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CiteULike: Journal of Algebra



CiteULike: Journal of Algebra



 



The dihedral group as a group of automorphisms

2012-11-26T11:58:11-00:00

Journal of Algebra, Vol. 375 (February 2013), pp. 1-12, doi:10.1016/j.jalgebra.2012.04.035

Suppose that D=ãα,βã is a dihedral group generated by two involutions α and β. Let D act on a finite group G in such a manner that CG(αβ)=1. We show that if CG(α) and CG(β) are both nilpotent of class c, then G is nilpotent and the class of G is bounded solely in terms of c. If both CG(α) and CG(β) are of exponent dividing e, then the exponent of G is bounded solely in terms of e and |D|. Previously, results of this kind were known only for groups acted on by Frobenius groups of automorphisms.
Pavel Shumyatsky



Base change and theta-correspondences for supercuspidal representations of

2012-11-26T11:58:02-00:00

Journal of Algebra, Vol. 375 (February 2013), pp. 13-21, doi:10.1016/j.jalgebra.2012.11.006

Let F be a p-adic field with p odd. Quadratic base change and theta-lifting are shown to be compatible for supercuspidal representations of SL(2,F). The argument involves the theory of types and the lattice-model of the Weil representation.
David Manderscheid



Action du groupe des opérateurs de gamétisation sur les identités ω-polynomiales

2012-11-26T11:57:52-00:00

Journal of Algebra, Vol. 375 (February 2013), pp. 22-32, doi:10.1016/j.jalgebra.2012.11.019

The gametization process reduces the study of non-commutative and non-associative algebras satisfying non-homogeneous polynomial identities with variables in X=x1,…,xn to algebras verifying simpler identities. However after a gametization, certain identities remain invariant and other identities, said universal invariant, are invariant for every gametization. Now in the case n=1, for all algebras satisfying a universal invariant polynomial identity studied until now, we know that the existence of an idempotent is not certain. Using an action of the gametization operators group on the non-commutative and non-associative algebra of polynomials KãXã, we give all identities which are invariant and universal invariant by gametization.
Cristián Mallol, Richard Varro



A note on the isomorphism conjectures for Leavitt path algebras

2012-11-26T11:57:44-00:00

Journal of Algebra, Vol. 375 (February 2013), pp. 33-40, doi:10.1016/j.jalgebra.2012.11.017

We relate two conjectures which have been raised for classification of Leavitt path algebras. For purely infinite simple unital Leavitt path algebras, it is conjectured that K0 classifies them completely (Abrams et al., 2008, 2011  and ). For arbitrary unital Leavitt path algebras, it is conjectured that classifies them completely (Hazrat, in press [12]). We show that for two finite graphs with no sinks (which their associated Leavitt path algebras include the purely infinite simple ones) if their -groups of their Leavitt path algebras are isomorphic then their K0-groups are isomorphic as well. We also provide a short proof of the fact that for a finite graph, its associated Leavitt path algebra is strongly graded if and only if the graph has no sinks.
R Hazrat



Syzygies of differentials of forms

2012-11-26T11:57:35-00:00

Journal of Algebra, Vol. 375 (February 2013), pp. 41-58, doi:10.1016/j.jalgebra.2012.11.008

Given a standard graded polynomial ring R=k[x1,…,xn] over a field k of characteristic zero and a graded k-subalgebra A=k[f1,…,fm]⊂R, one relates the module ΩA/k of Kähler k-differentials of A to the transposed Jacobian module of the forms f1,…,fm by means of a Leibniz mapΩA/k→D whose kernel is the torsion of ΩA/k. Letting D denote the R-submodule generated by the (image of the) syzygy module of ΩA/k and Z the syzygy module of D, there is a natural inclusion D⊂Z coming from the chain rule for composite derivatives. The main goal is to give means to test when this inclusion is an equality – in which case one says that the forms f1,…,fm are polarizable. One surveys some classes of subalgebras that are generated by polarizable forms. The problem has some curious connections with constructs of commutative algebra, such as the Jacobian ideal, the conormal module and its torsion, homological dimension in R and syzygies, complete intersections and Koszul algebras. Some of these connections trigger questions which have interest in their own.
Isabel Bermejo, Philippe Gimenez, Aron Simis



Some bismash products that are not group algebras II

2012-11-26T11:57:26-00:00

Journal of Algebra, Vol. 375 (February 2013), pp. 59-72, doi:10.1016/j.jalgebra.2012.10.022

We consider a family of Hopf algebras, each constructed as a bismash product from a factorisation of the symmetric group SN. These are semisimple as algebras over C and we show that they do not have the structure (as an algebra) of any group algebra if N⩾5. Previously, the author has established this in the special cases where N has the form p+1 or p+2 for p prime; here the general result is obtained by very different methods.
Michael Collins



The theory of prime ideals of Leavitt path algebras over arbitrary graphs

2012-11-26T11:57:18-00:00

Journal of Algebra, Vol. 375 (February 2013), pp. 73-96, doi:10.1016/j.jalgebra.2012.11.004

Given an arbitrary graph E and a field K, the prime ideals as well as the primitive ideals of the Leavitt path algebra LK(E) are completely characterized in terms of their generators. The stratification of the prime spectrum of LK(E) is indicated with information on its individual stratum. Necessary and sufficient conditions are given on the graph E under which every prime ideal of the Leavitt path algebra LK(E) is primitive. Leavitt path algebras with Krull dimension zero are characterized and those with various prescribed Krull dimension are constructed. The minimal prime ideals of LK(E) are described in terms of the graphical properties of E and using this, complete descriptions of the height one as well as the co-height one prime ideals of LK(E) are given.
Kulumani Rangaswamy



Demazure crystals and tensor products of perfect Kirillov–Reshetikhin crystals with various levels

2012-11-26T11:57:08-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 1-26, doi:10.1016/j.jalgebra.2012.10.020

In this paper, we study a tensor product of perfect Kirillov–Reshetikhin crystals (KR crystals for short) whose levels are not necessarily equal. We show that, by tensoring with a certain highest weight element, such a crystal becomes isomorphic as a full subgraph to a certain disjoint union of Demazure crystals contained in a tensor product of highest weight crystals. Moreover, we show that this isomorphism preserves their Z-gradings, where the Z-grading on the tensor product of KR crystals is given by the energy function, and that on the other side is given by the minus of the action of the degree operator.
Katsuyuki Naoi



On a generalization of M-group

2012-11-26T11:56:59-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 27-41, doi:10.1016/j.jalgebra.2012.10.018

In this paper, we show that if for every nonlinear complex irreducible character Ï of a finite group G, some multiple of Ï is induced from an irreducible character of some proper subgroup of G, then G is solvable. This is a generalization of Taketaʼs theorem on the solvability of M-group.
Tung Le, Jamshid Moori, Hung Tong-Viet



Dehornoy-like left orderings and isolated left orderings

2012-11-26T11:56:50-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 42-58, doi:10.1016/j.jalgebra.2012.10.016

We introduce a Dehornoy-like ordering of groups, which is a generalization of the Dehornoy ordering of the braid groups. Under an assumption which we call Property F, we show that Dehornoy-like orderings have properties similar to the Dehornoy ordering, and produce isolated left orderings. We also construct new examples of Dehornoy-like orderings and isolated orderings and study their more precise properties.
Tetsuya Ito



Conformal designs and D.H. Lehmerʼs conjecture

2012-11-26T11:56:39-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 59-65, doi:10.1016/j.jalgebra.2012.10.019

In 1947, Lehmer conjectured that the Ramanujan Ï-function Ï(m) is non-vanishing for all positive integers m, where Ï(m) are the Fourier coefficients of the cusp form Î of weight 12. It is known that Lehmerʼs conjecture can be reformulated in terms of spherical t-design, by the result of Venkov. In this paper, we show that Ï(m)=0 is equivalent to the fact that the homogeneous space of the moonshine vertex operator algebra (Vâ®)m+1 is a conformal 12-design. Therefore, Lehmerʼs conjecture is now reformulated in terms of conformal t-designs.
Tsuyoshi Miezaki



Schur-finiteness in λ-rings

2012-11-26T11:56:29-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 66-78, doi:10.1016/j.jalgebra.2012.09.039

We introduce the notion of a Schur-finite element in a λ-ring.
C Mazza, C Weibel



Peirce inner ideals in general position

2012-11-26T11:56:21-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 79-103, doi:10.1016/j.jalgebra.2012.10.021

Three elements U, V, and W in the complete lattice I(A) of weakâ-closed inner ideals in a JBWâ-triple A are said to be in general position when (U,V) form a rigidly collinear pair, and (U,W) and (V,W) form orthogonal pairs. A complete description of the supremum U∨V∨W in I(A) in terms of the Peirce spaces corresponding to U, V, and W is given for the case in which all three inner ideals are Peirce and three other conditions are satisfied. By considering an example in the Albert JBWâ-triple H3(O) it is shown that none of these is redundant and, therefore, that the result obtained is the best possible.
Martin Edwards



Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

2012-11-26T11:56:12-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 104-121, doi:10.1016/j.jalgebra.2012.11.002

We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau–Ginzburg model coming from a toric degeneration.
Nathan Ilten, Jacob Lewis, Victor Przyjalkowski



Macaulay–Lex rings

2012-11-26T11:56:03-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 122-131, doi:10.1016/j.jalgebra.2012.10.025

We present simple proofs of Macaulayʼs theorem and Clements–Lindströmʼs theorem. We generalize Shakinʼs theorem by proving that a stable ideal I of S is Macaulay–Lex if and only if I is a piecewise lexsegment ideal. We also study Macaulay–Lex ideals of the form , where 2⩽e1⩽â¯â©½en⩽∞ and tiAbed Abedelfatah



Groups with all centralizers subnormal of defect at most two

2012-11-26T11:55:54-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 132-140, doi:10.1016/j.jalgebra.2012.10.026

We study the class of all groups in which the centralizer of each element is a subnormal subgroup. In particular, we focus on the case when the defect of every centralizer is at most 2. We show that a group without involutions satisfies this property if and only if it is 3-Engel.
Costantino Delizia, Primož Moravec, Chiara Nicotera



Griess algebras generated by 3 Ising vectors of central 2A-type

2012-11-26T11:55:46-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 141-166, doi:10.1016/j.jalgebra.2012.10.023

We study the Griess algebra generated by three Ising vectors e,f, and g in a CFT type vertex operator algebra V with V1=0 such that . We call such a configuration of central 2A-type. Under this assumption, we show that there are only 5 possible structures of Griess algebras and they correspond exactly to the Griess algebras GVB(nX) of the five VOA VB(nX), nX∈1A,2B,3A,4B,2C, constructed by Höhn–Lam–Yamauchi.
Ching Lam, Che Su



Conjugacy classes of Renner monoids

2012-11-26T11:55:38-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 167-180, doi:10.1016/j.jalgebra.2012.10.024

In this paper we describe conjugacy classes of a Renner monoid R with unit group W, the Weyl group. We show that every element in R is conjugate to an element ue where u∈W and e is an idempotent in a cross section lattice. Denote by W(e) and Wâ(e) the centralizer and stabilizer of e∈Î in W, respectively. Let W(e) act by conjugation on the set of left cosets of Wâ(e) in W. We find that ue and ve (u,v∈W) are conjugate if and only if uWâ(e) and vWâ(e) are in the same orbit. As consequences, there is a one-to-one correspondence between the conjugacy classes of R and the orbits of this action. We then obtain a formula for calculating the number of conjugacy classes of R, and describe in detail the conjugacy classes of the Renner monoid of some J-irreducible monoids. We then generalize Munn conjugacy on a rook monoid to any Renner monoid and show that Munn conjugacy coincides with semigroup conjugacy, action conjugacy, and character conjugacy. We also show that the number of inequivalent irreducible representations of R over an algebraically closed field of characteristic zero equals the number of Munn conjugacy classes in R.
Zhuo Li, Zhenheng Li, Youʼan Cao



The Abhyankar–Moh Theorem for plane valuations at infinity

2012-11-26T11:55:29-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 181-194, doi:10.1016/j.jalgebra.2012.11.001

We introduce the class of plane valuations at infinity and prove an analogue to the Abhyankar–Moh (semigroup) Theorem for it.
C Galindo, F Monserrat



General heart construction for twin torsion pairs on triangulated categories

2012-11-26T11:55:20-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 195-215, doi:10.1016/j.jalgebra.2012.10.027

In our previous article, we constructed an abelian category from any torsion pair on a triangulated category. This generalizes the heart of a t-structure and the ideal quotient by a cluster tilting subcategory. Recently, generalizing the quotient by a cluster tilting subcategory, Buan and Marsh showed that an integral preabelian category can be constructed as a quotient, from a rigid object in a triangulated category with some conditions. In this article, by considering a pair of torsion pairs, we make a simultaneous generalization of these two constructions.
Hiroyuki Nakaoka



On Jordan decomposition of characters for

2012-11-26T11:55:12-00:00

Journal of Algebra, Vol. 374 (January 2013), pp. 216-230, doi:10.1016/j.jalgebra.2012.10.015

As shown by Bonnafé, a step in proving a Jordan decomposition of characters of finite special linear groups is the parametrization of unipotent characters of centralizers of semi-simple elements in projective linear groups. We show the same kind of result in the case of finite special unitary groups. The proof leads to a mild adaptation of Bonnaféʼs methods expounded in [B99]. The outcome is a Jordan decomposition of characters compatible with Lusztigʼs twisted induction.
Marc Cabanes



Embeddings of semisimple complex Lie groups and cohomological components of modules

2012-11-26T11:54:55-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 1-29, doi:10.1016/j.jalgebra.2012.09.030

Let be an embedding of semisimple complex Lie groups, a pair of nested Borel subgroups and the associated embedding of flag manifolds. Let be an equivariant invertible sheaf on and O(λ) be its restriction to G/B. Consider the G-equivariant pullback We establish a necessary and sufficient condition for nonvanishing of . Also, we prove a theorem on the structure of the set of pairs of dominant weights with cohomological. Here V(μ) and denote the respective highest weight modules. Simplified specializations are formulated for regular and diagonal embeddings. In particular, we give an alternative proof of a recent theorem of Dimitrov and Roth. Beyond the regular and diagonal cases, we study equivariantly embedded rational curves and we also show that the generators of the algebra of ad-invariant polynomials on a semisimple Lie algebra can be obtained as cohomological components. Our methods rely on Kostantʼs theory of Lie algebra cohomology.
Valdemar Tsanov



Motivic rigidity of Severi–Brauer varieties

2012-11-26T11:54:46-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 30-38, doi:10.1016/j.jalgebra.2012.08.028

Let D be a central division algebra over a field F. We study in this note the rigidity of the motivic decompositions of the Severi–Brauer varieties of D, with respect to the ring of coefficients and to the base field. We first show that if the ring of coefficient is a field, these decompositions only depend on its characteristic. In a second part we show that if D remains division over a field extension E/F, the motivic decompositions of several Severi–Brauer varieties of D remain the same when extending the scalars to E.
Charles De Clercq



Intertwining operators and fusion rules for vertex operator algebras arising from symplectic fermions

2012-11-26T11:54:38-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 39-64, doi:10.1016/j.jalgebra.2012.09.022

We determine fusion rules (dimensions of the spaces of intertwining operators) among simple modules for the vertex operator algebra obtained as an even part of the symplectic fermionic vertex operator superalgebra. By using these fusion rules we show that the fusion algebra of this vertex operator algebra is isomorphic to the group algebra of the Klein four group over Z.
Toshiyuki Abe, Yusuke Arike



Inseparable local uniformization

2012-11-26T11:54:29-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 65-119, doi:10.1016/j.jalgebra.2012.09.023

It is known since the works of Zariski in the early 40s that desingularization of varieties along valuations (called local uniformization of valuations) can be considered as the local part of the desingularization problem. It is still an open problem if local uniformization exists in positive characteristic and dimension larger than three. In this paper, we prove that Zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions, and therefore any valuation can be uniformized by a purely inseparable alteration.
Michael Temkin



Linear characters of over Dedekind domains

2012-11-26T11:54:20-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 120-129, doi:10.1016/j.jalgebra.2012.08.029

For an important class of arithmetic Dedekind domains o including the ring of integers of not totally complex number fields, we describe explicitly the group of linear characters of SL(2,o). For this, we introduce and determine, for arbitrary Dedekind domains o, the group of linear characters of SL(2,o) whose kernel is a congruence subgroup.
Hatice Boylan, Nils-Peter Skoruppa



Hilbert depth of graded modules over polynomial rings in two variables

2012-11-26T11:54:11-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 130-152, doi:10.1016/j.jalgebra.2012.09.026

In this article we mainly consider the positively Z-graded polynomial ring R=F[X,Y] over an arbitrary field F and Hilbert series of finitely generated graded R-modules. The central result is an arithmetic criterion for such a series to be the Hilbert series of some R-module of positive depth. In the generic case, that is deg(X) and deg(Y) being coprime, this criterion can be formulated in terms of the numerical semigroup generated by those degrees.
Julio Moyano-Fernández, Jan Uliczka



Linear algebraic groups as parameterized Picard–Vessiot Galois groups

2012-11-26T11:54:01-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 153-161, doi:10.1016/j.jalgebra.2012.09.037

We show that a linear algebraic group is the Galois group of a parameterized Picard–Vessiot extension of k(x), x′=1, for certain differential fields k, if and only if its identity component has no one-dimensional quotient as a linear algebraic group.
Michael Singer



Liftings and quasi-liftings of DG modules

2012-11-26T11:53:52-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 162-182, doi:10.1016/j.jalgebra.2012.09.036

We prove lifting results for DG modules that are akin to Auslander, Ding, and Solbergʼs famous lifting results for modules.
Saeed Nasseh, Sean Sather-Wagstaff



Reducibility of certain induced representations of and

2012-11-26T11:53:43-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 183-206, doi:10.1016/j.jalgebra.2012.10.011

We give equivalent conditions for certain parabolically induced representations of E6 and E7 to be irreducible by calculating the action of certain Weyl group element and detecting poles of the first two Langlands–Shahidi L-functions at s=0.
Jing Lau



Three dimensional canonical singularities in codimension two in positive characteristic

2012-11-26T11:53:35-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 207-222, doi:10.1016/j.jalgebra.2012.10.007

We investigate local structure of a three dimensional variety X defined over an algebraically closed field k of characteristic p>0 with at most canonical singularities. Under the assumption that p⩾3 and a general hyperplane cut of X has at most rational singularities, we show that local structure of X in codimension two is well understood in the level of local equations. Consequently, we find that i) any singularity of such a variety X in codimension two is compound Du Val, ii) it has a crepant resolution, iii) it is analytically a product of a rational double point and a nonsingular curve when p⩾3 with two exceptions in p=3, and iv) holds outside some finite points of X for any resolution of singularities .
Masayuki Hirokado, Hiroyuki Ito, Natsuo Saito



Effectively categorical abelian groups

2012-11-26T11:53:26-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 223-248, doi:10.1016/j.jalgebra.2012.09.020

We study effective categoricity of computable abelian groups of the form ⊕i∈ÏH, where H is a subgroup of (Q,+). Such groups are called homogeneous completely decomposable. It is well-known that a homogeneous completely decomposable group is computably categorical if and only if its rank is finite. We study -categoricity in this class of groups, for n>1. We introduce a new algebraic concept of S-independence which is a generalization of the well-known notion of p-independence. We develop the theory of S-independent sets. We apply these techniques to show that every homogeneous completely decomposable group is -categorical. We prove that a homogeneous completely decomposable group of infinite rank is -categorical if and only if it is isomorphic to the free module over the localization of Z by a computably enumerable set of primes P with the semi-low complement (within the set of all primes). We apply these results and techniques to study the complexity of generating bases of computable free modules over localizations of integers, including the free abelian group.
Rodney Downey, Alexander Melnikov



Identification of simple representations for affine q-Schur algebras

2012-11-26T11:53:17-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 249-275, doi:10.1016/j.jalgebra.2012.10.013

We define an explicit action of the quantum loop algebra on the tensor space and show that these actions are compatible with the natural embedding . As an application, we identify simple representations of affine q-Schur algebras Sâµ(n,r)C arising from simple polynomial representations of (Frenkel and Mukhin, 2002) [11] with those arising from simple modules of the affine Hecke algebra Hâµ(r)C of type A (Rogawski, 1985) [26].
Bangming Deng, Jie Du



Group algebras and Lie nilpotence

2012-11-26T11:53:09-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 276-283, doi:10.1016/j.jalgebra.2012.09.043

Let â be an involution of a group algebra FG induced by an involution of the group G. For , we classify the groups G with no 2-elements and with no nonabelian dihedral groups involved whose Lie algebra of â-skew elements is nilpotent.
A Giambruno, C Polcino Milies, Sudarshan Sehgal



New irreducible modules for Heisenberg and affine Lie algebras

2012-11-26T11:53:00-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 284-298, doi:10.1016/j.jalgebra.2012.09.035

We study Z-graded modules of nonzero level with arbitrary weight multiplicities over Heisenberg Lie algebras and the associated generalized loop modules over affine Kac–Moody Lie algebras. We construct new families of such irreducible modules over Heisenberg Lie algebras. Our main result establishes the irreducibility of the corresponding generalized loop modules providing an explicit construction of many new examples of irreducible modules for affine Lie algebras. In particular, to any function Ï:N→± we associate a Ï-highest weight module over the Heisenberg Lie algebra and a Ï-imaginary Verma module over the affine Lie algebra. We show that any Ï-imaginary Verma module of nonzero level is irreducible.
Viktor Bekkert, Georgia Benkart, Vyacheslav Futorny, Iryna Kashuba



Lower central series of free algebras in symmetric tensor categories

2012-11-26T11:52:51-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 299-311, doi:10.1016/j.jalgebra.2012.10.001

We continue the study of the lower central series of a free associative algebra, initiated by Feigin and Shoikhet (2007) [FS]. We generalize via Schur functors the constructions of the lower central series to any symmetric tensor category; specifically we compute the modified first quotient , and second and third quotients B2, and B3 of the series for a free algebra T(V) in any symmetric tensor category, generalizing the main results of Feigin and Shoikhet (2007) [FS] and Arbesfeld and Jordan (2010) [AJ]. In the case Am|n:=T(Cm|n), we use these results to compute the explicit Hilbert series. Finally, we prove a result relating the lower central series to the corresponding filtration by two-sided associative ideals, confirming a conjecture from Etingof et al. (2009) [EKM], and another one from Arbesfeld and Jordan (2010) [AJ], as corollaries.
Asilata Bapat, David Jordan



Simplicity and maximal commutative subalgebras of twisted generalized Weyl algebras

2012-11-26T11:52:43-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 312-339, doi:10.1016/j.jalgebra.2012.10.009

In this paper we prove three theorems about twisted generalized Weyl algebras (TGWAs). First, we show that each non-zero ideal of a TGWA has non-zero intersection with the centralizer of the distinguished subalgebra R. This is analogous to earlier results known to hold for crystalline graded rings. Second, we give necessary and sufficient conditions for the centralizer of R to be commutative (hence maximal commutative), generalizing a result by V. Mazorchuk and L. Turowska. And third, we generalize results by D.A. Jordan and V. Bavula on generalized Weyl algebras by giving necessary and sufficient conditions for certain TGWAs to be simple, in the case when R is commutative. We illustrate our theorems by considering some special classes of TGWAs and providing concrete examples. We also discuss how simplicity of a TGWA is related to the maximal commutativity of R and the (non-)existence of non-trivial Zn-invariant ideals of R.
JT Hartwig, J Öinert



Closed subsets in duals of commutative table algebras

2012-11-26T11:52:34-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 340-355, doi:10.1016/j.jalgebra.2012.09.040

A lower bound on the number of elements outside a closed subset of a C-basis of a commutative C-algebra with dual a table algebra is derived, as is an equivalent condition for when the lower bound is met. As corollaries, lower bounds are obtained on the number of primitive idempotent matrices of rank greater than 1 in the adjacency algebra of a commutative, imprimitive association scheme; and, for a given normal subgroup N of a finite group G, on the number of irreducible characters of G whose kernels do not contain N, and on the number of conjugacy classes of G not contained in N. Also found are equivalent conditions for when these lower bounds are met.
Harvey Blau, Gang Chen



On the derived category of a graded commutative noetherian ring

2012-11-26T11:52:25-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 356-376, doi:10.1016/j.jalgebra.2012.09.038

For any graded commutative noetherian ring, where the grading group is abelian and where commutativity is allowed to hold in a quite general sense, we establish an inclusion-preserving bijection between, on the one hand, the twist-closed localizing subcategories of the derived category, and, on the other hand, subsets of the homogeneous spectrum of prime ideals of the ring. We provide an application to weighted projective schemes.
Ivo DellʼAmbrogio, Greg Stevenson



The proportion of Weierstrass semigroups

2012-11-26T11:52:17-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 377-391, doi:10.1016/j.jalgebra.2012.09.041

We solve a problem of Komeda concerning the proportion of numerical semigroups which do not satisfy Buchweitzʼ necessary criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve. A result of Eisenbud and Harris gives a sufficient condition for a semigroup to occur as a Weierstrass semigroup. We show that the family of semigroups satisfying this condition has density zero in the set of all semigroups. In the process, we prove several more general results about the structure of a typical numerical semigroup.
Nathan Kaplan, Lynnelle Ye



The primary components of positive critical binomial ideals

2012-11-26T11:52:08-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 392-413, doi:10.1016/j.jalgebra.2012.10.014

A natural candidate for a generating set of the (necessarily prime) defining ideal of an n-dimensional monomial curve, when the ideal is an almost complete intersection, is a full set of n critical binomials. In a somewhat modified and more tractable context, we prove that, when the exponents are all positive, critical binomial ideals in our sense are not even unmixed for n⩾4, whereas for n⩽3 they are unmixed. We further give a complete description of their isolated primary components as the defining ideals of monomial curves with coefficients. This answers an open question on the number of primary components of Herzog–Northcott ideals, which comprise the case n=3. Moreover, we find an explicit, concrete description of the irredundant embedded component (for n⩾4) and characterize when the hull of the ideal, i.e., the intersection of its isolated primary components, is prime. Note that these last results are independent of the characteristic of the ground field. Our techniques involve the Eisenbud–Sturmfels theory of binomial ideals and Laurent polynomial rings, together with theory of Smith Normal Form and of Fitting ideals. This gives a more transparent and completely general approach, replacing the theory of multiplicities used previously to treat the particular case n=3.
Liam OʼCarroll, Francesc Planas-Vilanova



Integer-valued polynomials on algebras

2012-11-26T11:52:00-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 414-425, doi:10.1016/j.jalgebra.2012.10.003

Let D be a domain with quotient field K and A a D-algebra. A polynomial with coefficients in K that maps every element of A to an element of A is called integer-valued on A. For commutative A we also consider integer-valued polynomials in several variables. For an arbitrary domain D and I an arbitrary ideal of D we show I-adic continuity of integer-valued polynomials on A. For Noetherian one-dimensional D, we determine spectrum and Krull dimension of the ring IntD(A) of integer-valued polynomials on A. We do the same for the ring of polynomials with coefficients in Mn(K), the K-algebra of n×n matrices, that map every matrix in Mn(D) to a matrix in Mn(D).
Sophie Frisch



On supersimple groups

2012-11-26T11:51:51-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 426-438, doi:10.1016/j.jalgebra.2012.09.033

We show that an infinite group having a supersimple theory has a finite series of definable subgroups whose factors are infinite and either virtually FC or virtually simple modulo a finite FC-centre. We deduce that a group which is type-definable in a supersimple theory has a finite series of relatively definable subgroups whose factors are either abelian or simple groups. In this decomposition, the non-abelian simple factors are unique up to isomorphism.
Cédric Milliet



Addendum to “A seven-term exact sequence for the cohomology of a group extension” [J. Algebra 369 (1) (2012) 70–95]

2012-11-26T11:51:43-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 439-440, doi:10.1016/j.jalgebra.2012.10.017
Karel Dekimpe, Manfred Hartl, Sarah Wauters



Computing generators of the unit group of an integral abelian group ring

2012-11-26T11:51:34-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 441-452, doi:10.1016/j.jalgebra.2012.09.031

We describe an algorithm for obtaining generators of the unit group of the integral group ring ZG of a finite abelian group G. We used our implementation in Magma of this algorithm to compute the unit groups of ZG for G of order up to 110. In particular for those cases we obtained the index of the group of Hoechsmann units in the full unit group. At the end of the paper we describe an algorithm for the more general problem of finding generators of an arithmetic group corresponding to a diagonalisable algebraic group.
Paolo Faccin, Willem de Graaf, Wilhelm Plesken



Rational plane curves parameterizable by conics

2012-11-26T11:51:25-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 453-480, doi:10.1016/j.jalgebra.2012.09.034

We introduce the class of rational plane curves parameterizable by conics as an extension of the family of curves parameterizable by lines (also known as monoid curves). We show that they are the image of monoid curves via suitable quadratic transformations in projective plane. We also describe all the possible proper parameterizations of them, and a set of minimal generators of the Rees Algebra associated to these parameterizations, extending well-known results for curves parameterizable by lines.
Teresa Cortadellas Benítez, Carlos DʼAndrea



The classification of normalizing groups

2012-11-26T11:51:16-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 481-490, doi:10.1016/j.jalgebra.2012.08.033

Let X be a finite set such that |X|=n. Let Tn and Sn denote the transformation monoid and the symmetric group on n points, respectively. Given a∈TnâSn, we say that a group G⩽Sn is a-normalizing if The goal of this paper is to classify the normalizing groups and hence answer a question of Levi, McAlister, and McFadden. The paper ends with a number of problems for experts in groups, semigroups and matrix theory.
João Araújo, Peter Cameron, James Mitchell, Max Neunhöffer



Coclass theory for nilpotent semigroups via their associated algebras

2012-11-26T11:51:07-00:00

Journal of Algebra, Vol. 373 (January 2013), pp. 491-501, doi:10.1016/j.jalgebra.2012.09.042

Coclass theory has been a highly successful approach towards the investigation and classification of finite nilpotent groups. Here we suggest a variation of this approach for finite nilpotent semigroups: we propose to study coclass graphs for the contracted semigroup algebras of nilpotent semigroups. We exhibit a series of conjectures on the shape of these coclass graphs. If these are proven, then this reduces the classification of nilpotent semigroups of a fixed coclass to a finite calculation. We show that our conjectures are supported by the nilpotent semigroups of coclass 0 and 1. Computational experiments suggest that the conjectures also hold for the nilpotent semigroups of coclass 2 and 3.
Andreas Distler, Bettina Eick



The dihedral group as a group of automorphisms

2012-11-17T11:08:13-00:00

Journal of Algebra, Vol. 375 (February 2013), pp. 1-12, doi:10.1016/j.jalgebra.2012.04.035

Suppose that D=ãα,βã is a dihedral group generated by two involutions α and β. Let D act on a finite group G in such a manner that CG(αβ)=1. We show that if CG(α) and CG(β) are both nilpotent of class c, then G is nilpotent and the class of G is bounded solely in terms of c. If both CG(α) and CG(β) are of exponent dividing e, then the exponent of G is bounded solely in terms of e and |D|. Previously, results of this kind were known only for groups acted on by Frobenius groups of automorphisms.
Pavel Shumyatsky



Base change and theta-correspondences for supercuspidal representations of

2012-11-17T11:08:05-00:00

Journal of Algebra, Vol. 375 (February 2013), pp. 13-21, doi:10.1016/j.jalgebra.2012.11.006

Let F be a p-adic field with p odd. Quadratic base change and theta-lifting are shown to be compatible for supercuspidal representations of SL(2,F). The argument involves the theory of types and the lattice-model of the Weil representation.
David Manderscheid