pinecone/core-2020-05-10-deduplication huggingface dataset:
Types of Reductive Monoids

AbstractLet M be a reductive monoid with a reductive unit group G. Clearly there is a natural G×G action on M. The orbits are the J-classes (in the sense of semigroup theory) and form a finite lattice. The general problem of finding the lattice remains open. In this paper we study a new class of reductive monoids constructed by multilined closure. We obtain a general theorem to determine the lattices of these monoids. We find that the (J,σ)-irreducible monoids of Suzuki type and Ree type belong to this new class. Using the general theorem we then list all the lattices and type maps of the (J,σ)-irreducible monoids of Suzuki type and Ree type
The Lattice of J-Classes of (J,σ)-Irreducible Monoids, II

AbstractA reductive monoid is an algebraic monoid with a reductive unit group. Generally, we have the question of finding the orbits of the unit group of a reductive monoid acting on both sides of the monoid. Putcha and Renner give a recipe to determine the orbits for J-irreducible monoids according to the Dynkin diagrams. We obtain a similar recipe for the question to (J,σ)-irreducible monoids (not J-irreducible) of type D2n. However, there is no similar answer for types An (n≥4) and E26. The fixed points of any (J,σ)-irreducible monoid under σ is a finite reductive monoid. We obtain that any such finite reductive monoid is J-irreducible. Then we find the orbits of these monoids under the two sided action of their unit groups
Homological Algebra Modulo a Regular Sequence with Special Attention to Codimension Two

AbstractLet M be a finite module over a ring R obtained from a commutative ring Q by factoring out an ideal generated by a regular sequence. The homological properties M over R and over Q are intimately related. Their links are analyzed here from the point of view of differential graded homological algebra over a Koszul complex that resolves R over Q. One outcome of this approach is a transparent derivation of some central results of the theory. Another is a new insight into codimension two phenomena, yielding an explicit finitistic construction of the generally infinite minimal R-free resolution of M. It leads to theorems on the structure and classification of finite modules over codimension two local complete intersections that are exact counterparts of Eisenbud's results for modules over hypersurfaces
Conormal Geometry of Maximal Minors

AbstractLet A be a Noetherian local domain, N be a finitely generated torsion-free module, and M a proper submodule that is generically equal to N. Let A[N] be an arbitrary graded overdomain of A generated as an A-algebra by N placed in degree 1. Let A[M] be the subalgebra generated by M. Set C≔Proj(A[M]) and r≔dimC. Form the (closed) subset W of Spec(A) of primes p where A[N]p is not a finitely generated module over A[M]p, and denote the preimage of W in C by E. We prove this: (1) dimE=r−1 if either (a) Nis free and A[N] is the symmetric algebra, or (b) W is nonempty and A is universally catenary, and (2) E is equidimensional if (a) holds and A is universally catenary. Our proof was inspired by some recent work of Gaffney and Massey, which we sketch; they proved (2) when A is the ring of germs of a complex-analytic variety, and applied it to improve a characterization of Thom's Af-condition in equisingularity theory. From (1), we recover, with new proofs, the usual height inequality for maximal minors and an extension of it obtained by the authors in 1992. From the latter, we recover the authors' generalization to modules of Böger's criterion for integral dependence of ideals. Finally, we introduce an application of (1), being made by Thorup, to the geometry of the dual variety of a projective variety, and use it to obtain an interesting example where the conclusion of (1) fails and A[N] is a finitely generated module over A[M]
A New Lattice Construction: The Box Product

AbstractIn a recent paper, the authors have proved that for lattices A and B with zero, the isomorphismConcA⊗B≅ConcA⊗ConcBholds, provided that the tensor product satisfies a very natural condition (of being capped) implying that A⊗B is a lattice. In general, A⊗B is not a lattice; for instance, we proved that M3⊗F(3) is not a lattice.In this paper, we introduce a new lattice construction, the box product for arbitrary lattices. The tensor product construction for complete lattices introduced by G. N. Raney in 1960 and by R. Wille in 1985 and the tensor product construction of A. Fraser in 1978 for semilattices bear some formal resemblance to the new construction.For lattices A and B, while their tensor product A⊗B (as semilattices) is not always a lattice, the box product, A□B, is always a lattice. Furthermore, the box product and some of its ideals behave like an improved tensor product. For example, if A and B are lattices with unit, then the isomorphismConcA□B≅ConcA⊗ConcBholds. There are analogous results for lattices A and B with zero and for a bounded lattice A and an arbitrary lattice B.A join semilattice S with zero is called {0}-representable, if there exists a lattice L with zero such that ConcL≅S. The above isomorphism results yield the following consequence: The tensor product of two {0}-representable semilattices is {0}-representable
Weak Hopf Algebras I. Integral Theory and C-Structure

AbstractWe give an introduction to the theory of weak Hopf algebras proposed as a coassociative alternative of weak quasi-Hopf algebras. We follow an axiomatic approach keeping as close as possible to the “classical” theory of Hopf algebras. The emphasis is put on the new structure related to the presence of canonical subalgebras AL and AR in any weak Hopf algebra A that play the role of non-commutative numbers in many respects. A theory of integrals is developed in which we show how the algebraic properties of A, such as the Frobenius property, or semisimplicity, or innerness of the square of the antipode, are related to the existence of non-degenerate, normalized, or Haar integrals. In case of C*-weak Hopf algebras we prove the existence of a unique Haar measure h∈A and of a canonical grouplike element g∈A implementing the square of the antipode and factorizing into left and right elements g=gLg−1R, gL∈AL, gR∈AR. Further discussion of the C*-case will be presented in Part II
Isomorphisms of Integral Group Rings of Infinite Groups

AbstractThis paper deals with the isomorphism problem for integral group rings of infinite groups. In the first part we answer a question of Mazur by giving conditions for the isomorphism problem to be true for integral group rings of groups that are a direct product of a finite group and a finitely generated free abelian group. It is also shown that the isomorphism problem for infinite groups is strongly related to the normalizer conjecture. Next we show that the automorphism conjecture holds for infinite finitely generated abelian groups G if and only if ZG has only trivial units. In the second part we partially answer a problem of Sehgal. It is shown that the class of a finitely generated nilpotent group G is determined by its integral group ring provided G has only odd torsion. When G has nilpotency class two then the finitely generated restriction is not needed. This, together with a result of Ritter and Sehgal, settles the isomorphism problem for finitely generated nilpotency class two groups. A link is pointed out between this problem and the dimension subgroup problem
The G-Exponent of a Pseudovariety of Semigroups

AbstractAsh's proof of the Pointlike Conjecture provides an algorithm for calculating the group-pointlike subsets of a finite semigroup S. We denote by PG(S) the subsemigroup of P(S), the elements of which are the group-pointlike subsets of S. This paper is concerned with some properties of the operator PG, which assigns to the pseudovariety V the pseudovariety PGV generated by the semigroups PG(S) with S in V. As PGV is a subpseudovariety of PV, some of them come from properties of the power operator. We show that P3GV=S for any pseudovariety of semigroups V that contains a semigroup such that the subsemigroup generated by its idempotents is non-permutative
On Two Presentations of the Affine Weyl Groups of Classical Types

AbstractThe main result of the paper is to get the transition formulae between the alcove form and the permutation form of w∈Wa, where Wa is an affine Weyl group of classical type. On the other hand, we get a new characterization for the alcove form of an affine Weyl group element which has a much simpler form compared with that in J. Y. Shi (J. London Math. Soc.35 (1987), 42–55). As applications, we give an affirmative answer to a conjecture of H. Eriksson and K. Eriksson (preprint, 1998) concerning the characterization of the inverse table of w∈Wa; we also describe the number πs(w) in terms of the permutation form of w∈Wa
(1+1+2)-Generated Equivalence Lattices

AbstractStrietz [Proceedings of the Lattice Theory Conference, Ulm, 1975, pp. 257–259; Studia Sci. Math. Hungar.12 (1977), 1–17] and Zádori [Colloq. Math. Soc. János Bolyai43, “Lectures in Universal Algebra, Proceedings of the Conference, Szeged, 1983,” North-Holland, Amsterdam/New York, 1986, pp. 573–586] have shown that Equ(A), the lattice of all equivalences of a finite set A with |A|≥7, has a four-element generating set such that exactly two of the generators are comparable. In other words, these lattices are (1+1+2)-generated. We extend this result for many infinite sets A; even for all sets if there are no inaccessible cardinals. Namely, we prove that if A is a set consisting of at least seven elements and there is no inaccessible cardinal ≤|A|, then the complete lattice Equ(A) is (1+1+2)-generated. This result is sharp in the sense that Equ(A) has neither a three-element generating set nor a four-element generating set with more than one pair of comparable generators
Tree Products of Conjugacy Separable Groups

AbstractA group G is said to be conjugacy separable if for each pair of elements x, y∈G such that x and y are not conjugate in G, there exists a finite homomorphic image Ḡ of G such that the images of x, y are not conjugate in Ḡ. In this note, we show that the tree products of finitely many conjugacy separable and subgroup separable groups amalgamating central subgroups with trivial intersections are conjugacy separable. We then apply our results to polycyclic-by-finite groups and free-by-finite groups
Bass Numbers in the Graded Case, a-Invariant Formulas, and an Analogue of Faltings' Annihilator Theorem

AbstractLet R=⊕n≥0Rn be a positively graded commutative Noetherian ring. A graded R-module is called *indecomposable (respectively *injective) if it is indecomposable (respectively injective) in the category of graded R-modules. A graded prime ideal of R is called irrelevant if it contains R+≔⊕n>0Rn.Let M=⊕n∈ZMn be a finitely generated graded R-module. The first main result of the paper presents a refinement of the theory of the Bass numbers of M with respect to an irrelevant graded prime ideal p of R. This refinement is related to the siting (in terms of degrees) of *indecomposable *injective direct summands, having associated prime p, of terms in the minimal *injective resolution of M.The second main result has a consequence which can be described in terms of the Castelnuovo regularity r of M: any *indecomposable *injective direct summand, having irrelevant associated prime, of any term in the minimal *injective resolution of M, must vanish in all degrees greater than r. Results of this type provide “uniform upper bounds” for the “degrees of influence” of such irrelevant primes, and these bounds are exploited in the later sections of the paper to (a) generalize known “a-invariant formulas” due to E. Hyry and N. V. Trung; (b) to throw new light on the fact that, for an ideal c0 of R0, the sequence (gradeMnc0)n is ultimately constant; and (c) to describe the ultimate constant value of that sequence by means of an analogue of Faltings' annihilator theorem for local cohomology
On the Automorphism Groups of Quasiprimitive Almost Simple Graphs

AbstractLet Γ be a graph and let G be a subgroup of automorphisms of Γ. Then G is said to be locally primitive on Γ if, for each vertex v, the stabilizer Gv induces a primitive group of permutations on the set of vertices adjacent to v. This paper investigates pairs (Γ,G) for which G is locally primitive on Γ, G is an almost simple group (that is, L≤G≤Aut(L) for some nonabelian simple group L), and the simple socle L is transitive on vertices. Each such graph is a cover of a possibly smaller graph Γ&#x0304; on which G is also locally primitive, and for which in addition AutΓ&#x0304; is quasiprimitive on vertices (that is, every nontrivial normal subgroup of AutΓ&#x0304; is vertex-transitive). It is proved that AutΓ&#x0304; is also an almost simple group. In the general case in which AutΓ is not quasiprimitive on vertices, we show that either every intransitive minimal normal subgroup of AutΓ centralizes L, or L is of Lie type and AutΓ involves an explicitly known same characteristic module for L
Lifting Idempotents in Associative Pairs

AbstractIn 1977, Nicholson developed the theory of suitable rings (Trans. Amer. Math. Soc.229 (1977), 269–278), namely, those in which idempotents lift modulo every left (right) ideal. In this paper the concepts of lifting idempotents modulo every left ideal in an associative pair (A+,A−) and lifting (von Neumann) regular elements modulo every left ideal of A+ (resp. A−) are introduced and shown to be equivalent. We study the behavior of a pair and its standard embedding with respect to the property of being idempotent-lifting. It is also proved that the Jacobson radical can be characterized as the largest one-sided ideal containing no nonzero regular elements. Finally, we show that lifting orthogonal idempotents is possible in this class of pairs
Invariants of Real Forms of Affine Kac-Moody Lie Algebras

AbstractThe theory of Vogan diagrams, which are Dynkin diagrams with an overlay of certain additional information, allows one to give a rapid classification of finite-dimensional real semisimple Lie algebras and to make use of this classification in practice. This paper develops a corresponding theory of Vogan diagrams for “almost compact” real forms of indecomposable nontwisted affine Kac–Moody Lie algebras. In this case also, the equivalence classes of Vogan diagrams correspond to the isomorphism classes of almost compact real forms. Although the real forms of such algebras had already been classified, the theory of Vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification
The Construction of Noetherian Operators

AbstractWe give an elementary and constructive, purely algebraic proof for the existence of noetherian differential operators for primary submodules of finite-dimensional free modules over polynomial algebras. By means of these operators the submodule can be described by differential conditions on the associated characteristic variety. This important result and the terminology are due to V. P. Palamodov. However, his, L. Ehrenpreis' and later J.-E. Björk's proofs of the existence theorem use complicated algebraic and analytic techniques and are not constructive as far as we see. The idea to characterize primary ideals by their associated differential operators is due to W. Gröbner. But M. Noether's Fundamentalsatz is based on similar ideas and is obviously the origin of Palamodov's terminology
Sur le groupe d'automorphismes du groupe libre I. Transvections

AbstractIn this study we define a new family of free automorphisms called transvections which generalise the Whitehead automorphisms. This leads to a new presentation of the group of free automorphisms and to a partial generalisation of McCool's well known result on the presentation of certain of its subgroups. We study the “linear” characteristics of transvections with the use of Fox free differential calculus. The groups generated by these automorphisms form the family of transvection groups that one could name the pre-classical groups. We give some simple examples of these groups, which we shall study further throughout the following papers
Skew Weyl Modules for GLn and Degree Reduction for Schur Algebras

AbstractWe prove a strong characteristic-free analogue of the classical adjoint formula 〈sλ, sμf〉=〈sλ/μ, f〉 in the ring of symmetric functions. This is done by showing that the representative of a suitably chosen functor involving a tensor product is the skew Weyl module. By “strong” we mean that this representative preserves not only Hom groups, but higher Ext groups also—a fact which can be used to compute some homological invariants of Weyl modules for GLn via recursion on degree. We use the following main tools: existence of Weyl filtrations in tensor products of Weyl modules, the Akin–Buchsbaum–Weyman constructions of Weyl modules and certain vanishing properties of Ext groups
A Normal Form for a Class of Monoids Including the Singular Braid Monoids

AbstractThe first part of this paper investigates a class of homogeneously presented monoids. Constructions which enable division and multiplication to be computed are described. The word problem and the division problem are solved, and a unique normal form is given for monoids in this class. The second part deals with the motivating examples of this class—the positive singular Artin monoids. Finally, the singular Artin monoids are introduced. The word and division problems are solved for all singular Artin monoids of finite type, including, for example, the singular braid monoids introduced by Baez and Birman
A Star-Variety With Almost Polynomial Growth

AbstractLet F be a field of characteristic zero. In this paper we construct a finite dimensional F-algebra with involution M and we study its ∗-polynomial identities; on one hand we determine a generator of the corresponding T-ideal of the free algebra with involution and on the other we give a complete description of the multilinear ∗-identities through the representation theory of the hyperoctahedral group. As an outcome of this study we show that the ∗-variety generated by M, var(M,∗) has almost polynomial growth, i.e., the sequence of ∗-codimensions of M cannot be bounded by any polynomial function but any proper ∗-subvariety of var(M,∗) has polynomial growth. If G2 is the algebra constructed in Giambruno and Mishchenko (preprint), we next prove that M and G2 are the only two finite dimensional algebras with involution generating ∗-varieties with almost polynomial growth
Identities with Skew Derivations

AbstractWe solve affirmatively a problem, raised by Kharchenko, on identities with compositions of skew derivations: We define the notion of trivial identities with compositions of skew derivations, which is unique in a certain sense. It is proved that if a prime ring R satisfies a nontrivial identity with compositions of skew derivations, then R also satisfies a generalized polynomial identity (without skew derivations). We actually work in a more general context, in which higher (skew) derivations in the literature known to the author are all covered
On Nonmodular Periodic Finitary Linear Groups

AbstractVery recently M. S. Lucido proved that for any prime p, a periodic finitary linear p′-group G of characteristic p is isomorphic to some finitary linear group G0 of characteristic zero. Moreover G0 can be chosen to be irreducible whenever G is irreducible. Here we offer an alternative proof, one that is somewhat shorter, more explicit and more elementary. We then generalize this result by proving that the same conclusion holds for periodic finitary skew linear p′-groups of characteristic p; that is, we replace fields by division rings in the hypotheses. This is less elementary
Exponent Reduction for Radical Abelian Algebras

AbstractLet k be a field. A radical abelian algebra over k is a crossed product (K/k,α), where K=k(T) is a radical abelian extension of k, T is a subgroup of K* which is finite modulo k*, and α∈H2(G,K*) is represented by a cocycle with values in T. The main result is that if A is a radical abelian algebra over k, and m=exp(A⊗kk(μ)), where μ denotes the group of all roots of unity, then k contains the mth roots of unity. Applications are given to projective Schur division algebras and projective Schur algebras of nilpotent type
Root Vectors Arising from Auslander–Reiten Quivers

AbstractConsider the canonical isomorphism between the Ringel–Hall algebra H(Λ) and the positive part U+ of the Drinfield–Jimbo quantum group Uq(g), where the finite dimensional hereditary algebra Λ and the semisimple Lie algebra g enjoy a common Dynkin diagram. We get an algorithm to decompose the root vectors into combinations of monomials of the canonical generators Ei in the quantum group depending only on the structure of the Auslander–Reiten quiver of Λ. The algorithm is deduced by using a similar method as we did in X. Chen and J. Xiao [1999, Compositio Math.117, No. 2, 161–187], particularly, the derivation defined there
Variational Calculus of Supervariables and Related Algebraic Structures

AbstractWe establish a formal variational calculus of supervariables, which is a combination of the bosonic theory of Gel'fand–Dikii and the fermionic theory in our earlier work. Certain interesting new algebraic structures are found in connection with Hamiltonian superoperators in terms of our theory. In particular, we find connections between Hamiltonian superoperators and Novikov–Poisson algebras that we introduced in our earlier work in order to establish a tensor theory of Novikov algebras. Furthermore, we prove that an odd linear Hamiltonian superoperator in our variational calculus induces a Lie superalgebra, which is a natural generalization of the super-Virasoro algebra under certain conditions
Solution of the Bernstein Problem in the Non-regular Case

AbstractThis paper describes explicitly all non-regular non-degenerate simplicial stochastic Bernstein algebras. Consequently, the Bernstein problem (S. N. Bernstein, Science Ukraine1 (1992), 14–19) in the non-degenerate case is settled, since the regular and exceptional cases have already been examined by Y. Lyubich in the 1970s. Notice that from this result it is possible to explicitly describe every non-regular simplicial algebra (A,Δ) since the simplicial subalgebra (〈supp(A2)〉, [supp(A2)]) is non-degenerate. Also we prove the relevant Lyubich's conjecture (1992, Yu I. Lyubich, Biomathematics22, 232) in an affirmative way: all normal simplicial stochastic Bernstein algebras are regular
Quadratic Lie Superalgebras with the Completely Reducible Action of the Even Part on the Odd Part

AbstractA quadratic Lie superalgebra is a Lie superalgebra g=g0⊕g1 with a non-degenerate supersymmetric consistent (i.e., even) g-invariant bilinear form B; B is called an invariant scalar product of g. We obtain an inductive classification of quadratic Lie superalgebras g=g0⊕g1 such that the action of g0 on g1 is completely reducible and g0 is a reductive Lie algebra. In the case of quadratic Lie superalgebras g=g0⊕g1 such that the action of g0 on g1 is completely reducible, we give an affirmative answer to the following open question (H. Benamor and S. Benayadi, Comm. Algebra27, No. 1 (1999), 67–88): can every B-irreducible non-simple quadratic Lie superalgebra be obtained by double extension? Next, we get an inductive classification of solvable quadratic Lie superalgebras g=g0⊕g1 such that the action of g0 on g1 is completely reducible. Finally, we give the classification of quadratic semisimple Lie superalgebras with the completely reducible action of the even part on the odd part