On the Unit Groups of Commutative Modular Group Algebras of KT-Groups

ABSTRACT

In this article, we prove that the inner projection of a projective curve with higher linear syzygies has also higher linear syzygies. Specifically, if a very ample line bundle ? on a smooth projective curve X satisfies property N _p for p  ≥  1 and H ¹ (? ^? 2) =  0 , then ?( ?  q ) satisfies property N _{p ? 1} for any point q  ∈  X . We also give simple proofs of well-known theorems about syzygies and raise some questions related to the line bundles of degree 2 g which do not satisfy property N ₁ .     

Minimal Idempotents of Twisted Group Algebras of Cyclic 2-Groups

For a field K of characteristic different from 2, we find the explicit form of the minimal idempotents of the twisted group algebra K_tg of a cyclic 2-group g over K.AMS Subject Classification (1991): primary 16S35, secondary 16U60.Partially supported by the Fund “NIMP’’ of Plovdiv University.     

Commutative Group Algebras of Cardinality {\cal N} 1

We let FG be the group algebra of an abelian group G over a field F with characteristic p. Also, we define G_p and S(FG) as the groups of all p-primary normed elements in G and FG, respectively. We prove that if G_p is Hausdorff and both F and G have cardinalities not exceeding ₁, then S(FG)/G_p is a direct sum of cyclics. Thus G_p is a direct factor of S(FG), and in particular G is a direct factor of the group of all normalized units V(FG), provided that the torsion part of G is a p-group. This answers a question posed by us in Hokkaido Math. J. (2000). Moreover we establish that if G is p-splitting, then any F-isomorphism of the group algebras FG and FH implies that H is p-splitting. We also show that if G is of power ₁ whose p-component G_p is a direct sum of torsion-complete groups and F has power p, then the F-isomorphism of FG and FH for any group H yields an isomorphism between G_p and H_p. In particular, when G is of power ₁ and is p-mixed of torsion-free rank 1 whose G_p is torsion-complete, we have G H. If F is in power p and G, with cardinality ₁, is a direct sum of p-local algebraically compact groups such that FG FH as F-algebras for some group H, then G H. These statements extend results due to Beers-Richman-Walker (1983), and also partially solve a well-known question raised by May in 1979.     

关于扭Hopf代数的一些注记

In this paper,we get some properties of the antipode of a twisted Hopf algebra.We proved that the graded global dimension of a twisted Hopf algebra coincides with the graded projective dimension of its trivial module k,which is also equal to the projective dimension of k.     

Twisted calculus

A twisted ring is a ring endowed with a family of endomorphisms satisfying certain relations. One may then consider the notions of twisted module and twisted differential module. We study them and show that, under some general hypothesis, the categories of twisted modules and integrable twisted differential modules are equivalent. As particular cases, one recovers classical results from the theory of finite difference equations or q-difference equations.     

On a Filtered Multiplicative Bases of Group Algebras II

We give an explicit list of all p-groups G with a cyclic subgroup of index p ², such that the group algebra KG over the field K of characteristic p has a filtered multiplicative K-basis. We also prove that such a K-basis does not exist for the group algebra KG, in the case when G is either a non-Abelian powerful p-group or a two generated p-group (p2) with a central cyclic commutator subgroup. This paper is a continuation of the paper which appeared in Arch. Math. (Basel) 74 (2000), 217–285.     

Twisted Group Algebras, Normal Subgroups and Derived Equivalences

We study Rickard equivalences between p-blocks of twisted group algebras and their local structure, in connection with Dade's conjectures. We prove that an extended form of Broué's conjecture implies Dade's Inductive Conjecture in the Abelian defect group case; this is a consequence of the fact that Rickard equivalences induced by complexes of graded bimodules preserve the relevant Clifford theoretical invariants. As an application, we show that these conjectures hold for p-extensions of blocks with cyclic defect groups.     

Isomorphism of Modular Group Algebras of p-Mixed Abelian Groups

Let A be a torsion-free abelian group and F a free subgroup of A. We prove that if A/F is a reduced p-group and A/(F + C) is reduced for every p-pure subgroup C of A, then A is free.

Let KG be the group algebra of an abelian group G over a field K of prime characteristic p. Denote by S(KG) the p-component of the group V(KG) of normalized units of KG (of augmentation 1). Let H be an arbitrary group and KH ? KG as K-algebras. We prove the following. First, assume that G is a splitting group, the p-component G _p of G is simply presented, and the field K is perfect. Then H _p  ? G _p . If, in addition, G is p-mixed, then G _p is a direct factor of S(KG), and G is a direct factor of V(KG), each with the same simply presented complement. Secondly, we introduce a class of special p-mixed abelian groups and prove that, if G belong to this class, then any group basis of the group algebra KG splits. Besides, H is p-mixed and splits. Thirdly, if G is a special p-mixed abelian group and G _p is a reduced totally projective p-group, then H ? G. These results correct some essential inaccuracies and incompleteness in the proofs of results in this direction of Danchev [3-8 Danchev , P. V. ( 1998 ). Isomorphism of commutative group algebras of mixed splitting groups . Compt. Rend. Acad. Bulg. Sci. 51 : 13 – 16 . Danchev , P. V. ( 2000 ). Isomorphism of modular group algebras of totally projective abelian groups . Communications in Algebra 28 : 2521 – 2531 . Danchev , P. V. ( 2001 ). On a question of W. L. May concerning the isomorphism of modular group algebras . Communications in Algebra 29 : 1953 – 1958 . Danchev , P. V. ( 2001 ). Normed units in Abelian group rings . Glasg. Math. J. 43 : 365 – 373 . Danchev , P. V. ( 2002 ). Invariants for group algebras of splitting abelian groups with simply presented components . Compt. Rend. Acad. Bulg. Sci. 55 : 5 – 8 . Danchev , P. V. ( 2004 ). A note on the isomorphic modular group algebras of abelian groups with simply presented p-components . Compt. Rend. Acad. Bulg. Sci. 57 : 13 – 14 . ].     

Module Correspondences for Twisted Group Algebras

Abstract

Let G and A be finite groups such that (|G|, |A|) = 1. Let K be an algebraically closed field with Char K = 0. Denote by K ^α G the twisted group algebra of G over K with factor set α. In this paper we prove that if A acts homogeneously on K ^α G, then there exists an action of A on G, and there is a one-to-one correspondence between the set of A-invariant irreducible K ^α G-modules and the set of irreducible K ^α C _G (A)-modules.     

Algebras of p-Groups of Class 2 Over the Rationals

In this work we analyze p-groups of class 2 G and H, with same rational group algebras. We prove that if QG = QH, then their commutators are equal and the centers, 𝒵(G) and 𝒵(H), have their orders preserved. We apply our results to Frattini Central p-groups, and we present an example of two groups of order p ⁷, with no isomorphic centers and different central cyclic components intersecting the cyclic components of the respective commutators groups.     

Laura Skew Group Algebras

We prove that if A is an artin algebra, G is a finite group acting on A such that |G| is invertible in A, and R = A[G]^b is a basic algebra associated with the skew group algebra, then A is left supported (or right supported, or laura, or left glued, or right glued, or weakly shod, or shod) if and only if so is R.     

三维半群代数

首先给出代数闭域上三维半群代数的幂等元集和Jacobson根,并且刻画了三维半群代数的同构类.通过计算箭图,研究了三维代数的表示型.进一步,证明一个三维(或者二维)半群代数是胞腔的,当且仅当它是交换的.作为推论,得到一个左零带所对应的半群代数是胞腔的,当且仅当这个左零带是一个半格.     

On Strongly Associative Group Algebras

Given a partial action α of a group G on the group algebra FH, where H is a finite group and F is a field whose characteristic p divides the order of H, we investigate the associativity question of the partial crossed product FH*_α G. If FH*_α G is associative for any G and any α, then FH is called “strongly associative.” Using a result of Dokuchaev and Exel (2005 Dokuchaev , M. , Exel , R. ( 2005 ). Associativity of crossed products by partial actions, enveloping actions and partial representations . Trans. Amer. Math. Soc. 357 : 1931 – 1952 .[Crossref], [Web of Science ®] , [Google Scholar]) we characterize the strongly associative modular group algebras FH for several classes of groups H.     

On the Dimensions of Commutative Subalgebras and Subgroups of Nilpotent Algebras and Lie Groups of Class 2

We obtain the functions that bound the dimensions of finite dimensional nilpotent associative or Lie algebras of class 2 over an algebraically closed field in terms of the dimensions of their commutative subalgebras. As a result, we also compute a similar function for complex nilpotent Lie groups of class 2.     

Twisted Semigroup Crossed Products and Twisted Toeplitz Algebras of Ordered Groups

Let г^＋ be the positive cone of a totally ordered abelian group г, and σa cocycle in г. We study the twisted crossed products by actions of г＋ as endomorphisms of C^＊-algebras, and use this to generalize the theorem of Ji.     

Homological Properties of Modules Over Group Algebras

Let G be a locally compact group, and let L¹ (G) be the Banachalgebra which is the group algebra of G. We consider a varietyof Banach left L¹ (G)-modules over L¹ (G), and seek to determineconditions on G that determine when these modules are eitherprojective or injective or flat in the category. The answerstypically involve G being compact or discrete or amenable. Forexample, in the case where G is discrete and 1 < p < ,we find that the module ^p (G) is injective whenever G is amenable,and that, if it is amenable, then G is ‘pseudo-amenable’,a property very close to that of amenability. 2000 MathematicsSubject Classification 46H25, 43A20.     

Real Commutative Division Algebras

The category of all two-dimensional real commutative division algebras is shown to split into two full subcategories. One of them is equivalent to the category of the natural action of the cyclic group of order 2 on the open right half plane . The other one is equivalent to the category of the natural action of the dihedral group of order 6 on the set of all ellipses in which are centered at the origin and have reciprocal axis lengths. Cross-sections for the orbit sets of these group actions are easily described. Together with they classify all real commutative division algebras up to isomorphism. Moreover we describe all morphisms between the objects in this classifying set, thus obtaining a complete picture of the category of all real commutative division algebras, up to equivalence. This supplements earlier contributions of Kantor and Solodovnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras, Nauka, Moscow, 1973; Benkart et al., Hadronic J., 4: 497–529, 1981; and Althoen and Kugler, Amer. Math. Monthly, 90: 625–635, 1983, who achieved partial results on the classification of the real commutative division algebras. Dedicated to Claus Michael Ringel on the occasion of his 60th birthday.     

Leavitt Path Algebras as Partial Skew Group Rings

We realize Leavitt path algebras as partial skew group rings and give new proofs, based on partial skew group ring theory of the Cuntz–Krieger uniqueness theorem and simplicity criteria for Leavitt path algebras.