scholarly journals On the realization and classification of symmetric algebras as cohomology rings

1983 ◽  
Vol 87 (1) ◽  
pp. 144-144 ◽  
Author(s):  
Larry Smith
Keyword(s):  
Cohomology Rings ◽  
Symmetric Algebras

scholarly journals Classification of Symmetric Special Biserial Algebras With At Most One Non-Uniserial Indecomposable Projective

2015 ◽  
Vol 58 (3) ◽  
pp. 739-767 ◽  
Author(s):  
Nicole Snashall ◽  
Rachel Taillefer
Keyword(s):  
Natural Generalization ◽  
Stable Equivalence ◽  
Symmetric Algebras ◽  
Special Biserial Algebras ◽  
Weakly Symmetric ◽  
Dihedral Type ◽  
Indecomposable Projective Module ◽  
Algebras Of Dihedral Type ◽  
Euclidean Type

AbstractWe consider a natural generalization of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one non-uniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the weakly symmetric algebras of Euclidean type n, as studied by Bocian et al., as well as some algebras of dihedral type.

scholarly journals CLASSIFICATION OF GRADED LEFT-SYMMETRIC ALGEBRAIC STRUCTURES ON WITT AND VIRASORO ALGEBRAS

2011 ◽  
Vol 22 (02) ◽  
pp. 201-222 ◽  
Author(s):  
XIAOLI KONG ◽  
HONGJIA CHEN ◽  
CHENGMING BAI
Keyword(s):  
Virasoro Algebra ◽  
Indecomposable Module ◽  
Multiplication Operators ◽  
Algebraic Structures ◽  
Witt Algebra ◽  
Left Multiplication ◽  
Symmetric Algebras ◽  
Nonlinear Math ◽  
Math Phys

We find that a compatible graded left-symmetric algebraic structure on the Witt algebra induces an indecomposable module V of the Witt algebra with one-dimensional weight spaces by its left-multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebraic structures on the Witt algebra are classified. All of them are simple and they include the examples given by [Comm. Algebra32 (2004) 243–251; J. Nonlinear Math. Phys.6 (1999) 222–245]. Furthermore, we classify the central extensions of these graded left-symmetric algebras which give the compatible graded left-symmetric algebraic structures on the Virasoro algebra. They coincide with the examples given by [J. Nonlinear Math. Phys.6 (1999) 222–245].

scholarly journals Projective bundles over toric surfaces

2016 ◽  
Vol 27 (04) ◽  
pp. 1650032 ◽  
Author(s):  
Suyoung Choi ◽  
Seonjeong Park
Keyword(s):  
Toric Variety ◽  
Cohomology Ring ◽  
Line Bundles ◽  
Projective Bundle ◽  
Cohomology Rings ◽  
Toric Surfaces ◽  
Projective Bundles ◽  
Higher Dimensional ◽  
Quasitoric Manifolds

Let [Formula: see text] be the Whitney sum of complex line bundles over a topological space [Formula: see text]. Then, the projectivization [Formula: see text] of [Formula: see text] is called a projective bundle over [Formula: see text]. If [Formula: see text] is a nonsingular complete toric variety, then so is [Formula: see text]. In this paper, we show that the cohomology ring of a nonsingular projective toric variety [Formula: see text] determines whether it admits a projective bundle structure over a nonsingular complete toric surface. In addition, we show that two [Formula: see text]-dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds are diffeomorphic if their cohomology rings are isomorphic as graded rings. Furthermore, we study the smooth classification of higher dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds.

scholarly journals The homotopy classification of four-dimensional toric orbifolds

2021 ◽  
pp. 1-23
Author(s):  
Xin Fu ◽  
Tseleung So ◽  
Jongbaek Song
Keyword(s):  
Homotopy Type ◽  
Homotopy Equivalent ◽  
Homotopy Classification ◽  
Cohomology Rings ◽  
Moore Space ◽  
Integral Cohomology ◽  
Cw Complex ◽  
Primary Torsion

Let X be a 4-dimensional toric orbifold. If $H^{3}(X)$ has a non-trivial odd primary torsion, then we show that X is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.

scholarly journals DERIVED EQUIVALENCE CLASSIFICATION OF SYMMETRIC ALGEBRAS OF POLYNOMIAL GROWTH

2010 ◽  
Vol 53 (2) ◽  
pp. 277-291 ◽  
Author(s):  
THORSTEN HOLM ◽  
ANDRZEJ SKOWROŃSKI
Keyword(s):  
Polynomial Growth ◽  
Derived Equivalence ◽  
Symmetric Algebras

AbstractWe complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence.

scholarly journals Derived equivalence classification of weakly symmetric algebras of Euclidean type

2004 ◽  
Vol 191 (1-2) ◽  
pp. 43-74 ◽  
Author(s):  
Rafał Bocian ◽  
Thorsten Holm ◽  
Andrzej Skowroński
Keyword(s):  
Derived Equivalence ◽  
Symmetric Algebras ◽  
Weakly Symmetric ◽  
Euclidean Type

scholarly journals Non-negatively curved GKM orbifolds

2021 ◽  
Author(s):  
Oliver Goertsches ◽  
Michael Wiemeler
Keyword(s):  
Finite Groups ◽  
Orbit Spaces ◽  
Cohomology Rings ◽  
Negatively Curved ◽  
Rational Cohomology ◽  
Isometric Actions ◽  
Torus Manifolds

AbstractIn this paper we study non-negatively curved and rationally elliptic GKM$$_4$$ 4 manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are quotients of isometric actions of finite groups on non-negatively curved torus orbifolds. Moreover, we give a simplified proof of a characterisation of products of simplices among orbit spaces of locally standard torus manifolds. This characterisation was originally proved in Wiemeler (J Lond Math Soc 91(3): 667–692, 2015) and was used there to obtain a classification of non-negatively curved torus manifolds.

scholarly journals Classification of Toric Manifolds over an n-Cube with One Vertex Cut

2018 ◽  
Vol 2020 (16) ◽  
pp. 4890-4941
Author(s):  
Sho Hasui ◽  
Hideya Kuwata ◽  
Mikiya Masuda ◽  
Seonjeong Park
Keyword(s):  
Fixed Point ◽  
Toric Variety ◽  
The Other ◽  
Toric Manifold ◽  
Smooth Manifolds ◽  
Cohomology Rings ◽  
Toric Manifolds ◽  
Compact Torus ◽  
Manifold With Corners

Abstract A complete nonsingular toric variety (called a toric manifold) is over $P$ if its quotient by the compact torus is homeomorphic to $P$ as a manifold with corners. Bott manifolds are toric manifolds over an $n$-cube $I^n$ and blowing them up at a fixed point produces toric manifolds over $\operatorname{vc}(I^n)$ an $n$-cube with one vertex cut. They are all projective. On the other hand, Oda’s three-fold, the simplest non-projective toric manifold, is over $\operatorname{vc}(I^3)$. In this paper, we classify toric manifolds over $\operatorname{vc}(I^n)$$(n\ge 3)$ as varieties and as smooth manifolds. It consequently turns out that there are many non-projective toric manifolds over $\operatorname{vc}(I^n)$ but they are all diffeomorphic, and toric manifolds over $\operatorname{vc}(I^n)$ in some class are determined by their cohomology rings as varieties.

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