scholarly journals THE UNIVERSAL COVER OF A MONOMIAL TRIANGULAR ALGEBRA WITHOUT MULTIPLE ARROWS

2008 ◽  
Vol 07 (04) ◽  
pp. 443-469 ◽  
Author(s):  
PATRICK LE MEUR
Keyword(s):  
Fundamental Group ◽  
Universal Cover ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Algebraically Closed Field ◽  
Triangular Algebra ◽  
Underlying Space ◽  
Finite Dimensional ◽  
Ordinary Quiver

Let A be a basic connected finite dimensional algebra over an algebraically closed field k. Assuming that A is monomial and that the ordinary quiver Q of A has no oriented cycle and no multiple arrows, we prove that A admits a universal cover with group the fundamental group of the underlying space of Q.

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.

Invariants of finite-dimensional algebras recognized by the semigroup of conjugacy classes of left ideals

2017 ◽  
Vol 16 (10) ◽  
pp. 1750182
Author(s):  
Arkadiusz Mȩcel ◽  
Jan Okniński
Keyword(s):  
Jacobson Radical ◽  
Semigroup Algebra ◽  
Conjugacy Classes ◽  
Maximal Subgroups ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Ordinary Quiver

We study the semigroup structure on the set [Formula: see text] of conjugacy classes of left ideals of a finite-dimensional algebra [Formula: see text] over an algebraically closed field [Formula: see text], equipped with the natural multiplication inherited from [Formula: see text], and the structure of the contracted semigroup algebra [Formula: see text]. It is shown that [Formula: see text] has a finite chain of ideals with either nilpotent or completely [Formula: see text]-simple factors with trivial maximal subgroups, so in particular it is locally finite. The ordinary quiver [Formula: see text] of [Formula: see text] is proved to be a subquiver of [Formula: see text], if [Formula: see text] is finite. Moreover, in this case, the structure of [Formula: see text] determines, up to isomorphism, the structure of the algebra [Formula: see text] modulo its Jacobson radical. Combining these results we show that if the semigroup [Formula: see text] is finite, then it determines the structure of any (not necessarily basic) triangular algebra [Formula: see text] which admits a normed presentation.

scholarly journals τ-Complexity and Tilting Modules

2016 ◽  
Vol 2016 ◽  
pp. 1-4 ◽  
Author(s):  
Lijing Zheng ◽  
Chonghui Huang ◽  
Qianhong Wan
Keyword(s):  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Finite Dimensional

Let A be a finite dimensional algebra over an algebraic closed field k. In this note, we will show that if T is a separating and splitting tilting A-module, then τ-complexities of A and B are equal, where B=EndA(T).

scholarly journals Torsion Pairs and Ringel Duality for Schur Algebras

2021 ◽  
Author(s):  
Karin Erdmann ◽  
Stacey Law
Keyword(s):  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Schur Algebras ◽  
Morita Equivalent ◽  
Finite Dimensional ◽  
Endomorphism Algebras ◽  
Hereditary Algebras ◽  
Torsion Pairs ◽  
Functorial Approach

AbstractLet A be a finite-dimensional algebra over an algebraically closed field. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective A–modules P into those of the torsion submodules of P. As an application, we show that blocks of both the classical and quantum Schur algebras S(2,r) and Sq(2,r) in characteristic p > 0 are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain 2pk simple modules for some k.

Separating Splitting Tilting Modules and Hereditary Algebras

1987 ◽  
Vol 30 (2) ◽  
pp. 177-181 ◽  
Author(s):  
Ibrahim Assem
Keyword(s):  
Exact Sequence ◽  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Finitely Generated ◽  
Tilting Module ◽  
Tilting Modules ◽  
Dimensional Algebra ◽  
Direct Sums ◽  
Finite Dimensional ◽  
Hereditary Algebras

AbstractLet A be a finite-dimensional algebra over an algebraically closed field. By module is meant a finitely generated right module. A module T^ is called a tilting module if and there exists an exact sequence 0 → A^ → T' → T" → 0 with T'. T" direct sums of summands of T. Let B = End T^·T^ is called separating (respectively, splitting) if every indecomposable A-module M (respectively, B-module N) is such that either Hom^(T,M) = 0 or (respectively, N ⊗ T = 0 or . We prove that A is hereditary provided the quiver of A has no oriented cycles and every separating tilting module is splitting.

Degrees in Auslander–Reiten components with almost split sequences of at most two middle termss

2015 ◽  
Vol 14 (07) ◽  
pp. 1550106 ◽  
Author(s):  
Claudia Chaio
Keyword(s):  
Closed Field ◽  
Finite Dimensional Algebra ◽  
Finite Degree ◽  
Artin Algebra ◽  
Dimensional Algebra ◽  
Finite Representation ◽  
Finite Representation Type ◽  
Finite Dimensional ◽  
Almost Split Sequences ◽  
Irreducible Morphisms

We consider A to be an artin algebra. We study the degrees of irreducible morphisms between modules in Auslander–Reiten components Γ having only almost split sequences with at most two indecomposable middle terms, that is, α(Γ) ≤ 2. We prove that if f : X → Y is an irreducible epimorphism of finite left degree with X or Y indecomposable, then there exists a module Z ∈ Γ and a morphism φ ∈ ℜn(Z, X)\ℜn+1(Z, X) for some positive integer n such that fφ = 0. In particular, for such components if A is a finite dimensional algebra over an algebraically closed field and f = (f1, f2)t : X → Y1 ⊕ Y2 is an irreducible epimorphism of finite left degree then we show that dl(f) = dl(f1) + dl(f2). We also characterize the artin algebras of finite representation type with α(ΓA) ≤ 2 in terms of a finite number of irreducible morphisms with finite degree.

DEGENERATION-LIKE ORDERS IN TRIANGULATED CATEGORIES

2005 ◽  
Vol 04 (05) ◽  
pp. 587-597 ◽  
Author(s):  
BERNT TORE JENSEN ◽  
XIUPING SU ◽  
ALEXANDER ZIMMERMANN
Keyword(s):  
Partial Order ◽  
Natural Generalization ◽  
Derived Categories ◽  
Derived Category ◽  
Closed Field ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional

In an earlier paper we defined a relation ≤Δ between objects of the derived category of bounded complexes of modules over a finite dimensional algebra over an algebraically closed field. This relation was shown to be equivalent to the topologically defined degeneration order in a certain space [Formula: see text] for derived categories. This space was defined as a natural generalization of varieties for modules. We remark that this relation ≤Δ is defined for any triangulated category and show that under some finiteness assumptions on the triangulated category ≤Δ is always a partial order.

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 Duality and socle generators for residual intersections

2019 ◽  
Vol 2019 (756) ◽  
pp. 183-226 ◽  
Author(s):  
David Eisenbud ◽  
Bernd Ulrich
Keyword(s):  
Gorenstein Ring ◽  
Jacobian Determinant ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Duality Results ◽  
Finite Dimensional ◽  
Algebra Over A Field ◽  
Residue Theory

AbstractWe prove duality results for residual intersections that unify and complete results of van Straten, Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt.Suppose that I is an ideal of codimension g in a Gorenstein ring, and {J\subset I} is an ideal with {s=g+t} generators such that {K:=J:I} has codimension s. Let {{\overline{I}}} be the image of I in {{\overline{R}}:=R/K}.In the first part of the paper we prove, among other things, that under suitable hypotheses on I, the truncated Rees ring {{\overline{R}}\oplus{\overline{I}}\oplus\cdots\oplus{\overline{I}}{}^{t+1}} is a Gorenstein ring, and that the modules {{\overline{I}}{}^{u}} and {{\overline{I}}{}^{t+1-u}} are dual to one another via the multiplication pairing into {{{\overline{I}}{}^{t+1}}\cong{\omega_{\overline{R}}}}.In the second part of the paper we study the analogue of residue theory, and prove that, when {R/K} is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of {I^{t+1}/JI^{t}\cong{\omega_{R/K}}} is generated by a Jacobian determinant.

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