scholarly journals Counting the Number of Elements in the Mutation Classes of $\tilde A_n-$Quivers

10.37236/585 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Janine Bastian ◽  
Thomas Prellberg ◽  
Martin Rubey ◽  
Christian Stump
Keyword(s):  
Equivalence Classes ◽  
Alternative Proof ◽  
Derived Equivalence ◽  
Tilted Algebras ◽  
Dynkin Type ◽  
Explicit Formulae

In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type $\tilde A_n$ in the derived equivalence classes. In particular, we obtain the number of elements in the mutation classes of quivers of type $\tilde A_n$. As a by-product, this provides an alternative proof for the number of quivers mutation equivalent to a quiver of Dynkin type $D_n$ which was first determined by Buan and Torkildsen.

scholarly journals Derived Equivalence Classification of the Cluster-Tilted Algebras of Dynkin Type E

2011 ◽  
Vol 16 (2) ◽  
pp. 527-551 ◽  
Author(s):  
Janine Bastian ◽  
Thorsten Holm ◽  
Sefi Ladkani
Keyword(s):  
Derived Equivalence ◽  
Tilted Algebras ◽  
Dynkin Type

scholarly journals Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type D

2014 ◽  
Vol 410 ◽  
pp. 277-332 ◽  
Author(s):  
Janine Bastian ◽  
Thorsten Holm ◽  
Sefi Ladkani
Keyword(s):  
Derived Equivalence ◽  
Type D ◽  
Tilted Algebras ◽  
Dynkin Type

scholarly journals On the finiteness of the derived equivalence classes of some stable endomorphism rings

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1157-1183 ◽  
Author(s):  
Jenny August
Keyword(s):  
Equivalence Class ◽  
Minimal Model ◽  
Equivalence Classes ◽  
Finite Dimensional Algebra ◽  
Derived Equivalence ◽  
Endomorphism Rings ◽  
Dimensional Algebra ◽  
Rigid Objects ◽  
Finite Dimensional ◽  
Frobenius Category

Abstract We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction $$f :X \rightarrow {\mathrm{Spec}\;}\,R$$ f : X → Spec R , where X has only Gorenstein terminal singularities, there is an associated finite dimensional algebra $$A_{{\text {con}}}$$ A con known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence class of $$A_{\text {con}}$$ A con and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f. This provides evidence towards a key conjecture in the area.

Implicational formulas in intuitionistic logic

1974 ◽  
Vol 39 (4) ◽  
pp. 661-664 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Propositional Calculus ◽  
Modus Ponens ◽  
Equivalence Classes ◽  
Alternative Proof ◽  
Hilbert Algebra ◽  
Hilbert Algebras ◽  
Free Generators ◽  
Finite Set ◽  
Image Position ◽  
The Right

In [1] Diego showed that there are only finitely many nonequivalent formulas in n variables in the positive implicational propositional calculus P. He also gave a recursive construction of the corresponding algebra of formulas, the free Hilbert algebra In on n free generators. In the present paper we give an alternative proof of the finiteness of In, and another construction of free Hilbert algebras, yielding a normal form for implicational formulas. The main new result is that In is built up from n copies of a finite Boolean algebra. The proofs use Kripke models [2] rather than the algebraic techniques of [1].Let V be a finite set of propositional variables, and let F(V) be the set of all formulas built up from V ⋃ {t} using → alone. The algebra defined on the equivalence classes , by settingis a free Hilbert algebra I(V) on the free generators . A set T ⊆ F(V) is a theory if ⊦pA implies A ∈ T, and T is closed under modus ponens. For T a theory, T[A] is the theory {B ∣ A → B ∈ T}. A theory T is p-prime, where p ∈ V, if p ∉ T and, for any A ∈ F(V), A ∈ T or A → p ∈ T. A theory is prime if it is p-prime for some p. Pp(V) denotes the set of p-prime theories in F(V), P(V) the set of prime theories. T ∈ P(V) is minimal if there is no theory in P(V) strictly contained in T. Where X = {A1, …, An} is a finite set of formulas, let X → B be A1 →····→·An → B (ϕ → B is B). A formula A is a p-formula if p is the right-most variable occurring in A, i.e. if A is of the form X → p.

scholarly journals ALGORITHMS FOR SINGULARITIES AND REAL STRUCTURES OF WEAK DEL PEZZO SURFACES

2014 ◽  
Vol 13 (05) ◽  
pp. 1350158
Author(s):  
NIELS LUBBES
Keyword(s):  
Equivalence Classes ◽  
Point Of View ◽  
Alternative Proof ◽  
Del Pezzo Surfaces ◽  
Isolated Singularities ◽  
Classification Of Singularities ◽  
Del Pezzo ◽  
Real Forms

In this paper, we consider the classification of singularities [P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction. I, II, III, Proc. Camb. Philos. Soc.30 (1934) 453–491] and real structures [C. T. C. Wall, Real forms of smooth del Pezzo surfaces, J. Reine Angew. Math.1987(375/376) (1987) 47–66, ISSN 0075-4102] of weak Del Pezzo surfaces from an algorithmic point of view. It is well-known that the singularities of weak Del Pezzo surfaces correspond to root subsystems. We present an algorithm which computes the classification of these root subsystems. We represent equivalence classes of root subsystems by unique labels. These labels allow us to construct examples of weak Del Pezzo surfaces with the corresponding singularity configuration. Equivalence classes of real structures of weak Del Pezzo surfaces are also represented by root subsystems. We present an algorithm which computes the classification of real structures. This leads to an alternative proof of the known classification for Del Pezzo surfaces and extends this classification to singular weak Del Pezzo surfaces. As an application we classify families of real conics on cyclides.

scholarly journals The Singularity Categories of the Cluster-Tilted Algebras of Dynkin Type

2014 ◽  
Vol 18 (2) ◽  
pp. 531-554 ◽  
Author(s):  
Xinhong Chen ◽  
Shengfei Geng ◽  
Ming Lu
Keyword(s):  
Tilted Algebras ◽  
Dynkin Type

Tilted algebras and configurations of self-injective algebras of Dynkin type

2016 ◽  
Vol 45 (8) ◽  
pp. 3278-3296
Author(s):  
Hideto Asashiba ◽  
Ken Nakashima
Keyword(s):  
Tilted Algebras ◽  
Dynkin Type
Sign in / Sign up

Export Citation Format

Share Document