Note on Conic Bundles over Henselian Discrete Valued Fields with Real Closed Residue Field

ABSTRACT

Let ? be a complete set of Sylow subgroups of a finite group G, that is, ? contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup H of a finite group G is said to be ?-permutable if H permutes with every member of ?. The purpose of this article is to study the influence of ?-permutability of all maximal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of a finite group G on the structure of G. Our results improve and extend the main results of Asaad (1998 Asaad , M. ( 1998 ). On maximal subgroups of Sylow subgroups of finite groups . Comm. Algebra 26 ( 11 ): 3647 – 3652 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), Asaad and Heliel (2003 Asaad , M. , Heliel , A. A. ( 2003 ). On permutable subgroups of finite groups . Arch. Math. 80 : 113 – 118 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Asaad et al. (1991 Asaad , M. , Ramadan , M. , Shaalan , A. ( 1991 ). Influence of π-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group . Arch. Math. 56 : 521 – 527 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Li et al. (2003 Li , Y. , Wang , Y. , Wei , H. ( 2003 ). The influence of π-quasinormality of maximal subgroups of Sylow subgroups of a finite group . Arch. Math. 81 ( 3 ): 245 – 252 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), Ramadan (1992 Ramadan , M. ( 1992 ). Influence of normality on maximal subgroups of Sylow subgroups of a finite group . Acta Math. Hungar. 59 ( 1–2 ): 107 – 110 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), and Srinivasan (1980 Srinivasan , S. ( 1980 ). Two sufficient conditions for supersolvability of finite groups . Israel J. Math. 35 : 210 – 214 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]).     

Simple Modules and Primitive Ideals of Non-Noetherian Generalized Down-Up Algebras

Generalized down-up algebras were first introduced in Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 : 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]). Their simple weight modules were classified in Cassidy and Shelton (2004 Cassidy , T. , Shelton , B. ( 2004 ). Basic properties of generalized down-up algebras . J. Algebra 279 : 402 – 421 .[Crossref], [Web of Science ®] , [Google Scholar]) in the noetherian case, and in Praton (2007 Praton , I. ( 2007 ). Simple weight modules of non-noetherian generalized down-up algebras . Comm. Algebra 35 : 325 – 337 .[Taylor &; Francis Online] , [Google Scholar]) in the non-noetherian case. Here we concentrate on non-noetherian down-up algebras. We show that almost all simple modules are weight modules. We also classify the corresponding primitive ideals.     

Standard Koszul standardly stratified algebras

In this paper, we prove that every standard Koszul (not necessarily graded) standardly stratified algebra is also Koszul. This generalizes a similar result of [3 Ágoston, I., Dlab, V., Lukács, E. (2003). Quasi-hereditary extension algebras. Algebras Represent. Theory 6:97–117.[Crossref], [Web of Science ®] , [Google Scholar]] on quasi-hereditary algebras.     

Sequences Determining Mixed Multiplicities and Reductions of Ideals

Abstract

Using the notion of (FC)-sequences in Viêt (2000 Viêt, D. Q. 2000. Mixed multiplicities of arbitrary ideals in local rings. Comm. Algebra, 28(8): 3803–3821. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), this paper presents some results concerning reductions and the vanishing and non-vanishing of mixed multiplicities of finite collection of arbitrary ideals in local rings.     

ℤ3-Twisted Representations of Lattice Vertex Operator Algebras

Abstract

Let L be a positive definite even lattice and let g ∈ Aut L be a fixed point free automorphism of order 3. We determine the twisted Zhu's algebra A _? (V _L ) for the lattice vertex operator algebra V _L , where ? is an automorphism of V _L induced from g. As a result, we show that the set of all irreducible ?-twisted modules for V _L (up to isomorphism) are exactly those constructed by Dong and Lepowsky (1996 Dong, C. and Lepowsky, J. 1996. The algebraic structure of relative twisted vertex operators. J. Pure and Applied Algebra, 110: 259–295. [Crossref], [Web of Science ®] , [Google Scholar]) and Lepowsky (1985 Lepowsky, J. 1985. Calculus of twisted vertex operators. Proc. Natl. Acad. Sci. USA, 82: 8295–8299. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]).     

The Larson–Sweedler theorem for weak multiplier Hopf algebras

The Larson–Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra [15 Larson, R. G., Sweedler, M. E. (1969). An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91:75–93.[Crossref], [Web of Science ®] , [Google Scholar]]. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés [44 Vecsernyés, P. (2003). Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras. J. Algebra 270:471–520. See also arXiv: 0111045v3 [math.QA] for an extended version.[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by Böhm et al. in [4 Böhm, G., Gómez-Torecillas, J., López-Centella, E. (2015). Weak multiplier bialgebras. Weak multiplier bialgebras. 367(12):8681–8872. See also arXiv: 1306.1466 [math.QA]. [Google Scholar]]. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. Weak multiplier Hopf algebras are introduced and studied in [40 Van Daele, A., Wang, S. (2015). Weak multiplier Hopf algebras I. The main theory. J. Ange. Math. (Crelles J.) 705:155–209, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle-2013-0053, July 2013. See also arXiv:1210.4395v1 [math.RA].[Web of Science ®] , [Google Scholar]]. Integrals on (regular) weak multiplier Hopf algebras are treated in [43 Van Daele, A., Wang, S. (2016). Weak multiplier Hopf algebras III. Integrals and duality. Preprint University of Leuven (Belgium) and Southeast University of Nanjing (China), See arXiv: 1701.04951.v3 [math.RA]. [Google Scholar]]. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting, see [13 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids I. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]] and [14 Kahng, B.-J., Van Daele, A. A class of C*-algebraic locally compact quantum groupoids II. Preprint Canisius College Buffalo (USA) and University of Leuven (Belgium). [Google Scholar]]. Our treatment of this material is motivated by the prospect of such a theory.     

Stratifying Systems via Relative Projective Modules

ABSTRACT

In this paper we point out that the “Process of standardization”, given in Dlab and Ringel (1992 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras . Repr. Theory and Related Topics, London Math. Soc. LNS 168 : 200 – 224 . [Google Scholar]), and also the “Comparison method” given in Platzeck and Reiten (2001 Platzeck , M. I. , Reiten , I. ( 2001 ). Modules of finite projective dimension for standardly stratified algebras . Comm. in Algebra 29 : 973 – 986 . [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) can be generalized. To do so, we introduce the concept of relative projective stratifying system and prove a result from which the Theorem 2 in Dlab and Ringel (1992 Dlab , V. , Ringel , C. M. ( 1992 ). The module theoretical approach to quasi-hereditary algebras . Repr. Theory and Related Topics, London Math. Soc. LNS 168 : 200 – 224 . [Google Scholar]) and Proposition 2.1 in Ringel (1991 Ringel , C. M. ( 1991 ). The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences . Math. Z. 208 : 209 – 223 .[Crossref], [Web of Science ®] , [Google Scholar]) follows.

     

Nil Lie p-algebras of slow growth

The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [27 Petrogradsky, V. M. (2006). Examples of self-iterating Lie algebras. J. Algebra 302(2):881–886.[Crossref], [Web of Science ®] , [Google Scholar]], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [39 Shestakov, I. P., Zelmanov, E. (2008). Some examples of nil Lie algebras. J. Eur. Math. Soc. (JEMS) 10(2):391–398.[Crossref], [Web of Science ®] , [Google Scholar]]. There are a few more examples of self-similar finitely generated restricted Lie algebras with a nil p-mapping, but, as a rule, that algebras have no clear basis and require technical computations. Now we construct a family L(Ξ) of 2-generated restricted Lie algebras of slow polynomial growth with a nil p-mapping, where a field of positive characteristic is arbitrary and Ξ an infinite tuple of positive integers. Namely, GKdimL(Ξ)≤2 for all such algebras. The algebras are constructed in terms of derivations of infinite divided power algebra Ω. We also study their associative hulls A?End(Ω). Algebras L and A are ?²-graded by a multidegree in the generators. If Ξ is periodic then L(Ξ) is self-similar. As a particular case, we construct a continuum subfamily of non-isomorphic nil restricted Lie algebras L(Ξ_α), α∈?⁺, with extremely slow growth. Namely, they have Gelfand-Kirillov dimension one but the growth is not linear. For this subfamily, the associative hulls A have Gelfand-Kirillov dimension two but the growth is not quadratic. The virtue of the present examples is that they have clear monomial bases.     

DIRECT SUM DECOMPOSITION OF THE PRODUCT OF PREINJECTIVE MODULES OVER RIGHT PURE SEMISIMPLE HEREDITARY RINGS

ABSTRACT

In his recent work, [1] Simson, D. 2000. An Artin Problem for Division Ring Extensions and the Pure Semisimplicity Conjecture, II. J. Algebra, 227: 670–705. [Crossref], [Web of Science ®] , [Google Scholar] and [2] Simson, D. 2001. On Small Right Pure Semisimple Rings and the Structure of their Auslander-Reiten Quiver. Communic. in Algebra, 29 in press[Web of Science ®] , [Google Scholar], on the pure semisimplicity conjecture Simson raised two problems about the structure of the direct sum decomposition of the direct product modulo the direct sum of indecomposable preinjective modules over right pure semisimple hereditary rings. The main goal of this paper is the proof of a theorem that resolves one of these problems and provides a partial answer to the other.     

Koszulity for graded skew PBW extensions

Pre-Koszul and Koszul algebras were defined by Priddy [15 Priddy, S. (1970). Koszul resolutions. Trans. Am. Math. Soc. 152:39–60.[Crossref] , [Google Scholar]]. There exist some relations between these algebras and the skew PBW extensions defined in [8 Gallego, C., Lezama, O. (2011). Gröbner bases for ideals of σ-PBW extensions. Comm. Algebra 39(1):50–75.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. In [24 Suárez, H., Reyes, A. (submitted for publications). Koszulity for skew PBW extensions over fields. [Google Scholar]] we gave conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that if R is a finite presented Koszul 𝕂-algebra then every graded skew PBW extension of R is Koszul.     

THE NICHOLS ZOELLER THEOREM FOR HOPF ALGEBRAS IN THE CATEGORY OF YETTER DRINFELD MODULES

In 1989 Nichols and Zoeller [NZ] Nichols, W. D. and Zoeller, M. B. 1989. A Hopf algebra freeness theorem. Amer. J. Math., 111: 381–385. [Crossref], [Web of Science ®] , [Google Scholar] showed that finite dimensional k-Hopf algebras are free over Hopf subalgebras. An analog result for Yetter Drinfeld Hopf algebras was not known. In this paper the existence of such a basis will be proved. Moreover the existence of a basis in a certain categorial sense cannot be expected.     

The representations of nested composition algebras

In this paper, we investigate the basis graph of the monoid algebra of a submonoid of the monoid of mappings from N = {1,…,n} to itself, defined by a nested sequence of compositions of N. Each such monoid is a left regular band (LRB), that is, a semigroup S satisfying x² = x and xyx = xy for all x,y∈S. This class is su?ciently rich that every path algebra of an acyclic quiver can be embedded in such a monoid algebra. The multiplication in the monoid algebra has a particularly simple quasi-multiplicative form, allowing definition over the integers. Combining this with a formula for Ext-groups for LRBs due to Margolis et al. [6 Margolis, S., Saliola, F., Steinberg, B. (2015). Combinatorial topology and the global dimension of algebras arising in combinatorics. J. Eur. Math Soc. 17(12):3037–3080.[Crossref], [Web of Science ®] , [Google Scholar]], we get a simple criterion for the nested composition algebras to be hereditary.     

A note on some structure theorems of modules in category 0

ABSTRACT

Using the local subgroup strategy of An and O'Brien (1997 An , J. , O'Brien , E. A. ( 1997 ). A local strategy to decide the Alperin and Dade conjectures . J. Alg. 189 : 34 – 57 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), An and O'Brien (1999 An , J. , O'Brien , E. A. ( 1999 ). The Alperin and Dade conjectures for the Fischer simple group Fi₂₃ . Internat. J. Alg. Comput. 9 : 621 – 670 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), we classify the radical subgroups and chains of the Fischer simple group Fi ₂₂ and verify the Alperin weight conjecture and the Uno reductive conjecture for this group; the latter is a refinement of the Dade reductive and Isaacs–Navarro conjectures.

     

Quotients of Koszul Algebras and 2-d-Determined Algebras

Vatne [13 Vatne , J. E. ( 2012 ). Quotients of Koszul algebras with almost linear resolution. Preprint, arXiv:1103.3572 . [Google Scholar]] and Green and Marcos [9 Green , E. L. , Marcos , E. N. (2011). d-Koszul algebras, 2-d-determined algebras and 2-d-Koszul algebras. J. Pure Appl. Algebra 215(4):439–449.[Crossref], [Web of Science ®] , [Google Scholar]] have independently studied the Koszul-like homological properties of graded algebras that have defining relations in degree 2 and exactly one other degree. We contrast these two approaches, answer two questions posed by Green and Marcos, and find conditions that imply the corresponding Yoneda algebras are generated in the lowest possible degrees.     

Table algebras with exactly one irreducible character whose degree and multiplicity are not equal

Commutative standard table algebras with exactly one multiplicity not equal to 1 are characterized by the wreath product of some special table algebras in [1 Antonou, A. (2015). Commutative standard table algebras with at most one nontrivial multiplicity. Commun. Algebra 43:2516–2523.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. A natural and much more general question is the characterization of standard table algebras (not necessarily commutative) with exactly one irreducible character whose degree and multiplicity are not equal and the degree is 1. We will give a characterization of such table algebras, including the main result of [1 Antonou, A. (2015). Commutative standard table algebras with at most one nontrivial multiplicity. Commun. Algebra 43:2516–2523.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] as a special case. Applications to association schemes are also discussed.     

Integral Domains in Which Each Ideal Is a W-Ideal

ABSTRACT

In this paper, we investigate integral domains in which each ideal is a w-ideal (i.e. the d- and w-operations are the same), called the DW-domains. In some sense this study is similar to that one given in Houston and Zafrullah (1988 Houston , E. , Zafrullah , M. (1988). Integral domains in which each t-ideal is divisorial. Michigan Math. J. 35:291–300. [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) [Houston, E., Zafrullah, M. (1988). Integral domains in which each t-ideal is divisorial. Michigan Math. J. 35:291–300.] for the TV-domains. We prove that a domain R is a DW-domain if and only if each maximal ideal of R is a w-ideal, and if R is a domain such that R _M is a DW-ideal for each maximal ideal M of R, then so is R, and the equivalence holds when R is v-coherent. We describe the w-operation on pull–backs in order to provide original examples.

     

On weakly clean rings

A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by Anderson and Camillo [2 Anderson, D. D., Camillo, V. P. (2002). Commutative rings whose elements are a sum of a unit and idempotent. Commun. Algebra 30(7):3327–3336.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and were extensively investigated by Ahn and Anderson [1 Ahn, M.-S., Anderson, D. D. (2006). Weakly clean rings and almost clean rings. Rocky Mountain J. Math. 36:783–798.[Crossref], [Web of Science ®] , [Google Scholar]], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings.