By gelfand transformation, the theory of c oeu algebras is also regarded as noncommutative topology. Two rings and are called morita equivalent if the categories of left right and modules are equivalent cf. Finally, we look at the corresponding notions for groupoids with structure and. We develop the notion of a morita context between two operator algebras \a\ and \b\. The standard model in noncommutative geometry and morita. In 2 we also noticed the construction of a strictly related category of moritaconnes. Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. Muhly and baruch solel received 2 september 1998 1.
The reader willing to learn more about these results can find necessary information in connes book. Then we consider strong morita equivalence for calgebras and morita equivalence for walgebras. Next, the book presents auslanderreiten theory, including almost split sequences and the auslanderreiten transform, and gives a proof of kacs generalization of gabriels theorem. Department of mathematics university of toronto toronto, ontario m5s 3g3, canada alan weinstein department of math.
Goren, mcgill university this is an other great mathematics book cover the following topics. Reciprocally, we would like to be able to recover the space with structure from the given algebra. Namely, im wondering what the similarities and differences are between the theory of ordinary schemes on the one hand, and the theory of. A morita equivalence is an equivalence in a 2category in the bicategory rng \mathrmrng of rings, bimodules as 1morphisms and bimodule homomorphisms intertwiners as 2morphisms. In the original sense, a morita equivalence between two rings is two bimodules. Symplectic geometry, groupoids, and integrable systems pp 2911 cite as. My main reference for these notes was chapter ii of basss book algebraic.
The theory, called noncommutative geometry, rests on two essential points. The most important examples of moritaequivalent rings are. Motivation for the preprojective algebra mathoverflow. In algebra, the importance of morita equivalence lies in the fact that morita equivalent al. Categories of operator modules morita equivalence and. The algebra r satisfies the dixmiermoeglin equivalence if the annihilators of irreducible representations are precisely those prime ideals that are locally closed in the spec r and if, moreover, these prime ideals are precisely those whose extended centres are algebraic extensions of the base field. Morita equivalence in algebra and geometry ralf meyer abstract. A theorem in ring theory says that the center of a ring is isomorphic to the center of its category of modules and that morita equivalent rings have isomorphic centers. The purpose of this book is to extend this correspondence to the noncommutative case in the framework of real analysis. An introduction to quiver representations about this title. Examples and morita equivalence rolf farnsteiner let. Im preparing to give another talk at the student seminar we are running at the university of alberta this semester.
Very basic noncommutative geometry vanderbilt university. Poisson geometry, deformation quantisation and group. Grothendieck toposes as unifying bridges in mathematics. Then that theory has a nontrivial autoequivalence exchanging the two sorts, and thats certainly a morita equivalence. Morita equivalence of noncommutative supertori request pdf. This may be too elementary, but anderson, fuller rings and categories of modules has chapter 6 giving basic properties of morita equivalence.
It is generally accepted that the classical analogue of a calgebra or noncommutative algebra is a poisson manifold. If the algebra in question also has a topology in which the algebraic operations are continuous and perhaps some other added structure, it is an obvious task to make a morita theory for the subcategory of modules which are compatible with these added structures. In algebra, ring theory is the study of ringsalgebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Then that theory has a nontrivial autoequivalence exchanging the two sorts, and thats certainly a. One punchline of our book is a version of the dixmierdouady theorem in. The material in the opening chapters is developed starting slowly with topics such as homological algebra, morita equivalence, and gabriels theorem. An algebra over ror c is the set of good r or cvalued functions on a space with structure. Twisted forms in geometry and algebra colorful algebra. This correspondence started with hilberts nullstellensatz that establishes a onetoone correspondence between the points of an algebraic variety, and the maximal ideals of its coordinate ring.
A good reference for the above sadly excluding the part about the preprojective algebra. Morita equivalence of calgebras, first introduced by m. In the third chapter of the book, the focus shifts to b as. There are many similarities between poisson geometry and the theory of associative algebras see e. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings group rings, division rings, universal enveloping algebras, as well as an. Pdf c algebras by example download read online free. It is compact if dimha,d algebra or noncommutative algebra is a poisson manifold. Morita equivalence and moduli problems mathoverflow.
For su ciently nice quotients, even though the two notions are not the same, they are related by morita equivalence, which is the. Geometric and algebraic topological methods in quantum. Advances in algebra and geometry university of hyderabad conference 2001, 279295, hindustan book agency, india, 2003. To appear in advances in algebra and geometry university of hyderabad conference 2001. The general concept of morita equivalence was developed by k. We start with morita equivalence and morita duality in pure algebra. Some basic properties of schemes one can formulate purely in derived terms. The theory and applications of c oeu algebras are related to fields ranging from operator theory, group representations and quantum mechanics, to noncommutative geometry and dynamical systems. Finitely generated abelian groups, vector spaces, the jordan canonical form, morita equivalence, injective and projective limits. Poisson geometry lies at the cusp of noncommutative algebra and differential geometry, with natural and important links to classical physics and quantum mechanics. Morita equivalence and duality queen mary college mathematics notes. The associated convolution algebra is called a peterweyl iwahori algebra. We show that, if we want h f to be a morita equivalence bimodule between a f and the associated cli ord algebra, two terms must be added to the dirac operator. Morita equivalence, picard groupoids and noncommutative field.
Dellambrogio, goncalo tabuada, a quillen model structure for classical morita theory and a tensor categorification of the brauer group, arxiv. The book may not go into the same depth as, say, artins algebra, but rather the foundational concepts for the study of algebra and geometry are emphasized in a variety of settings. An introduction to morita theory school of mathematics. Grothendieck toposes can serve as bridges for transferring properties between moritaequivalent theories in the following sense.
The interplay between topology, functions, geometry, and algebra kenji ueno, koji shiga, shigeyuki morita this is the second of three volumes that originated from a series of lectures in mathematics given to high school students by professors of kyoto university in japan. The results of noncommutative geometry that we use are explained in the paper. Chapter 2 is devoted to the basics of representation theory. To give a very much simplified example, consider the firstorder theory with two sorts, no function or relation symbols, and no axioms. The standard model in noncommutative geometry and morita equivalence. An equivalence relation on the class of all rings defined as follows. Morita equivalence of manysorted algebraic theories.
In algebraic geometry that is achieved by considering homomorphisms from the algebra to a eld or integral domain. We focus our attention on the category of hilbert modules over an operator algebra and on the category of operator modules over an operator algebra. Geometry according to which every concept or statement in algebra corresponds to, and can be equally formulated by, a similar concept and statement in geometry. Details and proofs can be found in fedosovs book 17.
W e study the notion of morita equiv alence in v arious categories. We point out the relations between noncommutative field theories and deformed vector bundles which give the morita equivalence bimodules. The morita equivalence section of mcconnell, robson noncommutative noetherian rings goes into more depth. It is named after japanese mathematician kiiti morita who defined equivalence and a similar notion of duality in 1958. Morita equivalence bimodule for the deformed algebras. Mesland 25 introduced what is in our opinion the most sophisticated notion of category. Morita equivalence and moritaazumaya duality theorems and the authors explain the necessary concepts in su cient detail. Moritaequivalence from the perspective provided by the theory of topostheoretic.
This book presents an introduction to the subject from a small group of leading researchers, and the result is a volume accessible to graduate students or experts from other fields. An introduction to morita theory matt booth october 2015 nov. We study the notion of morita equivalence in various categories. Thus to understand its relation with classical geometry one should. Noncommutative geometry has as its limiting case the classical geometry, but geometry expressed in algebraic terms. Rfunctors between module categories correspond to bimodules. Part of the mathematical sciences research institute publications book series msri. Functoriality and morita equivalence of operator algebras and poisson manifolds associated to groupoids by n. The existence of many natural spaces for which the classical settheoretic tools. Pdf the standard model in noncommutative geometry and.
The concept should be covered in any decent textbook on algebra and ring theory, e. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and. The second description leads to a noncommutative algebra, as the product, determined by the groupoid law of the equivalence relation, has the form of a convolution product like the product of matrices. So, more precisely, we expect to find an equivalence relation for poisson manifolds that plays the same role as morita equivalence does for calgebras. Both the iwahorihecke algebra and a peterweyl iwahori algebra have a natural c algebra structure, and the morita equivalence preserves irreducible hermitian and unitary modules. Introduction the correspondence between geometric spaces and commutative algebras is a familiar and basic idea of algebraic geometry. The aim of this note is to study categories of regular modules over not necessarily unital ralgebras, where a module mover an ralgebra. We show any peterweyl iwahori algebra is morita equivalent to the iwahorihecke algebra.
Other chapters explore some of the tools of the trade. Can someone give some examples of morita equivalent rings different from the classical one. Then w e consider strong morita equiv alence for calgebras and morita equiv alence for walgebras. Author of linear equations, lie groups, algebra, universal algebra, solid geometry, morita equivalence and duality, skew field constructions, free rings and their relations p. Citeseerx morita equivalence in algebra and geometry. Noncommutative geometry and algebra my main interest is the noncommutative world in all its aspects. Morita equivalence, picard groupoids and noncommutative. Abhyankar, algebraic geometry for scientists and engineers. Let k be a field and let r be a left noetherian kalgebra. Noncommutative geometry and algebra university of washington. Basic noncommutative geometry by masoud khalkhali zurich. The subject of calgebras received a dramatic revitalization in the 1970s by the introduction of topological methods through the work of brown, douglas, and fillmore on extensions of calgebras and elliotts use of ktheory to provide a useful classification of af algebras. Morita equivalence and duality queen mary college mathematics notes cohn, p.
Newest derivedalgebraicgeometry questions mathoverflow. Ifv is such a space, and uis a subspace, there is another subspace w v such that w u v we say that the category of vector spaces is a semisimple category. Morita treats both contravariant equivalences which he calls. I have a very vague question, but also a fairly specific wish. Every quotient of the theory mv is morita equivalent to a quotient of the theory l u and conversely. Algebraic geometry is in many ways the mirror image of commutative algebra. Morita equivalence is studied in section 7, and the main theorem of this paper describing how h.
In abstract algebra, morita equivalence is a relationship defined between rings that preserves many ringtheoretic properties. Aug 31, 2008 author of linear equations, lie groups, algebra, universal algebra, solid geometry, morita equivalence and duality, skew field constructions, free rings and their relations p. We start with morita equivalence and morita duality in. Abstract algebraic categories 1 0 preliminaries 3 1 algebraic theories and algebraic categories 11 2 sifted and. Poisson geometry and morita equivalence henrique bursztyn. On the contrary, this duality has always existed and has been utilized in mathematics and its applications very often. But since complexes of modules over a classical algebra are the same thing as dg modules over that classical algebra thought of as a dg algebra with zero differential and concentrated in degree zero, one is led to the notion of derived morita invariance by thinking of your classical algebra as a very simple kind of dg algebra. First, we want to develop a notion of morita equivalence for ccorrespondences that guarantees that if two ccorrespondences e and f are morita equivalent, then the tensor algebras of e and f, t. Citeseerx document details isaac councill, lee giles, pradeep teregowda. An introduction to the classification of amenable c algebras book summary. The moritaequivalence between mvalgebras and lattice. W e start with morita equiv alenc e and morita dualit y in pure algebra.
Some basic properties of schemes one can formulate purely in derived. Category theory and calgebras a commented very partial. Download compressed postscript file this article has been prepared for the spring 1997 math 277 course at the university of california at berkeley taught by alan weinstein. Morita equivalent symplectic groupoids springerlink. Kadison, pure states and approximate identities, proc. The dixmiermoeglin equivalence, morita equivalence, and. Morita theory in abelian, derived and stable model categories. This is very important as the study of abstract algebra is precisely that if you do not have a wideselection of examples and contexts to draw from. Harm derksen, university of michigan, ann arbor, mi and jerzy weyman, university of connecticut, storrs, ct. It is generally accepted that the classical analogue of a calgebra or.
The standard model in noncommutative geometry and morita equivalence article pdf available in journal of noncommutative geometry 102 december 2014 with 37 reads how we measure reads. Morita equivalence and continuoustrace calgebras american. Graduate courses 20172018 department of mathematics. Morita equivalence as an example of how category theory can be. Phillipss result in the case that the algebras are simple. Ralf meyer, morita equivalence in algebra and geometry. In fact, the notion of moritaequivalence formalizes in many situations the feeling of looking at the same thing in different ways, which explains why it is ubiquitous in mathematics.
Introduction the classical results of kiiti morita characterizing equivalence of categories of modules, see 12, have been generalized to onesorted algebraic theories in several articles. There was an ensuing discussion about how part of algebraic geometry consists of both applications of algebra to geometry,using algebraic methods to understand geometry and in the other direction thinking of algebra in a geometric way, applications of geometry to algebra. Since a hilbert cmodule yc is a hilbert space and kyc is then the usual algebra of compact operators 12, example 2. From this it follows that many other invariants, such as cohomology and deformation theory, are shared by all morita equiva. Morit a equiv alence in algebra and geometr y ralf meyer abstra ct. At the same time, we present in a compact way all the necessary up to date mathematical tools to be used in studying quantum problems. Noncommutative algebra 3 the category of finite dimensional vector spaces. Category theory and calgebras a commented very partial bibliography by ivo dellambrogio november 2012 to my knowledge, the language of category theory almost never appears in textbook treatments of the theory of calgebras, even in situations where it could simplify statements or otherwise help the exposition. The module operations are assumed to be completely bounded usually, completely contractive. On a physiological level this correspondence is perhaps related to a division in. By the morita equivalence, the theories mv and l u have the same classifying topos whence there is a bijection between the quotients of these theories.
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