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Showing papers in "Communications in Algebra in 1996"


Journal ArticleDOI
TL;DR: Hopf bigalois extensions as mentioned in this paper have been proposed for algebraic programs. But they do not cover the Hopf bigaleis extensions. Communications in Algebra: Vol. 24, No. 12, No., 12, pp. 3797-3825
Abstract: (1996). Hopf bigalois extensions. Communications in Algebra: Vol. 24, No. 12, pp. 3797-3825.

222 citations


Journal ArticleDOI
TL;DR: In this article, it is shown how to get rid of this oddness assumption as far as the determination of irreducible characters for quantum groups goes, see also [Kan95].
Abstract: Apart from being of interest in its own right, the representation theory for quantum groups at roots of unity enters into Lusztig’s programme (see e.g. [Lus94]) for determining the irreducible characters of semi-simple algebraic groups in characteristic p > 0. In [AJS94] this connection plays a key role. There our assumption is that the root of unity has odd order (bigger than the Coxeter number). Using the recent work of Kashiwara - Tanisaki, [KT95a] and [KT95b] together with the Kazhdan-Lusztig equivalence of categories, [KL93], [KL94] and [Lus94] it is possible to get rid of this oddness assumption as far as the determination of irreducible characters for quantum groups goes, see also [Kan95]. In this note we show how we can do this much more directly staying inside the theory for quantum groups. To be more specific we prove the following result (see below for more details and notations).

187 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the number of projective complex characters of a finite simple group of Lie type over a finite field of order q and d(G(q)) is the minimal degree of faithful projective representations of G(q) is the same as that of a classical group.
Abstract: Let G(q) be a finite simple group of Lie type over a finite field of order q and d(G(q)) the minimal degree of faithful projective complex representations of G(q). For the case G(q) is a classical group we deter-mine the number of projective complex characters of G(q) of degree d(G(q)). In several cases we also determine the projective complex characters of the second and the third lowest degrees. As a corollary of these results we deduce the classification of quasi-simple irreducible complex linear groups of degree at most 2r r a prime divisor of the group order.

169 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps including a homology isomorphism.
Abstract: We study properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps including a homology isomorphism. ...

155 citations


Journal ArticleDOI
TL;DR: In this article, the authors obtained similar results for Lie algebras with order multipliers for groups of order pj whose multiplier has order, and for groups with order multiplier.
Abstract: In recent work, groups of order pj whose multiplier has order , have been classified when t(G) = 0 or 1 in [2] and when t(G) = 2 in [6]. It is the purpose of this paper to obtain similar results for Lie algebras.

124 citations


Journal ArticleDOI
TL;DR: In this paper, a characterization of the borel-like subalgebras of quantum enveloping algesas is presented, where the subalgebra is characterized as a subgraph.
Abstract: (1996). A characterization of the borel-like subalgebras of quantum enveloping algebras. Communications in Algebra: Vol. 24, No. 9, pp. 2811-2823.

108 citations



Journal ArticleDOI
TL;DR: In this paper, the authors generalize the characterization of Gorenstein flat modules over GNNs to n − FC rings (coherent rings with finite sdf−FP−injective dimension).
Abstract: In this paper, we generalize the characterization of Gorenstein flat modules over Gorenstein rings to n − FC rings (coherent rings with finite sdf−FP−injective dimension), and characterize n − FC rings in terms of Gorenstein flat and projective modules.

106 citations


Journal ArticleDOI
TL;DR: In this paper, the authors define a generalization of a root system as a set of vectors in a vector space with some symmetry property, and classify irreducible generalized root systems.
Abstract: We define a generalization of a root system as a set of vectors in a vector space with some symmetry property. The main difference with the usual root systems is the existence of isotropic roots. We classify irreducible generalized root systems. As follows from our classification all such root systems are root systems of contragredient Lie superalgebras which were classified by V.Kac in 1977.

100 citations


Journal ArticleDOI
TL;DR: For finite dimensional Lie algebras, covers are isomorphic as mentioned in this paper, and it is shown that covers need not be isomorphic to find all the extensions of a finite group.
Abstract: In dealing with the central extensions of a finite group G one finds that although covers need not be isomorphic, for each such H there exists a cover for which H is a. homomorphic image [1]. For finite dimensional Lie algebras, covers are isomorphic. We shall show that the second property also holds for Lie algebras. Thus to find all such extensions one needs to compute the cover and consider ideals contained in the multiplier (kernel of the homomorphism). Several examples are constructed. Our Lie algebras are taken over a field.

90 citations


Journal ArticleDOI
Nanqing Ding1
TL;DR: In this paper, it was shown that if R is a left coherent ring, then the weak global dimension w D(R) = 2 if and only if every (n − 2)th F-cosyzygy of a finitely presented right R-module has a flat envelope with the unique mapping property.
Abstract: We prove that (a) if R is a left coherent ring, then the weak global dimension w D(R) = 2) if and only if every (n – 2)th F–cosyzygy of a finitely presented right R–module has a flat envelope with the unique mapping property; (b) if R is a left coherent and right perfect ring, then the right global dimension rD(R) = 2) if and only if every (n – 2)th P–cosyzygy of a right R–module has a projective envelope with the unique mapping property; (c) if R is a commutative ring, then R is π—coherent (resp. coherent) and the exactness of 0 -> K -> F0 -> F1 with Fo and F1 (finitely) projective and K finitely generated implies the projectivity of K if and only if every finitely generated (resp, finitely presented) R–module has a (finitely) projective envelope with the unique mapping property.

Journal ArticleDOI
TL;DR: In this paper, the fundamental groups of A and the first Hochschild cohomology H 1 (A A) have been studied and relations between them and A A have been established.
Abstract: A finite dimensional algebra A (over an algebraically closed field) is called triangular if its ordinary quiver has no oriented cycles. To each presentation (Q I) of A is attached a fundamental group π1(Q I), and A is called simply connected if π1(Q I) is trivial for every presentation of A. In this paper, we provide tools for computations with the fundamental groups, as well as criteria for simple connectedness. We find relations between the fundamental groups of A and the first Hochschild cohomology H 1 (A A).


Journal ArticleDOI
TL;DR: In this paper, the following form of the Clemens conjecture in low degree was proved: if d ≤ 9 and F is a general quintic threefold in P 4, then the Hilbert scheme of rational, singular, reduced and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (−1) ⊕ (−1), and in p 4 with maximal rank.
Abstract: . We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P 4. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (−1) ⊕ (−1), and in P 4 with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.

Journal ArticleDOI
TL;DR: In this article, some irreducible graded modules with 1-dimensional homogeneous spaces over the Virasoro-like algebra and its q-analogs are constructed.
Abstract: In this paper, some irreducible graded modules with 1-dimensional homogeneous spaces over the Virasoro-like algebra and its q-analogs are constructed. The unitarizability of these modules, and the conditions under which two of such irreducible graded modules are ismorphic are determined. Some other kinds of irreducible graded modules with 1-dimensional homogeneous spaces over the Virasorolike algebra and its q-analogs are also given.



Journal ArticleDOI
Akira Masuoka1
TL;DR: In this article, some further classification results on semisimple hopf algebras are presented. But these results are based on the same model as in this paper.
Abstract: (1996). Some further classification results on semisimple hopf algebras. Communications in Algebra: Vol. 24, No. 1, pp. 307-329.

Journal ArticleDOI
TL;DR: Agarwal et al. as mentioned in this paper showed that exchange rings satisfying s-comparability are separative, thus answering the questions affirmatively in the s comparable case, and showed that this condition still implies separativity for exchange rings.
Abstract: There are several long-standing open problems which ask whether regular rings, and C∗-algebras of real rank zero, satisfy certain module cancellation properties. Ara, Goodearl, O'Meara and Pardo recently observed that both types of rings are exchange rings, and showed that separative exchange rings have these good cancellation properties, thus answering the questions affirmatively in the separative case. In this article, we prove that, for any positive integer s, exchange rings satisfying s-comparability are separative, thus answering the questions affirmatively in the s-comparable case. We also introduce the weaker, more technical, notion of generalized s-comparability, and show that this condition still implies separativity for exchange rings. On restricting to directly finite regular rings, we recover results of Ara, O'Meara and Tyukavkin.


Journal ArticleDOI
TL;DR: In this paper, the structure of the Hochschild cohomology ring of the group algebra k G was determined and it was shown that for any finite group G the Krull dimension of H H *(k G) equals the p-rank of G.
Abstract: Let k be a field of characteristic P >0 and let G be a finite abelian group. We determine the structure of the Hochschild cohomology ring of the group algebra k G. Moreover, we prove that for any finite group G the Krull dimension of H H *(k G) equals the p-rank of G.

Journal ArticleDOI
TL;DR: In this article, the authors discuss the properties of solvable complete Lie algebras, describe the structures of the root spaces, and prove that solvable Lie algesas of maximal rank are com-complete.
Abstract: In this paper, we will discuss the properties of solvable complete Lie algebra, describe the structures of the root spaces of solvable complete Lie algebra, prove that solvable Lie algebras of maximal rank are com-plete, and construct some new complete Lie algebras from Kac-Moody algebras.

Journal ArticleDOI
TL;DR: In this paper it was shown that if d is a nonzero derivation of a 2-torsion-free 3-prime near-ring N and an element a ∊ N is such that axd = xda for all x ∊ n, then a is a central element, then N is a commutative ring.
Abstract: The analog of Posner's theorem on the composition of two derivations in prime rings is proved for 3-prime near-rings. It is shown that if d is a nonzero derivation of a 2-torsionfree 3-prime near-ring N and an element a ∊ N is such that axd = xda for all x ∊ N, then a is a central element. As a consequence it is shown that if d\ and d2 are nonzero derivations of a 2-torsionfree 3-prime near-ring N and xd1yd2 = yd2xd1 for all x, y ∊ N, then N is a commutative ring. Thus two theorems of Herstein are generalized

Journal ArticleDOI
TL;DR: A formal deformation of a Leibniz algebra A over a commutative ring k, consisting of a formal series extending the Leibiz bracket of A, defined by the authors in this article, is studied from the point of view of algebraic varieties.
Abstract: A formal deformation of a Leibniz algebra A over a commutative ring k, consists of a formal series extending the Leibniz bracket of A, defined by fo We show that if f is a 2-cocycle for the cohomology of A with coefficients in its adjoint representation, a sufficient condition for the existence of F with f1=f, is the vanishing of the third cohomology group of A with coefficients in its adjoint representation. In the last section, we study deformations from the point of view of algebraic varieties.

Journal ArticleDOI
TL;DR: In this paper, a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where G has at most two fusion classes of order i for every i (23 examples), and any two elements of G of the same order are fused or inversenfused.
Abstract: Elements a,b of a group G are said to be fused if a = bσ and to be inverse-fused if a =(b-1)σ for some σ ϵ Aut(G). The fusion class of a ϵ G is the set {aσ | σ ϵ Aut(G)}, and it is called a fusion class of order i if a has order iThis paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where: (i) G has at most two fusion classes of order i for every i (23 examples); and (ii) any two elements of G of the same order are fused or inversenfused. The examples in case (ii) are: A5, A6,L2(7),L2(8), L3(4), Sz(8), M11 and M23An application is given concerning isomorphisms of Cay ley graphs.

Journal ArticleDOI
TL;DR: In this article, it was shown that for an odd prime p, every subgroup of G of order p lies in the center of G, then G is p-nilpotent.
Abstract: 1 Introduction and Preliminaries All groups considered in this note will be finite. Recall that a minimal subgroup of a finite group is a subgroup of prime order. Many authors have investigated the structure of a finite group G, under the assumption that all minimal subgroups of G are well-situated in the group. Ito [7;III, 5.3] proved that if G is a group of odd order and all minimal subgroups of G lie in the center of G, then G is nilpotent. An extension of Ito's result is the following statement [7;IV,p.435]: If for an odd prime p, every subgroup of G of order p lies in the center of G, then G is p-nilpotent. If all element of G of orders 2 and 4 lie in the center of G, then G is 2-nilpotent. Buckley [4] proved that if G is a group of odd order and all minimal subgroups of G are normal in G, then G is supersoluble. Later Shaalan [8] proved that if G is a finite group and every subgroup of G of prime order or order 4 is π-quasinormal in G, then G is supersoluble. Recall that a subgroup H of a group G is...


Journal ArticleDOI
TL;DR: In this article, the authors provide several new criteria for a ring to be a complete matrix ring, and their relative strengths are indicated by calculating the structures they impose on universal algebras.
Abstract: This paper provides several new criteria for a ring to be a complete matrix ring. Some applications demonstrate their efficacy; and their relative strengths are indicated by calculating the structures they impose on universal algebras.

Journal ArticleDOI
TL;DR: In this paper, the Filter dimension of algebras and modules is defined as a simplicity criterion of generalized weyl algesbras, which is a simplification criterion of the filter dimension of the weyl dimension.
Abstract: (1996). Filter dimension of algebras and modules, a simplicity criterion of generalized weyl algebras. Communications in Algebra: Vol. 24, No. 6, pp. 1971-1992.

Journal ArticleDOI
TL;DR: In this paper, crossed coproducts and cleft coextensions are used to define cross-coproduct coextension in algebraic algebras.
Abstract: (1996). Crossed coproducts and cleft coextensions. Communications in Algebra: Vol. 24, No. 4, pp. 1229-1243.