scholarly journals Approximations, ghosts and derived equivalences

2019 ◽  
Vol 150 (2) ◽  
pp. 813-840
Author(s):  
Yiping Chen ◽  
Wei Hu
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Noncommutative Geometry ◽  
Endomorphism Rings ◽  
Derived Equivalences ◽  
Symmetric Approximation ◽  
Quotient Rings ◽  
Tilting Objects

AbstractApproximation sequences and derived equivalences occur frequently in the research of mutation of tilting objects in representation theory, algebraic geometry and noncommutative geometry. In this paper, we introduce symmetric approximation sequences in additive categories and weakly n-angulated categories which include (higher) Auslander-Reiten sequences (triangles) and mutation sequences in algebra and geometry, and show that such sequences always give rise to derived equivalences between the quotient rings of endomorphism rings of objects in the sequences modulo some ghost and coghost ideals.

scholarly journals A Representation Theorem for Archimedean Quadratic Modules on ∗-Rings

2009 ◽  
Vol 52 (1) ◽  
pp. 39-52 ◽  
Author(s):  
Jakob Cimprič
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Representation Theorem ◽  
Commutative Rings ◽  
Real Algebraic Geometry ◽  
New Approach ◽  
Noncommutative Version ◽  
Quadratic Modules ◽  
Rings With Involution ◽  
Associative Rings

AbstractWe present a new approach to noncommutative real algebraic geometry based on the representation theory of C*-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative C*-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

scholarly journals Relative Calabi–Yau structures

2019 ◽  
Vol 155 (2) ◽  
pp. 372-412 ◽  
Author(s):  
Christopher Brav ◽  
Tobias Dyckerhoff
Keyword(s):  
Algebraic Geometry ◽  
Representation Theory ◽  
Riemann Surfaces ◽  
Fukaya Categories ◽  
Composition Law

We introduce relative noncommutative Calabi–Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi–Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi–Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

scholarly journals Lifting to Cluster-tilting Objects in Higher Cluster Categories

2012 ◽  
Vol 19 (04) ◽  
pp. 707-712
Author(s):  
Pin Liu
Keyword(s):  
Positive Integer ◽  
Cluster Category ◽  
Tilting Modules ◽  
Endomorphism Rings ◽  
Tilted Algebras ◽  
Cluster Categories ◽  
Cluster Tilting ◽  
Tilting Objects

Let d > 1 be a positive integer. In this note, we consider the d-cluster-tilted algebras, i.e., algebras which appear as endomorphism rings of d-cluster-tilting objects in higher cluster categories (d-cluster categories). We show that tilting modules over such algebras lift to d-cluster-tilting objects in the corresponding higher cluster category.

scholarly journals THE MONOID OF FAMILIES OF QUIVER REPRESENTATIONS

2002 ◽  
Vol 84 (3) ◽  
pp. 663-685 ◽  
Author(s):  
MARCUS REINEKE
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Representation Theory ◽  
Quantum Group ◽  
Subject Classification ◽  
Quiver Representations ◽  
Enveloping Algebras ◽  
Geometric Methods ◽  
Quantized Enveloping Algebras ◽  
Monoid Structure

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, that is, subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analysed by representation-theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiver representations, allow us to relate the monoid to certain degenerate forms of quantized enveloping algebras.2000 Mathematical Subject Classification: 16G20, 14L30, 17B37.

scholarly journals Application of Quotient Rings for Stability Analysis in Chemical Systems

2011 ◽  
Vol 66 (3-4) ◽  
pp. 231-241
Author(s):  
Sonja Sauerbrei ◽  
Anke Sensse ◽  
Markus Eiswirth
Keyword(s):  
Algebraic Geometry ◽  
Stability Analysis ◽  
Surface Reaction ◽  
Normal Forms ◽  
Jacobian Matrix ◽  
Stationary Solutions ◽  
Polynomial Rings ◽  
Chemical Systems ◽  
The Stability ◽  
Quotient Rings

Concepts from algebraic geometry (polynomial rings) can be used to determine analytically the stationary solutions in chemical reactions systems, more generally, systems of ordinary differential equations of polynomial form. The stability analysis via the Jacobian matrix often leads to complicated expressions which can hardly be analyzed. It is shown that these expressions can be simplified by forming quotient rings of the corresponding polynomial ring. The coefficients in the characteristic equation of the Jacobian can be represented by the normal forms obtained by generating the quotient rings so that their sign changes in dependence of a kinetic parameter and, hence, the stability can be determined. The procedure is illustrated using a well-known surface reaction.

On Generalized Schur's Lemma and Its Applications

2012 ◽  
Vol 19 (02) ◽  
pp. 337-352 ◽  
Author(s):  
Lizhong Wang
Keyword(s):  
Representation Theory ◽  
Endomorphism Ring ◽  
Symmetric Groups ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
Endomorphism Rings ◽  
Natural Embedding ◽  
Induced Module ◽  
Schur’S Lemma ◽  
Schur's Lemma

In this paper, we generalize Schur's lemma on the basis of endomorphism rings for permutation modules. Let H be a subgroup of G and let M be a module of H. Set N = NG(H). Then there is a natural embedding of End N(MN) into End G(MG). By taking H to be a p-subgroup of G, we can reformulate Green's theory on modular representation. A defect theory is defined on the endomorphism ring of any induced module and it is used to prove Green's correspondence and related results. This defect theory can unify some well known results in modular representation theory. By using generalized Schur's lemma, we can also give a method to determine the multiplicity of simple modules in any permutation module of symmetric groups. This makes it possible to prove various versions of Foulkes' conjecture in a uniform way.

On E. Verlinde’s Formula in the Context of Stable Bundles

1991 ◽  
Vol 06 (16) ◽  
pp. 2847-2858 ◽  
Author(s):  
Raoul Bott
Keyword(s):  
Algebraic Geometry ◽  
Point Of View ◽  
Stable Bundles ◽  
Quotient Rings

E. Verlinde’s formula for the dimension of the nonabelian θ-functions is discussed from an algebraic geometry point of view and related to certain quotient rings of the representative ring of sums.

scholarly journals Loop groups and noncommutative geometry

2017 ◽  
Vol 29 (09) ◽  
pp. 1750029
Author(s):  
Sebastiano Carpi ◽  
Robin Hillier
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Representation Theory ◽  
Noncommutative Geometry ◽  
Loop Group ◽  
Positive Energy ◽  
Loop Groups ◽  
Spectral Triples ◽  
Conformal Field ◽  
Coset Models

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level [Formula: see text] projective unitary positive-energy representations of any given loop group [Formula: see text]. The construction is based on certain supersymmetric conformal field theory models associated with [Formula: see text] in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.

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