scholarly journals THE CENTER OF THE CATEGORY OF BIMODULES AND DESCENT DATA FOR NONCOMMUTATIVE RINGS

2012 ◽  
Vol 11 (06) ◽  
pp. 1250102
Author(s):  
A. L. AGORE ◽  
S. CAENEPEEL ◽  
G. MILITARU
Keyword(s):  
Differential Geometry ◽  
Commutative Ring ◽  
Baxter Equation ◽  
Flat Connection ◽  
Dual Basis ◽  
Monoidal Categories ◽  
Noncommutative Rings ◽  
Descent Data ◽  
Weak Center ◽  
Braided Monoidal Categories

Let A be an algebra over a commutative ring k. We compute the center of the category of A-bimodules. There are six isomorphic descriptions: the center equals the weak center, and can be described as categories of noncommutative descent data, comodules over the Sweedler canonical A-coring, Yetter–Drinfeld type modules or modules with a flat connection from noncommutative differential geometry. All six isomorphic categories are braided monoidal categories: in particular, the category of comodules over the Sweedler canonical A-coring A ⊗ A is braided monoidal. We provide several applications: for instance, if A is finitely generated projective over k then the category of left End k(A)-modules is braided monoidal and we give an explicit description of the braiding in terms of the finite dual basis of A. As another application, new families of solutions for the quantum Yang–Baxter equation are constructed: they are canonical maps Ω associated to any right comodule over the Sweedler canonical coring A ⊗ A and satisfy the condition Ω3 = Ω. Explicit examples are provided.

scholarly journals Quasitriangular Hopf group algebras and braided monoidal categories

2014 ◽  
Vol 64 (4) ◽  
pp. 893-909
Author(s):  
Shiyin Zhao ◽  
Jing Wang ◽  
Hui-Xiang Chen
Keyword(s):  
Group Algebras ◽  
Monoidal Categories ◽  
Braided Monoidal Categories

scholarly journals BILINEAR MAPS ON ARTINIAN MODULES

2012 ◽  
Vol 11 (05) ◽  
pp. 1250090
Author(s):  
GEORGE M. BERGMAN
Keyword(s):  
Commutative Ring ◽  
Finite Length ◽  
Nonzero Element ◽  
Bilinear Maps ◽  
Bilinear Map ◽  
Noncommutative Rings ◽  
Artinian Modules

It is shown that if a bilinear map f : A × B → C of modules over a commutative ring k is nondegenerate (i.e. if no nonzero element of A annihilates all of B, and vice versa), and A and B are Artinian, then A and B are of finite length. Some consequences are noted. Counterexamples are given to some attempts to generalize the above result to balanced bilinear maps of bimodules over noncommutative rings, while the question is raised whether other such generalizations are true.

scholarly journals Some functors arising from the consideration of torsion theories over noncommutative rings

1976 ◽  
Vol 15 (3) ◽  
pp. 455-460 ◽  
Author(s):  
Jonathan S. Golan
Keyword(s):  
Distributive Lattice ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Noncommutative Rings ◽  
Torsion Theories ◽  
Associative Rings

To each associative (but not necessarily commutative) ring R we assign the complete distributive lattice R-tors of (hereditary) torsion theories over R-mod. We consider two ways of making this process functorial – once contravariantly and once covariantly – by selecting appropriate subcategories of the category of associative rings. Combined with a functor due to Rota, this gives us functors from these subcategories to the category of commutative rings.

scholarly journals Mathematics and Poetry • Unification, Unity, Union

Sci ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 84
Author(s):  
Florin Felix Nichita
Keyword(s):  
Differential Geometry ◽  
Baxter Equation ◽  
Dual Numbers ◽  
Euler’S Formula ◽  
Euler’S Identity

We consider a multitude of topics in mathematics where unification constructions play an important role: the Yang–Baxter equation and its modified version, Euler’s formula for dual numbers, means and their inequalities, topics in differential geometry, etc. It is interesting to observe that the idea of unification (unity and union) is also present in poetry. Moreover, Euler’s identity is a source of inspiration for the post-modern poets.

scholarly journals Instantons and vortices on noncommutative toric varieties

2014 ◽  
Vol 26 (09) ◽  
pp. 1430008 ◽  
Author(s):  
Lucio S. Cirio ◽  
Giovanni Landi ◽  
Richard J. Szabo
Keyword(s):  
Moduli Spaces ◽  
Toric Varieties ◽  
Partition Functions ◽  
Toric Geometry ◽  
Monoidal Categories ◽  
Klein Quadric ◽  
Noncommutative Algebraic Geometry ◽  
Fundamental Matter ◽  
Braided Monoidal Categories ◽  
Noncommutative Gauge Theories

We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.

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