fundamental inverse semigroup TE from a semilattice E provides an important tool in the study of inverse semigroups. We present here a semigroup FE that plays for a class of E-semiadequate semigroups the role that TE plays for inverse semigroups. Every inverse semigroup with semilattice of idempotents E is E-semiadequate. There are however many interesting E-semiadequate semigroups that are not inverse, we consider various such examples arising from Schützenberger products
Rees Matrix Covers and Semidirect Products of Regular Semigroups

AbstractIn a recent paper (Trans. Amer. Math. Soc. 349 (1997), 4265–4310), P.G. Trotter and the author introduced a “regular” semidirect product U∗V of e-varieties U and V. Among several specific situations investigated there was the case V=RZ, the e-variety of right zero semigroups. Applying a covering theorem of McAlister, it was shown there that in several important cases (for instance for the e-variety of inverse semigroups), U∗RZ is precisely the e-variety LU of “locally U” semigroups.The main result of the current paper characterizes membership of a regular semigroup S in U∗RZ in a number of ways; one in terms of an associated category SE and another in terms of S regularly dividing a regular Rees matrix semigroup over a member of U. The categorical condition leads directly to a characterization of the equality U∗RZ=LU in terms of a graphical condition on U, slightly weaker than “e-locality.” Among consequences of known results on e-locality, the conjecture CR∗RZ=LCR (with CR denoting the e-variety of completely regular semigroups), is therefore verified. The connection with matrix semigroups then leads to a range of Rees matrix covering theorems that, while slightly weaker than McAlister's, apply to a broader range of examples. K. Auinger and P. G. Trotter (Pseudovarieties, regular semigroups and semidirect products, J. London Math. Soc. (2)58 (1998), 284–296) have used our results to describe the pseudovarieties generated by several important classes of (finite) regular semigroups
Congruence Modularity Implies the Arguesian Law for Single Algebras with a Difference Term

AbstractRecently, a generalization of commutator theory has been developed for algebraic systems belonging to a congruence modular variety. This general commutator theory is used here both to provide a very simple proof of a classical result by R. Freese and B. Jónsson and to solve a problem raised by S. Tschantz. Some other known results are improved or are given a simple proof. Moreover, we pursue the goal of extending commutator theory to a somewhat larger setting; thus we give a more general version of the result by Freese and Jónsson (partially solving a problem we raised in P. Lipparini, 1994, Trans. Amer. Math. Soc.346, 177–202)
Generalized Reduced Enveloping Algebras for Restricted Lie Algebras

AbstractWe observe that any finite-dimensional indecomposable module for a restricted Lie algebra over an algebraically closed field is a module for a finite-dimensional quotient of the universal enveloping algebra. These algebras form a two-parameter family which generalizes the notion of a reduced enveloping algebra. We identify each such algebra as a reduced enveloping algebra for an associated Lie algebra and use this to compute support varieties and obtain some representation-theoretic consequences
On the Depth of the Invariants of the Symmetric Power Representations of SL2(Fp)

AbstractWe study the depth of the ring of invariants of SL2(Fp) acting on the nth symmetric power of the natural two-dimensional representation for n<p. These symmetric power representations are the irreducible representations of SL2(Fp) over Fp. We prove that, when the greatest common divisor of p−1 and n is less than or equal to 2, the depth of the ring of invariants is 3. We also prove that the depth is 3 for n=3, p≠7 and n=4, p≠5. However, for n=3, p=7 the depth is 4 and for n=4, p=5 the depth is 5. In these two exceptional cases, the ring of invariants is Cohen–Macaulay
On Duflo's Theorem That Minimal Primitive Ideals Are Centrally Generated

AbstractLet g be a complex semisimple Lie algebra. Duflo's theorem states that each minimal primitive ideal of the enveloping algebra U(g) is generated by its intersection with the center. We prove the following generalization. For the “relative enveloping algebra” A=U(g)/I relative to a parabolic subalgebra q of g, where I denotes the annihilator of the induced module U(g)⊗U([q,q])C, let Z denote the center of A, let Z̃ denote its normalization, and let Ã=AZ̃ be the slight extension of A obtained by normalizing the center. We present a theorem that states that under certain conditions, which are always satisfied if g=sln, we have that each minimal primitive ideal of Ã is generated by its intersection with the center Z̃. Duflo's theorem is the special case where q is a Borel subalgebra (then I=0, Z̃=Z, and Ã=A=U(g)). Note that the normalization of the center is really necessary for the theorem: The corresponding statement for A and Z instead of Ã and Z̃ fails for g=sl3
Etaleness and Normality

AbstractLet A be an excellent normal semilocal ring with strict Henselization Ahs. If A⊆D⊆Ahs is a normal semilocal intermediate ring semilocally dominated by Ahs, then D is a direct limit of semilocal-etale extensions of A. If A is an excellent normal local domain then the integral closure of A in a finite separable field extension L of the quotient field Q(A) is etale over A if and only if there is a Q(A) morphism L→Q(Ahs) and Q(Ahs) contains a normal closure of L. We apply these results to the normalizations of an arbitrary excellent reduced local ring A and its strict Henselization Ahs to obtain a characterization for local-etaleness of a local intermediate ring A⊆D⊆Ahs whose total quotient ring L is finitely generated over Q(A). Specifically, such an intermediate ring D is local-etale over A if and only if the normalization D̃ of D is semilocal and has the proper residual field structure relative to Ã and to D. We generalize this result to characterize local-etaleness of arbitrary injective reduced local morphisms A→D, where A is excellent, finding that such a morphism is local-etale if and only if the morphism Ã→D̃ is semilocal-etale and there is an isomorphism D̃≅D⊗AÃ
Ultraproducts of Z with an Application to Many-Valued Logics

AbstractUp to categorical equivalence, abelian lattice-ordered groups with strong unit coincide with Chang's MV-algebras—the Lindenbaum algebras of the infinite-valued Łukasiewicz calculus. While the property of being a strong unit is not definable even in first-order logic, MV-algebras form an equational class. On the other hand, the addition operation and the translation invariant lattice order of a lattice-ordered group are more amenable than the truncated addition operation of an MV-algebra. In this paper MV-algebraic and group-theoretical techniques are combined to classify and axiomatize all universal classes generated by an infinite totally ordered MV-algebra A such that the quotient of A by its unique maximal ideal is finite. The number of elements of this quotient, and that of the largest finite subalgebra of A turns out to be a complete classifier. The main tool for our results is given by order preserving embeddings of totally ordered groups G into ultrapowers of the additive group of integers, that also preserve the nondivisibility properties of prescribed elements of G
Doubly Z-Graded Lie Algebras Containing a Virasoro Algebra

AbstractIn this paper we study Z×Z-graded Lie algebras A=⊕i,j∈ZAi,j over a characteristic 0 field F with dimAi,j≤1 for each i and j, where Z denotes the integers. We shall assume that A satisfies the following three properties:&#x02022;(I)′=⊕≃Vir, the centerless Virasoro algebra;&#x02022;(II)[,]=0, and ad and ad act faithfully on the subspace⊕;&#x02022;(III)dim=dim=1, and is generated by and ′.The main result of the paper is the classification of these algebras. More precisely, A is isomorphic to a simple Block algebra L(a) which has basis {ei,j∣(i,j)∈Z×Z\{(−1/a,0),(−2/a,0)}} and subject to the bracket[formula]for some suitable constant a∈F\{0,±1,±2}, or is isomorphic to a one- or two-dimensional central extension of L(a) with a∈F\{0,±1,±2}, or A is isomorphic a one-dimensional derivation extension of L(±1) and L(±2), or a two-dimensional extension of L(±1) by a center element and a derivation
Cross Product Bialgebras, I

AbstractThe subject of this article is bialgebra factorizations or cross product bialgebras without cocycles. We establish a theory characterizing cross product bialgebras universally in terms of projections and injections. Especially all known types of biproduct, double cross product, and bicross product bialgebras can be described by this theory. Furthermore the theory provides new families of (cocycle-free) cross product bialgebras. Besides the universal characterization we find an equivalent (co)modular description of certain types of cross product bialgebras in terms of so-called Hopf data. With the help of Hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras. We are working in the general setting of braided monoidal categories, which allows us to apply our results in particular to the braided category of Hopf bimodules over a Hopf algebra. Majid's double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category. This resembles the case of the Drinfel'd double which can be constructed as a twist of a specific cross product
Group Gradings on Full Matrix Rings

AbstractWe study G-gradings of the matrix ring Mn(k), k a field, and give a complete description of the gradings where all the elements ei,j are homogeneous, called good gradings. Among these, we determine the ones that are strong gradings or crossed products. If G is a finite cyclic group and k contains a primitive |G|th root of 1, we show how all G-gradings of Mn(k) can be produced. In particular we give a precise description of all C2-gradings of M2(k) and show that for algebraically closed k, any such grading is isomorphic to one of the two good gradings
Basic Representations for Classical Affine Lie Algebras

AbstractPresented here is a construction of certain bases of basic representations for classical affine Lie algebras. The starting point is a Z-grading g=g−1+g0+g1 of a classical Lie algebra g and the corresponding decomposition g̃=g̃−1+g̃0+g̃1 of the affine Lie algebra g̃. By using a generalization of the Frenkel–Kac vertex operator formula for A(1)1 one can construct a spanning set of the basic g̃-module in terms of monomials in basis elements of g̃1 and certain group element e. These monomials satisfy certain combinatorial Rogers–Ramanujan type difference conditions arising from the vertex operator formula, and the main result is that these differences coincide with the energy function of a perfect crystal corresponding to the g0-module g1. The linear independence of the constructed spanning set of the basic g̃-module is proved by using a crystal base character formula for standard modules due to S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki
Modules, Comodules, and Cotensor Products over Frobenius Algebras

AbstractWe characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left Ae-modules. We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than comodules.We prove that the cotensor product M□N of a right A-module M and a left A-module N is isomorphic to the vector space of homomorphisms from a particular left Ae-module D to N⊗M, viewed as a left Ae-module. Some properties of D are described. Finally, we show that when A is a symmetric algebra, the cotensor product M□N and its derived functors are given by the Hochschild cohomology over A of N⊗M
Global Dimension 4 Extensions of Artin–Schelter Regular Algebras

AbstractThis paper classifies central and normal extensions from global dimension 3 Artin–Schelter regular algebras to global dimension 4 Artin–Schelter regular algebras. Let A be an AS regular algebra of global dimension 3, and let D be an extension of A by a normal graded element z, i.e., D/〈z〉=A. The algebra A falls under a classification due to Artin, Schelter, Tate, and Van den Bergh [Artin and Schelter, Adv. Math.66 (1987), pp. 171–216; Artin et al., in “The Grothendieck Festschrift,” Vol. 1, pp. 33–85, Birkhäuser, Basel, 1990; Artin et al., Invent Math.106 (1991), pp. 335–388] and is either quadratic or cubic. The quadratic algebras A are Koszul, and this fact was used by Le Bruyn, Smith, and Van den Bergh [Le Bruyn et al., Math. Z.222 (1996), 171–212] to classify the four-dimensional AS regular algebras D when A is quadratic and deg(z)=1. Alternative methods are needed when A is cubic or deg(z)>1. We prove in all such cases that the regularity of D and z is equivalent to the regularity of z in low degree (e.g., 2 or 3) and this is equivalent to easily verifiable matrix conditions on the relations for D
Correspondences Between Valued Division Algebras and Graded Division Algebras

AbstractIf D is a tame central division algebra over a Henselian valued field F, then the valuation on D yields an associated graded ring GD which is a graded division ring and is also central and graded simple over GF. After proving some properties of graded central simple algebras over a graded field (including a cohomological characterization of its graded Brauer group), it is proved that the map [D]↦[GD]g yields an index-preserving isomorphism from the tame part of the Brauer group of F to the graded Brauer group of GF. This isomorphism is shown to be functorial with respect to field extensions and corestrictions, and using this it is shown that there is a correspondence between F-subalgebras of D (with center tame over F) and graded GF-subalgebras of GD
Buchsbaum–Rim Sheaves and Their Multiple Sections

AbstractThis paper begins by introducing and characterizing Buchsbaum–Rim sheaves on Z=ProjR, where R is a graded Gorenstein K-algebra. They are reflexive sheaves arising as the sheafification of kernels of sufficiently general maps between free R-modules. Then we study multiple sections of a Buchsbaum–Rim sheaf Bϕ, i.e, we consider morphisms ψ: P→Bϕ of sheaves on Z dropping rank in the expected codimension, where H0*(Z,P) is a free R-module. The main purpose of this paper is to study properties of schemes associated to the degeneracy locus S of ψ. It turns out that S is often not equidimensional. Let X denote the top-dimensional part of S. In this paper we measure the “difference” between X and S, compute their cohomology modules and describe ring-theoretic properties of their coordinate rings. Moreover, we produce graded free resolutions of X (and S) which are in general minimal. Among the applications we show how one can embed a subscheme into an arithmetically Gorenstein subscheme of the same dimension and prove that zero-loci of sections of the dual of a null correlation bundle are arithmetically Buchsbaum
Toric Ideals Generated by Quadratic Binomials

AbstractA combinatorial criterion for the toric ideal arising from a finite graph to be generated by quadratic binomials is studied. Such a criterion guarantees that every Koszul algebra generated by squarefree quadratic monomials is normal. We present an example of a normal non-Koszul squarefree semigroup ring whose toric ideal is generated by quadratic binomials as well as an example of a non-normal Koszul squarefree semigroup ring whose toric ideal possesses no quadratic Gröbner basis. In addition, all the affine semigroup rings which are generated by squarefree quadratic monomials and which have 2-linear resolutions will be classified. Moreover, it is shown that the toric ideal of a normal affine semigroup ring generated by quadratic monomials is generated by quadratic binomials if its underlying polytope is simple
Bipartite Graphs Whose Edge Algebras Are Complete Intersections

AbstractLet R be a monomial subalgebra of k[x1,…,xN] generated by square free monomials of degree two. This paper addresses the following question: when is R a complete intersection? For such a k-algebra we can associate a graph G whose vertices are x1,…,xN and whose edges are {(xi,xj)|xixj∈R}. Conversely, for any graph G with vertices {x1,…,xN} we define the edge algebra associated with G as the subalgebra of k[x1,…,xN] generated by the monomials {xixj|(xi,xj) is an edge of G}. We denote this monomial algebra by k[G]. This paper describes all bipartite graphs whose edge algebras are complete intersections
Presentations for Subgroups of Monoids

AbstractThe main result of this paper gives a presentation for an arbitrary subgroup of a monoid defined by a presentation. It is a modification of the well known Reidemeister–Schreier theorem for groups. Some consequences of this result are explored. It is proved that a regular monoid with finitely many left and right ideals is finitely presented if and only if all its maximal subgroups are finitely presented. An inverse monoid with finitely many left and right ideals is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid. An example of a finitely presented monoid with a finitely generated but not finitely presented group of units is exhibited
Standard Young Tableaux in the Weyl Group Setting

AbstractA generalization of the notion of standard Young tableau has recently arisen from work on the representation theory of affine Hecke algebras. In the generalized setting, a standard tableau is defined to be any element of a finite Weyl group whose inversion set satisfies a certain pair of intersection conditions. In this paper, we prove that the set of generalized standard tableaux of fixed shape, when nonempty, is a certain interval in the weak ordering. In addition, we establish a nonemptiness criterion for the set of standard tableaux of prescribed shape. These results are obtained for shapes that satisfy an integrality condition
The Projective Geometry of the Gale Transform

AbstractThe Gale transform, an involution on sets of points in projective space, appears in a multitude of guises and in subjects as diverse as optimization, coding theory, theta functions, and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture (see Eisenbud and Popescu, 1999, Invent. Math.136, 419–449). In this paper we reexamine the Gale transform in the light of modern algebraic geometry. We give a more general definition in the context of finite (locally) Gorenstein subschemes. We put in modern form a number of the more remarkable examples discovered in the past, and we add new constructions and connections to other areas of algebraic geometry. We generalize Goppa's theorem in coding theory and we give new applications to Castelnuovo theory. We also give references to classical and modern sources
A Quantum Deformation of Invariants of Higher Binary Forms

AbstractWe use the theory of the quantum group Uq(gl(2,R)) to develop a quantum theory of invariants and show a decomposition of invariants into a Gordan–Capelli series. Higher binary forms are introduced on the basis of braided algebras. We define quantised invariants and give basic examples. We show that the symbolic method of Clebsch and Gordan works also in the quantised case. We discuss the deformed discriminant of the quadratic and the cubic form, the deformed invariants I1, I2 of the quartic form and further invariants without a classical analog