pinecone/core-2020-05-10-deduplication huggingface dataset:
On the Associative Analog of Lie Bialgebras

AbstractAn infinitesimal bialgebra is at the same time an associative algebra and coalgebra in such a way that the comultiplication is a derivation. This paper continues the basic study of these objects, with emphasis on the connections with the theory of Lie bialgebras. It is shown that non-degenerate antisymmetric solutions of the associative Yang–Baxter equation are in one to one correspondence with non-degenerate cyclic 2-cocycles. The associative and classical Yang–Baxter equations are compared: it is studied when a solution to the first is also a solution to the second. Necessary and sufficient conditions for obtaining a Lie bialgebra from an infinitesimal one are found, in terms of a canonical map that behaves simultaneously as a commutator and a cocommutator. The class of balanced infinitesimal bialgebras is introduced; they have an associated Lie bialgebra. Several well known Lie bialgebras are shown to arise in this way. The construction of Drinfeld's double from earlier work by the author (in press, in Contemp. Math., Amer. Math. Soc., Providence) for arbitrary infinitesimal bialgebras is complemented with the construction of the balanced double, for balanced ones. This construction commutes with the passage from balanced infinitesimal bialgebras to Lie bialgebras
Smallest Graded Betti Numbers

AbstractIt is known that given a Hilbert function H, there need not exist a module which has uniquely the smallest graded Betti numbers among all modules attaining H. In this paper we extend the previous example of this behavior to an infinite family and demonstrate with a second infinite family that even when the given Hilbert function is that of a complete intersection, a module with uniquely smallest graded Betti numbers need not exist. Finally we prove a conjecture of Geramita, Harima, and Shin concerning the non-existence of uniquely smallest graded Betti numbers among all Gorenstein rings attaining a given Hilbert function
Dimension n Representations of the Braid Group on n Strings

AbstractIn 1996, E. Formanek classified all the irreducible complex representations of Bn of dimension at most n−1, where Bn is the Artin braid group on n strings. In this paper we extend this classification to the representations of dimension n, for n≥9. We prove that all such representations are equivalent to the tensor product of a one-dimensional representation and a specialization of a certain one-parameter family of n-dimensional representations which was first discovered in 1996 by Tong, Yang, and Ma. In order to do this, we classify all the irreducible complex representations ρ of Bn for which rank(ρ(σi)−1)=2, where the σi are the standard generators
Kummer Elements and the mod-3 Invariant of Albert Algebras

AbstractLet k be a field with characteristic not 2 and 3. Assume that k contains the cube roots of unity. Let J be a Tits'-first-construction Albert division algebra over k. In this paper we relate Kummer elements in J with the mod-3 invariant g3(J). We prove that if x∈J is a Kummer element with x3=λ, then J≃J(D,λ) for some D, a degree-3 central division algebra over k. We show that if J1=J(A,μ) and J2=J(B,ν) are Tits' first-construction Albert division algebras with g3(J1)=g3(J2) then J2≃J(D,μ) for some degree-3 central division algebra D over k
The Classification of Elements in T×C*

AbstractIn this paper, we give an explicit description of the good, semi-good, and bad elements in T×C*. These three concepts were introduced by Xi for proving the Deligne–Langlands conjecture for complex affine Hecke algebras when the order of the nonzero complex parameter q is not too small. First, we define the root graph of a root system and the degenerate paths associated with an element (s,q)∈T×C*. Then we establish a criterion for (s,q) being good in terms of the non-existence of the degenerate paths associated with (s,q) in the corresponding root graph. Finally, we distinguish the bad elements from the semi-good ones by the vanishing of the Poincaré polynomial at q
Tractability of Algebraic Function Fields in One Variable of Genus Zero Over Global Fields

AbstractIf the condition described in Definition 1.1 on the simultaneous representation of quaternion algebras holds over a field F, then F is called tractable. In this paper, we show that a field F is tractable if and only if any purely transcendental extension of F is tractable. Next, if F is a global field of characteristic unequal to two, we consider tractability of an algebraic function field K in one variable of genus zero over F and determine precisely when K is tractable if F has at most one ordering
Classification des triples de Manin pour les algèbres de Lie réductives complexes Avec un appendice de Guillaume Macey

RésuméWe study real and complex Manin triples for a complex reductive Lie algebra g. First, we generalize results of E. Karolinsky (1996, Math. Phys. Anal. Geom3, 545–563; 1999, Preprint math.QA.9901073) on the classification of Lagrangian subalgebras. Then we show that, if g is noncommutative, one can attach to each Manin triple in g another one for a strictly smaller reductive complex Lie subalgebra of g. This gives a powerful tool for induction. Then we classify complex Manin triples in terms of what we call generalized Belavin–Drinfeld data. This generalizes, by other methods, the classification of A. Belavin and V. G. Drinfeld of certain r-matrices, i.e., the solutions of modified triangle equations for constants (cf. A. Belavin and V. G. Drinfeld, “Triangle Equations and Simple Lie Algebras,” Mathematical Physics Reviews, Vol. 4, pp. 93–165, Harwood Academic, Chur, 1984, Theorem 6.1). We get also results for real Manin triples. In passing, we retrieve a result of A. Panov (1999, Preprint math.QA.9904156) which classifies certain Lie bialgebra structures on a real simple Lie algebra
Lie Triple Systems, Restricted Lie Triple Systems, and Algebraic Groups

AbstractWe define a restricted structure for Lie triple systems in the characteristic p>2 setting, akin to the restricted structure for Lie algebras, and initiate a study of a theory of restricted modules. In general, Lie triple systems have natural embeddings into certain canonical Lie algebras, the so-called “standard” and “universal” embeddings, and any Lie triple system can be shown to arise precisely as the −1-eigenspace of an involution (an automorphism which squares to the identity) on some Lie algebra. We specialize to Lie triple systems which arise as the differentials of involutions on simple, simply connected algebraic groups over algebraically closed fields of characteristic p. Under these hypotheses we completely classify the universal and standard embeddings in terms of the Lie algebra and its universal central extension
Galois Algebras and Monoidal Functors between Categories of Representations of Finite Groups

AbstractWe use relations between Galois algebras and monoidal functors to describe monoidal functors between categories of representations of finite groups. We pay special attention to two kinds of these monoidal functors: monoidal functors to vector spaces and monoidal equivalences between categories of representations. The functors of the second kind induce isomorphisms of character tables. We show that pairs of groups with the same character table obtained in this way are a generalization of the construction proposed by B. Fischer (1988, Rend. Circ. Mat. Palermo (2) Suppl.19, 71–77)
The Center of Thin Gaussian Groups

AbstractThin Gaussian groups are a natural generalization of spherical Artin groups, namely groups of fractions of monoids in which the existence of least common multiples is kept as an hypothesis, but the relations between the generators are not supposed to necessarily be of Coxeter type. Here we completely describe the center of thin Gaussian groups by constructing a minimal generating set for the quasi-center. We deduce that every thin Gaussian group is an iterated crossed product of thin Gaussian groups with a cyclic center
Polynomial Properties in Unitriangular Matrices

AbstractLet Gn=Gn(q) be the group of the upper unitriangular matrices of size n over Fq, the finite field of q=pt elements. G. Higman has conjectured that, for each n, the number of conjugacy classes of elements of Gn is a polynomial expression in q. In this paper we prove that the number of conjugacy classes of Gn of cardinality qs, with s≤n−3, is a polynomial in q−1, with non-negative integral coefficients, fs(q−1), of degree less than or equal to the integer part of 2s+1. In addition, fs(q−1) depends only on s and not on n. We determine these polynomials arguing with the methods we gave previously (1995, J. Algebra177, 899–925). In fact, the coefficients of these polynomials are obtained by certain generating functions
Relating Properties of a Ring and Its Ring of Row and Column Finite Matrices

AbstractMackey and Ornstein proved that if R is a semi-simple ring then the ring of row and column finite matrices over R (RCFMΓ(R)) is a Baer ring for any infinite set Γ. A ring with identity is a Baer ring if every left (equivalent every right) annihilator is generated by an idempotent. This result is discussed in Kaplansky's book, “Rings of Operators.” This result is of course decades old. Here we prove that the converse is true. The proof is long and we develop techniques which allow us to obtain results of a more modern flavor about RCFMΓ(R), where R is a perfect or semi-primary ring. Finally, we obtain good enough results on annihilators in RCFM(Z) to show that this ring is coherent
Quantized Rank R Matrices

AbstractFirst some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized n×r matrices as well as certain quantized factor algebras Mr+1q(n) of Mq(n) are analyzed. For r=1,…,n−1,Mr+1q(n) is the quantized function algebra of rank r matrices obtained by working modulo the ideal generated by all (r+1)×(r+1) quantum subdeterminants and a certain localization of this algebra is proved to be isomorphic to a more manageable one. In almost all cases, the quantum parameter is a primitive mth root of unity. The degrees and centers of the algebras are determined when m is a prime and the general structure is obtained for arbitrary m
Frobenius Extensions and Weak Hopf Algebras

AbstractWe study a symmetric Markov extension of k-algebras N↪M, a certain kind of Frobenius extension with conditional expectation that is tracial on the centralizer and dual bases with a separability property. We place a depth two condition on this extension, which is essentially the requirement that the Jones tower N↪M↪M1↪M2 can be obtained by taking relative tensor products with centralizers A=CM1(N) and B=CM2(M). Under this condition, we prove that N↪M is the invariant subalgebra pair of a weak Hopf algebra action by A, i.e., that N=MA. The endomorphism algebra M1=EndNM is shown to be isomorphic to the smash product algebra M#A. We also extend results of W. Szymański (1994, Proc. Amer. Math. Soc. 120, 519–528), D. Nikshych and L. Vainerman (2000, Funct. Anal.171, 278–307), and L. Kadison and D. Nikschych (Comm. Algebra, 29(9) 2001)
On the Partially Ordered Monoid Generated by the Operators H, S, P, Ps on Classes of Algebras

AbstractLet H, S, P be the standard operators on classes of algebras of the same type and Ps be the operator of closure under subdirect products. In this paper the monoid generated by class operators H, S, P, Ps and the corresponding partially ordered set are described. It turns out that there are 22 different operators such that every composition of H, S, P, Ps coincides with one of them. The semigroups (monoids) of operators H, S, P, Ps for the varieties of Boolean algebras, distributive lattices, unary algebras (of countable unary type), Abelian groups, and groups are also describe
Almost GCD Domains of Finite t-Character

AbstractLet D be an integral domain. Two nonzero elements x,y∈D are v-coprime if (x)∩(y)=(xy). D is an almost-GCD domain (AGCD domain) if for every pair x,y∈D, there exists a natural number n=n(x,y) such that (xn)∩(yn) is principal. We show that if x is a nonzero nonunit element of an almost GCD domain D, then the set {M; M maximal t-ideal, x∈M} is finite, if and only if the set S(x):={y∈D; y nonunit, y divides xn for some n} does not contain an infinite sequence of mutually v-coprime elements, if and only if there exists an integer r such that every sequence of mutually v-coprime elements of S(x) has length ≤r. One of the various consequences of this result is that a GCD domain D is a semilocal Bézout domain if and only if D does not contain an infinite sequence of mutually v-coprime nonunit elements. Then, we study integrally closed AGCD domains of finite t-character of the type A+XB[X] and we construct examples of nonintegrally closed AGCD of finite t-character by local algebra techniques
Pointed Brauer Trees

AbstractIn this paper we establish a one-to-one correspondence between Brauer trees with an additional structure that we call “pointing” and equivalence classes of tilting complexes of the corresponding Brauer star algebra which are of a special “combinatorial” nature. Under the correspondence, the identification of the indecomposable summands of the tilting complex with edges of the Brauer tree is completely natural. The procedure gives an algorithm for producing large numbers of distinct tilting complexes for the same Brauer tree
On Multicommutators for Simple Algebraic Groups

AbstractThere are several examples of groups for which any pair of commutators can be written such that both of them have a common entry, and one can look for a similar property for n-tuples of commutators. Here we answer, for simple algebraic groups over any field, the weaker question, under which condition the set of n-tuples of commutators with one common entry is Zariski dense in the set of all n-tuples of commutators. Surprisingly, there is a uniform bound on n in terms of the so-called Coxeter number of G to answer the question positively. An analogue result is proved for Lie algebras of simple and simply connected algebraic groups
Accurate Elements and Super-Primitive Elements over Rings

AbstractLet R⊆R[α] be a simple algebraic extension of commutative rings with identity, and let I be the kernel of the R-algebra map R[X]→R[α] sending X to α, where R[X] is the polynomial ring in one indeterminate over R. Then we say that α is accurate over R if I is generated by nonzero polynomials of least degree in I. The main purpose of this paper is to give several conditions for α to be accurate over R. Our results generalize some work of Nagata and Mirbagheri and Ratliff. We also discuss some related topics, including consideration of relations between accurate elements and super-primitive elements
Local Cohomology of Stanley–Reisner Rings with Supports in General Monomial Ideals

AbstractWe study the local cohomology modules HiIΣ(k[Δ]) of the Stanley–Reisner ring k[Δ] of a simplicial complex Δ with support in the ideal IΣ⊂k[Δ] corresponding to a subcomplex Σ⊂Δ. We give a combinatorial topological formula for the multigraded Hilbert series, and in the case where the ambient complex is Gorenstein, compare this with a second combinatorial formula that generalizes results of Mustata and Terai. The agreement between these two formulae is seen to be a disguised form of Alexander duality. Other results include a comparison of the local cohomology with certain Ext modules, results about when it is concentrated in a single homological degree, and combinatorial topological interpretations of some vanishing theorems
Module Varieties over Canonical Algebras

AbstractThe main purpose of this paper is the study of module varieties over the class of canonical algebras, providing a rich source of examples of varieties with interesting properties. Our main tool is a stratification of module varieties, which was recently introduced by Richmond. This stratification does not require a precise knowledge of the module category. If it is finite, then it provides a method to classify irreducible components. We determine the canonical algebras for which this stratification is finite. In this case, we describe the algorithm for calculating the dimension of the variety and the number of irreducible components of maximal dimension. For an infinite family of examples we give easy combinatorial criteria for irreducibility, Cohen–Macaulay and normality
Tame and Wild Projective Curves and Classification of Vector Bundles

AbstractWe propose a new method of classifying vector bundles on projective curves, especially singular ones, according to their “representation type.” In particular, we prove that the classification problem of vector bundles, respectively of torsion-free sheaves, on projective curves is always finite, tame, or wild. We completely classify curves which are of finite, respectively tame, vector bundle type by their dual graph. Moreover, our methods yield a geometric description of all indecomposable vector bundles and torsion-free sheaves on finite and tame curves
Right Orderable Residually Finite p-Groups and a Kourovka Notebook Problem

AbstractRhemtulla proved that if a group is a residually finite p-group for infinitely many primes p, then it is two-sided orderable. In problem 10.30 of The Kourovka Notebook (14th ed.), N. Ya. Medvedev asked if there is a non-right-orderable group which is a residually finite p-group for at least two different primes p. Using a result of Witte, we will show that many subgroups of finite index in GL3(Z) give examples of such groups. On the other hand, we will show that no such example can exist among solvable by finite groups
The Geometry of Points on Quantum Projectivizations

AbstractSuppose S is an affine, noetherian scheme, X is a separated, noetherian S-scheme, E is a coherent OX-bimodule, and I⊂T(E) is a graded ideal. We study the geometry of the functor Γn of flat families of truncated B=T(E)/I-point modules of length n+1. We then use the results of our study to show that if Proj B is a quantum ruled surface, the point modules over B are parameterized by the closed points of PX2(E). When X=P1, we construct, for any B-point module, a graded OX−B-bimodule resolution
Approximations and the Little Finitistic Dimension of Artinian Rings

AbstractLet R be a ring and let simp-R be a representative set of all simple (right R-) modules. Denote by P<ω the class of all modules which are finitely generated and have finite projective dimension. The little finitistic dimension of R is defined by fdim(R)=sup{proj.dim(M)∣M∈P<ω}. Let (A,B) be the complete cotorsion theory cogenerated by P<ω. For each S∈simp-R, let fS: XS→S be a special A-precover of S. We prove that fdim (R)=max{proj.dim(XS)∣S∈simp-R} provided that R is right artinian. As a corollary, we extend to right artinian rings the well-known Auslander–Reiten sufficient condition for finiteness of the little finitistic dimension
The Top of the Lattice of Normal Subgroups of the Grigorchuk Group

AbstractA complete description of the lattice of all normal subgroups not contained in the stabilizer of the fourth level of the tree and, consequently, of index ≤212 in the Grigorchuk group G is given. This leads to the following sharp version of the congruence property: a normal subgroup not contained in the stabilizer at level n+1 contains the stabilizer at level n+3 (in fact such a normal subgroup contains the subgroup Nn+1), but, in general, it does not contain the stabilizer at level n+2. The determination of all normal subgroups at each level n≥4 is then reduced to the analysis of certain G-modules which depend only on n and the previous description, as for the analogous problem for the automorphism group of the regular rooted tree
Oriented Quantum Algebras and Invariants of Knots and Links

AbstractIn our previous work oriented quantum algebras were motivated and introduced in a very natural categorical setting within the context of knots and links, some examples were discussed, and a rudimentary theory of oriented quantum algebras was sketched. Invariants of knots and links can be computed from oriented quantum algebras. Here we continue the study of oriented quantum algebras. We view them from a more algebraic perspective and develop a more detailed theory for them and their associated invariants. We study a class of examples associated with the HOMFLY polynomial in depth
Integral Non-commutative Spaces

AbstractA non-commutative space X is a Grothendieck category Mod X. We say X is integral if there is an indecomposable injective X-module EX such that its endomorphism ring is a division ring and every X-module is a subquotient of a direct sum of copies of EX. A noetherian scheme is integral in this sense if and only if it is integral in the usual sense. We show that several classes of non-commutative spaces are integral. We also define the function field and generic point of an integral space and show that these notions behave as one might expect
Free Unit Groups in Group Algebras

AbstractLet K[G] denote the group algebra of a finite group G over a field K. If either char K=0 and G is nonabelian, or K is a nonabsolute field of characteristic π>0 and G/Oπ(G) is nonabelian, then it is well known that the group of units U(K[G]) contains a nonabelian free group. For the most part, this follows from the fact that GL2(K) contains such a free subgroup. In this paper, we refine the above result by showing that there are two cyclic subgroups X and Y of G of prime power order, and two units uX∈U(K[X]) and uY∈U(K[Y]), such that 〈uX,uY〉 contains a nonabelian free group. Indeed, we obtain a rather precise description of these units by using an aspect of Tits' theorem on free subgroups in linear groups
The Hit Problem for the Modular Invariants of Linear Groups

AbstractLet the mod 2 Steenrod algebra, A, and the general linear group, GLk:=GL(k,F2), act on Pk:=F2[x1,…,xk] with deg(xi)=1 in the usual manner. We prove that, for a family of some rather small subgroups G of GLk, every element of positive degree in the invariant algebra PkG is hit by A in Pk. In other words, (PkG)+⊂A+·Pk, where (PkG)+ and A+ denote respectively the submodules of PkG and A consisting of all elements of positive degree. This family contains most of the parabolic subgroups of GLk. It should be noted that the smaller the group G is, the harder the problem turns out to be. Remarkably, when G is the smallest group of the family, the invariant algebra PkG is a polynomial algebra in k variables, whose degrees are ≤8 and fixed while k increases.It has been shown by Hu·ng [Trans. Amer. Math. Soc.349 (1997), 3893–3910] that, for G=GLk, the inclusion (PkGLk)+⊂A+·Pk is equivalent to a weak algebraic version of the long-standing conjecture stating that the only spherical classes in Q0S0 are the elements of Hopf invariant 1 and those of Kervaire invariant 1
Realizations of Quantum Hom-Spaces, Invariant Theory, and Quantum Determinantal Ideals

AbstractFor a Hecke operator R, one defines the matrix bialgebra ER, which is considered as function algebra on the quantum space of endomorphisms of the quantum space associated to R. One generalizes this notion, defining the function algebra MRS on the quantum space of homomorphisms of two quantum spaces associated to two Hecke operators R and S, respectively. MRS can be considered as a quantum analog (or a deformation) of the function algebra on the variety of matrices of a certain degree. We provide two realizations of MRS as a quotient algebra and as a subalgebra of a tensor algebra, whence we derive interesting information about MRS, for instance the Koszul property, a formula for computing the Poincaré series. On MRS coact the bialgebras ER and ES. We study the two-sided ideals in MRS, invariant with respect to these actions, in particular, the determinantal ideals. We prove analogies of the fundamental theorems of invariant theory for these quantum groups and quantum hom-spaces