pinecone/core-2020-05-10-deduplication huggingface dataset:
Skew Derivations Whose Invariants Satisfy a Polynomial Identity

AbstractIf σ is an automorphism and δ is a q-skew σ-derivation of a ring R, then the subring of invariants is the set R(δ)={r∈R|δ(r)=0}. The main result of this paper isTheorem.Let R be a prime algebra with a q-skew σ-derivation δ, where δ and σ are algebraic. If R(δ)satisfies a P.I., then R satisfies a P.I.If δ is separable, then we also obtain the following result:Theorem.Let δ be a separable q-skew σ-derivation of an algebra R, where δ and σ are algebraic.•(i)•(ii)∩ σWhen R is a domain, it is necessary to assume neither that σ is algebraic nor that δ is q-skew as we proveTheorem.If R is a domain with an algebraic σ-derivation δ such that R(δ) satisfies a P.I., then R also satisfies a P.I
Invariant Theory of Canonical Algebras

AbstractBased on the first fundamental theorem of classical invariant theory we present a reduction technique for computing relative invariants for quivers with relations. This is applied to the invariant theory of canonical algebras and yields an explicit construction of the moduli spaces (together with the quotient morphisms from the corresponding representation spaces) for families of modules with a fixed dimension vector belonging to the central sincere separating subcategory. By means of a tilting process we extend these results to the invariant theory of concealed-canonical algebras, thus covering the cases of tame hereditary, tame concealed, and tubular algebras, respectively. Our approach yields, in particular, a uniform treatment to an essential part of the invariant theory of extended Dynkin quivers, a topic popular over the years, but stretches far beyond since also concealed-canonical algebras of tubular or wild representation type are covered
Cross Characteristic Projective Representations for Some Classical Groups

AbstractLet G(pn) be a finite simple group of Lie type and let V be a projective irreducible representation of G(pn) over a field of characteristic different from p. Lower bounds for the dimension of V have been computed in [V. Landázuri and G. M. Seitz, J. Algebra32 (1974), 418–433] and [G. M. Seitz and A. E. Zalesskii, J. Algebra158 (1993), 233–243] and have recently been improved in the case of linear groups by R. Guralnick and P. H. Tiep [Proc. London Math. Soc. (3)78 (1999), 116–138] and in the case of characteristic zero representations in [P. H. Tiep and A. Zalesskii, Comm. Algebra24 (1996), 2093–2167]. The purpose of this paper is to improve the bounds in [G. M. Seitz and A. E. Zalesskii, J. Algebra158 (1993), 233–243] for the case of orthogonal groups and to provide estimates for the gap between the smallest representations for orthogonal and unitary groups
Superrigidity Theorems in Positive Characteristic

AbstractWe generalize the superrigidity theorem of G. A. Margulis (1991, “Discrete Subgroups of Semisimple Lie Groups,” VII.5.19, Springer-Verlag, New York/Berlin) to the case of the more general non-reductive target group F. We also provide a proof of the cocycle superrigidity theorem in the case of algebraic groups defined over a local field of positive characteristic and the generalization of this proof to the case of the more general target group analogous to our generalization of the superrigidity theorem
Real 2-Regular Classes and 2-Blocks

AbstractSuppose that G is a finite group. We show that every 2-block of G has a defect class which is real. As a partial converse, we show that if G has a real 2-regular class with defect group D and if N(D)/D has no dihedral subgroup of order 8, then G has a real 2-block with defect group D. More generally, we show that every 2-block of G which is weakly regular relative to some normal subgroup N has a defect class which is real and contained in N. We give several applications of these results and also investigate some consequences of the existence of non-real 2-blocks
Finite Length and Pure-Injective Modules over a Ring of Differential Operators

AbstractLet k be an algebraically closed field of characteristic zero, On=k[[x1,…,xn]] the ring of formal power series over k, and Dn the ring of differential operators over On. Suppose that ρ is a prime ideal of On of height n−1; i.e., A=On/ρ is a curve. We prove that every indecomposable finite length module over Dn with support on ρ is uniserial with isomorphic or alternating composition factors. For the ring D(A) of differential operators over A we also classify indecomposable pure-injective modules and show that the Cantor–Bendixson rank of the Ziegler spectrum over D(A) is equal to 2
On a Class of Lattice Ordered Inverse Semigroups

AbstractIt is well known that the free group on a non-empty set can be totally ordered and, further, that each compatible latttice ordering on a free group is a total ordering. On the other hand, Saitô has shown that no non-trivial free inverse semigroup can be totally ordered. In this note we show, however, that every free inverse monoid admits compatible lattice orderings which are closely related to the total orderings on free groups.These orderings are natural in the sense that the imposed partial ordering on the idempotents coincides with the natural partial ordering. For this to happen in a lattice ordered inverse semigroup, the idempotents must form a distributive lattice. The method of construction of the lattice orderings on free inverse monoids can be applied to show that naturally lattice ordered inverse semigroups with a given distributive lattice E of idempotents can have arbitrary Green's relation structure. Analogous results hold for naturally ∧-semilatticed inverse semigroups. In this case, there is no restriction on the semilattice E of idempotents.We also show that every compatible lattice ordering on the free monogenic inverse monoid is of the type considered here. This permits us to prove that there are precisely eight distinct compatible lattice orderings on this semigroup. They belong to two families, each of which contains four members, of conjuguate lattice orderings
Triangular Matrix Representations

AbstractIn this paper we develop the theory of generalized triangular matrix representation in an abstract setting. This is accomplished by introducing the concept of a set of left triangulating idempotents. These idempotents determine a generalized triangular matrix representation for an algebra. The existence of a set of left triangulating idempotents does not depend on any specific conditions on the algebras; however, if the algebra satisfies a mild finiteness condition, then such a set can be refined to a “complete” set of left triangulating idempotents in which each “diagonal” subalgebra has no nontrivial generalized triangular matrix representation. We then apply our theory to obtain new results on generalized triangular matrix representations, including extensions of several well known results
An Infinite Analogue of Rings with Stable Rank One

AbstractReplacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings) and include examples like EndF(V), the ring of endomorphisms of a vector space V over some field F, and B(F), the ring of all row- and column-finite matrices over F. We show that the category of QB-rings is stable under the formation of corners, ideals, and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of QB-rings to be a QB-ring, and show that extensions of B-rings often lead to QB-rings. Specializing to the category of exchange rings we characterize the subset of exchange QB-rings as those in which every von Neumann regular element extends to a maximal regular element, i.e., a quasi-invertible element. Finally we show that the C*-algebras that are QB-rings are exactly the extremally rich C*-algebras studied by L. G. Brown and the second author
On the Orthogonal Basis of Symmetry Classes of Tensors

AbstractA necessary and sufficient condition for the existence of the orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with dicyclic group and dihedral group were studied by M. R. Darafsheh and M. R. Pournaki (in press, Linear and Multilinear Algebra) and R. R. Holmes and T. Y. Tam (1992, Linear and Multilinear Algebra32, 21–31). These authors used a certain permutation structure of these groups to prove the necessary condition. In this article we show that the necessary condition found in these previous works is independent of the permutation structures of these groups
On Groups of Automorphisms of Residually Finite Groups

AbstractWe show that certain properties of groups of automorphisms can be read off from the actions they induce on the finite characteristic quotients of their underlying group G. In particular, we obtain criteria for groups of automorphisms of a residually finite and (soluble minimax)-by-finite group G to be nilpotent or soluble. Moreover, we give explicit bounds on the class (the derived length, resp.) of such groups of automorphisms in terms of invariants of G. Finally, we consider similar questions when G is the free group of rank two
Affine Shuffles, Shuffles with Cuts, the Whitehouse Module, and Patience Sorting

AbstractType A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group Sn are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter q satisfies gcd(n,q−1)=1, the induced measures on conjugacy classes of the symmetric group coincide. This gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents. Using representation theoretic work on the Whitehouse module, a formula is obtained for the cycle structure of a riffle shuffle followed by a cut. It is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness. Generating functions for the first pile size in patience sorting from decks with repeated values are derived. This relates to random matrices
Galois Reconstruction of Finite Quantum Groups

AbstractLet C be a (small) category and let F: C→Malgf be a functor, where Malgf is the category of finite-dimensional measured algebras over a field k (or Frobenius algebras). We construct a universal Hopf algebra Aaut(F) such that F factorizes through a functor F̄: C→Mcoalgf(Aaut(F)), where Mcoalgf(Aaut(F)) is the category of finite-dimensional measured Aaut(F)-comodule algebras. This general reconstruction result allows us to recapture a finite-dimensional Hopf algebra A from the category Mcoalgf(A) and the forgetful functor ω: Mcoalgf(A)→Malgf: we have A≅Aaut(ω). Our universal construction is also done in a C*-algebra framework, and we get compact quantum groups in the sense of Woronowicz
Covers Induced by Ext

AbstractWe prove a generalization of the flat cover conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−,C)-cover, for any class of pure-injective modules C, and that (2) each module has a Ker Tor(−,B)-cover, for any class of left R-modules B.For Dedekind domains, we describe Ker Ext(−,C) explicitly for any class of cotorsion modules C; in particular, we prove that (1) holds, and that Ker Ext(−,C) is a cotilting torsion-free class. For right hereditary rings, we prove the consistency of the existence of special Ker Ext(−,G)-precovers for any set of modules G
Galois Theory of Thick Subcategories in Modular Representation Theory

AbstractD. J. Benson, J. F. Carlson, and J. Rickard [1997, Fund. Math.153, 59–80] classified the tensor-closed thick subcategories of finite-dimensional representations of finite groups over algebraically closed fields. In this paper, we remove the algebraically closed hypothesis by applying some Galois theory. Our methods apply more generally to finite-dimensional cocommutative Hopf algebras over a field. Thus they allow us to drop the algebraically closed hypothesis in the classification of thick subcategories of modules over finite-dimensional sub-Hopf algebras of the Steenrod algebra as well
Complexity of Representations of Quantised Function Algebras and Representation Type

AbstractLet G be a simply connected, connected, semisimple algebraic group over C, B± be opposite Borel subgroups of G, and b=Lie(B+). Let Oϵ[G] be the quantised function algebra at a root of unity ϵ and let U≥0ϵ be the quantised enveloping algebra of b at a root of unity. We study the finite dimensional factor algebras Oϵ[G](g) and U≥0ϵ(b) for g∈G and b∈B−, which were introduced by De Concini, Kac, and Procesi (1992, in “Geometry and Analysis,” pp. 41–65, Tata Inst. Fund. Res., Bombay) and De Concini and Lyubashenko (1994, Adv. Math.108, 205–262). In particular we describe the complexity of these factor algebras in terms of the Weyl group of G and deduce results on their representation type. Finally we completely describe those factor algebras of finite representation type
Mixed Ladder Determinantal Varieties

AbstractWe investigate ladder determinantal varieties defined by ideals of minors of possibly different sizes in the different regions (the steps) of one-sided ladders L. These varieties are an important generalization of the classical ladder determinantal varieties (i.e., with equal-size minors) since they are very closely related to Schubert varieties, this being the first main result of this paper. We show that they correspond to opposite cells in Schubert varieties in flag varieties of type An. In consequence, one deduces the normality and the Cohen–Macaulayness of these one-sided ladder determinantal varieties with ideals of mixed-size minors, as well as the fact that they have rational singularities. Next we show that, up to products by affine spaces, each of these varieties is a basic open set in a classical ladder determinantal variety (i.e., with equal-size minors) and that it contains as a basic open set another classical ladder determinantal variety. This result, along with a general localisation lemma used to show it, enables us to compute the divisor class group and singular locus of the coordinate rings of these varieties, as well as to determine when they are Gorenstein
Twisted Hopf Algebras, Ringel–Hall Algebras, and Green's Categories With an appendix by the referee

AbstractThe concept and some basic properties of a twisted Hopf algebra are introduced and investigated. Its unique difference from a Hopf algebra is that the comultiplication δ: A→A⊗A is an algebra homomorphism, not for the componentwise multiplication on A⊗A, but for the twisted multiplication on A⊗A given by Lusztig's rule.Also, it is proved that any object A in Green's category has a twisted Hopf algebra structure, any morphism between objects is a twisted Hopf algebra homomorphism, the antipode s of A is self-adjoint under the Lusztig form (−,−) on A, and the Green polynomials Ma,b(t) share a so-called cyclic-symmetry.As examples, the twisted Ringel–Hall algebras, Ringel's twisted composition algebras, Lusztig's free algebras ′F and non-degenerate algebras F, the positive part U+ of the Drinfeld–Jimbo quantized enveloping algebras U, and Rosso's quantum shuffle algebra T(V) all are twisted Hopf algebras. The antipode and its inverse for a twisted Ringel–Hall are explicitly given, and all δ-primitive elements are determined
Weak Hopf Algebras II. Representation Theory, Dimensions, and the Markov Trace

AbstractIf A is a weak C*-Hopf algebra then the category of finite-dimensional unitary representations of A is a monoidal C*-category with its monoidal unit being the GNS representation Dε associated to the counit ε. This category has isomorphic left dual and right dual objects, which leads, as usual, to the notion of a dimension function. However, if ε is not pure the dimension function is matrix valued with rows and columns labeled by the irreducibles contained in Dε. This happens precisely when the inclusions AL⊂A and AR⊂A are not connected. Still, there exists a trace on A which is the Markov trace for both inclusions. We derive two numerical invariants for each C*-WHA of trivial hypercenter. These are the common indices I and δ, of the Haar, respectively Markov, conditional expectations of either one of the inclusions AL/R⊂A or ÁL/R⊂Á. In generic cases I>δ. In the special case of weak Kac algebras we reproduce D. Nikshych's result (2000, J. Operator Theory, to appear) by showing that I=δ and is always an integer
Extending Ring Topologies

AbstractThroughout this abstract, (R,T) denotes a compact (Hausdorff) topological ring. The authors extend T to ring topologies on R which are totally bounded or even pseudocompact; a principal tool is the Bohr compactification of a topological ring. They show inter alia: If the Jacobson radical J(R) of R satisfies w(R/J(R))>ω then there is a pseudocompact ring topology on R strictly finer than T; if in addition w(R)=w(R/J(R))=α with cf(α)>ω then there are exactly 22|R|-many such topologies.The ring (R,T) is said to be a van der Waerden ring if T is the only totally bounded ring topology on R. Theorem 4.13 asserts that if R is semisimple, then R is a van der Waerden ring if and only if in the Kaplansky representation R=Πn<ω(Rn)αn of R as a product of full matrix rings over finite fields each αn is finite.Other classes of van der Waerden rings are constructed, and it is shown that there are non-compact totally bounded rings (S,U) such that U is the only totally bounded ring topology on S
Quantum Divided Power Algebra, Q-Derivatives, and Some New Quantum Groups

AbstractThe discussions in the present paper arise from exploring intrinsically the structural nature of the quantum n-space. A kind of braided category GB of Λ-graded θ-commutative associative algebras over a field k is established. The quantum divided power algebra over k related to the quantum n-space is introduced and described as a braided Hopf algebra in GB (in terms of its 2-cocycle structure), over which the so-called special q-derivatives are defined so that several new interesting quantum groups, especially the quantized polynomial algebra in n variables (as the quantized universal enveloping algebra of the abelian Lie algebra of dimension n) and the quantum group associated to the quantum n-space, are derived from our approach independently of using the R-matrix. As a verification of its validity for our discussion, the quantum divided power algebra is equipped with the structure of a Uq(sln)-module algebra via certain q-differential operators' realization. Particularly, one of the four kinds of root vectors of Uq(sln) in the sense of Lusztig can be specified precisely under the realization
The Maximal Orders Which Contain a Given Idealiser Ring

AbstractLet C be a commutative Dedekind domain, let S be a maximal C-order in a simple Artinian ring Q, and let K be a proper essential right ideal of S with SK=S. We shall study those maximal C-orders in Q which contain the idealiser ring R of K and the way in which invertible ideals of R act on them by conjugation. In a special case, which includes that in which R is hereditary, we shall show that these maximal orders are in one-to-one correspondence with the ideals of S which contain the bound of K, and we shall give a complete description of the invertible ideals of R
Generalizing Morita Duality: A Homological Approach

AbstractLet R and S be arbitrary associative rings. Given a bimodule RWS, we denote by Δ? and Γ? the functors Hom?(−,W) and Ext1?(−,W), where ?=R or S. The functors ΔR and ΔS are right adjoint with the evaluation maps δ as unities. A module M is Δ-reflexive if δM is an isomorphism. In this paper we give, for a weakly cotilting bimodule RWS, the notion of Γ-reflexivity. We construct large Abelian subcategories MR and MS where the functors ΓR and ΓS are left adjoint and a “cotilting theorem” holds
Twistings and Hopf Galois Extensions

AbstractLet k be a commutative ring, let H be a k-Hopf algebra, and let A be a right H-comodule algebra. A twisting of A is a map τ: H⊗A→A such that (A,∗τ,ρA) is also an H-comodule algebra, where the product ∗τ is defined by a∗τb=∑a0τ(a1⊗b). In this note, we observe that there is a map of pointed sets from the twistings of A to the H-measurings from AcoH to A and study the set of twistings that map to the trivial measuring. If A/AcoH is Galois and H is finitely generated projective, then the twistings that map to the trivial measuring can be described as a set of invertible twisted cocycles: ϕ: H⊗H→A. An equivalence relation on the set of twisted cocycles corresponds to isomorphism classes of Galois extensions
Automorphisms of Repetitive Algebras

AbstractLet B̂ be the repetitive algebra of a finite dimensional algebra B over a field K by the B-bimodule DB=Hom(B,K), and let ν be the Nakayama automorphism of B̂. We determine the positive automorphisms ϕ of B̂ such that the orbit algebra B̂/(ϕν) is isomorphic to a splittable extension algebra of B by the B-bimodule (DB)β for an automorphism β of B, and we characterize weakly symmetric algebras and symmetric algebras of the form B̂/(ϕν) with a positive automorphism ϕ of B̂. As an application, we characterize some class of weakly symmetric algebras with non-periodic generalized standard Auslander–Reiten components
Noncommutative Rational Double Points

AbstractIn this paper we study noncommutative analogues of rational double points. The approach is to consider the action of a finite group G on certain noncommutative analogues of k[[x,y]] which were studied by Artin and Stafford (“Regular Local Rings of Dimension 2,” manuscript). An explicit description in terms of generators and relations is given for a large class of such algebras when G is cyclic. Finally, we show that these algebras are AS-Gorenstein of dimension two, have finite representation type and, in many cases, are regular in codimension one
A Generalization of the Terwilliger Algebra

AbstractP. M. Terwilliger (1992, J. Algebraic Combin.1, 363–388) considered the C-algebra generated by a given Bose Mesner algebra M and the associated dual Bose Mesner algebra M*. This algebra is now known as the Terwilliger algebra and is usually denoted by T. Terwilliger showed that each vanishing intersection number and Krein parameter of M gives rise to a relation on certain generators of T. These relations determine much of the structure of T, thought not all of it in general. To illuminate the role these relations play, we consider a certain generalization T of T. To go from T to T, we replace M and M* with a pair of dual character algebras C and C*. We define T by generators and relations; intuitively T is the associative C-algebra with identity generated by C and C* subject to the analogues of Terwilliger's relations. T is infinite dimensional and noncommutative in general. We construct an irreducible T-module which we call the primary module; the dimension of this module is the same as that of C and C*. We find two bases of the primary module; one diagonalizes C and the other diagonalizes C*. We compute the action of the generators of T on these bases. We show T is a direct sum of two sided ideals T0 and T1 with T0 isomorphic to a full matrix algebra. We show that the irreducible module associated with T0 is isomorphic to the primary module. We compute the central primitive idempotent of T associated with T0 in terms of the generators of T