THE AUSLANDER–REITEN SEQUENCES ENDING AT GABRIEL–ROITER FACTOR MODULES OVER TAME HEREDITARY ALGEBRAS

2007 ◽  
Vol 06 (06) ◽  
pp. 951-963 ◽  
Author(s):  
BO CHEN
Keyword(s):  
Closed Field ◽  
Hereditary Algebra ◽  
Algebraically Closed Field ◽  
Type Formula ◽  
Finite Dimensional ◽  
Hereditary Algebras ◽  
Almost All ◽  
Euclidean Type ◽  
Tame Hereditary Algebras

Let Λ = kQ be a finite dimensional hereditary algebra over an algebraically closed field k with Q a quiver of Euclidean type [Formula: see text], [Formula: see text], or [Formula: see text]. We study the Auslander–Reiten sequences terminating at Gabriel–Roiter factor modules and show that for almost all but finitely many Gabriel–Roiter factor modules, the Auslander–Reiten sequences have indecomposable middle terms.

COMPLEXITY OF TRIVIAL EXTENSIONS OF ITERATED TILTED ALGEBRAS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250067 ◽  
Author(s):  
MARJU PURIN
Keyword(s):  
Representation Type ◽  
Trivial Extension ◽  
Closed Field ◽  
Hereditary Algebra ◽  
Algebraically Closed Field ◽  
Tilted Algebra ◽  
Finite Dimensional ◽  
Tilted Algebras

We study the complexity of a family of finite-dimensional self-injective k-algebras where k is an algebraically closed field. More precisely, let T be the trivial extension of an iterated tilted algebra of type H. We prove that modules over the trivial extension T all have complexities either 0, 1, 2 or infinity, depending on the representation type of the hereditary algebra H.

scholarly journals TILTING MUTATION FOR m-REPLICATED ALGEBRAS

2011 ◽  
Vol 10 (04) ◽  
pp. 649-664 ◽  
Author(s):  
HONGBO LV ◽  
SHUNHUA ZHANG
Keyword(s):  
Closed Field ◽  
Cluster Category ◽  
Tilting Module ◽  
Hereditary Algebra ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Left Part ◽  
The Relationship

Let A be a finite-dimensional hereditary algebra over an algebraically closed field k, A(m) be the m-replicated algebra of A and [Formula: see text] be the m-cluster category of A. In this paper, we introduce the notion of mutation team in mod A(m), and prove that each faithful almost complete tilting module over A(m) has a mutation team by showing that the sequence of the complements satisfies the properties of the mutation team. We also prove that for each partial mutation team in the m-left part of mod A(m), there exists a faithful almost complete tilting module having the partial mutation team as the set of indecomposable complements. As an application, we prove that m-cluster mutation in [Formula: see text] can be realized as tilting mutation in mod A(m), and we also give the relationship between connecting sequences in mod A(m) and higher AR-angles in the m-cluster category [Formula: see text].

scholarly journals Radically filtered quasi-hereditary algebras and rigidity of tilting modules

2017 ◽  
Vol 163 (2) ◽  
pp. 265-288
Author(s):  
AMIT HAZI
Keyword(s):  
Closed Field ◽  
Tilting Module ◽  
Tilting Modules ◽  
Hereditary Algebra ◽  
Algebraically Closed Field ◽  
Hereditary Algebras ◽  
Composition Factors

AbstractLetAbe a quasi-hereditary algebra. We prove that in many cases, a tilting module is rigid (i.e. has identical radical and socle series) if it does not have certain subquotients whose composition factors extend more than one layer in the radical series or the socle series. We apply this theorem to show that the restricted tilting modules forSL4(K) are rigid, whereKis an algebraically closed field of characteristicp≥ 5.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

scholarly journals Involutions on finite-dimensional algebras over real closed fields

2004 ◽  
Vol 77 (1) ◽  
pp. 123-128 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Characteristic Zero ◽  
Real Closed Field ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Real Closed Fields ◽  
Finite Dimensional ◽  
Closed Fields ◽  
Finite Dimensional Algebras

AbstractIt is shown that the following conditions on a finite-dimensional algebra A over a real closed field or an algebraically closed field of characteristic zero are equivalent: (i) A admits a special involution, in the sense of Easdown and Munn, (ii) A admits a proper involution, (iii) A is semisimple.

Second Cohomology of the Modular Lie Superalgebra of Cartan Type K

2009 ◽  
Vol 16 (02) ◽  
pp. 309-324 ◽  
Author(s):  
Wenjuan Xie ◽  
Yongzheng Zhang
Keyword(s):  
Cohomology Group ◽  
Lie Superalgebra ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cartan Type ◽  
Modular Lie Superalgebra ◽  
Finite Dimensional ◽  
Type K

Let 𝔽 be an algebraically closed field and char 𝔽 = p > 3. In this paper, we determine the second cohomology group of the finite-dimensional Contact superalgebra K(m,n,t).

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

scholarly journals Multiplicities in the Tensor Product of Finite-Dimensional Representations of Discrete Groups

1978 ◽  
Vol 21 (1) ◽  
pp. 17-19
Author(s):  
Dragomir Ž. Djoković
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Unitary Representations ◽  
Vector Spaces ◽  
Closed Field ◽  
Discrete Groups ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

Let G be a group and ρ and σ two irreducible unitary representations of G in complex Hilbert spaces and assume that dimp ρ= n < ∞. D. Poguntke [2] proved that is a sum of at most n2 irreducible subrepresentations. The case when dim a is also finite he attributed to R. Howe.We shall prove analogous results for arbitrary finite-dimensional representations, not necessarily unitary. Thus let F be an algebraically closed field of characteristic 0. We shall use the language of modules and we postulate that allour modules are finite-dimensional as F-vector spaces. The field F itself will be considered as a trivial G-module.

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