Abetween complete and cocomplete abelian categories by the conditions lrx x, reps. Category theory organizes mathematics using morphisms that transmit structure and determination. We study the global dimensions of the coherent functors over two categories that are linked by a pair of adjoint functors. Fixed points of adjoint functors enriched in a quantaloid. Very roughly speaking, representation theory studies symmetry in linear spaces. My ulterior motive is to gain a better understanding of representation theory by developing its categorical foundations. In technical terms, representation theory studies representations of associative algebras. Frobenius reciprocity and the haagerup tensor product.
Preliminaries the nakayama functor is well known in the representation theory of nitedimensional algebras see, for example, 1, 15. Category theory stanford encyclopedia of philosophy. In this theory, one considers representations of the group algebra a cg of a. The basis for the theory is laid by first showing that the objecttoobject heteromorphisms between the objects of different categories. Fcategories and f functors in the representation theory of lie algebras.
Each local adjunction gives rise to an ordinary adjunction of functors between categories of hilbert space representations. Zuckermans cohomological construction and the localization theory of bernstein and beilinson. These constructions made an enormous impact on our understanding of representation theory during the last decades see, for example, 19. Originally the main lenses were universal mapping properties and natural transformations. In trying to think of an intuitive answer to a question on adjoints, i realised that i didnt have a nice conceptual understanding of what an adjoint pair actually is i know the definition several of them, ive read the nlab page and any good answers will be added there, ive worked with them, ive found examples of functors with and without adjoints, but i couldnt explain what an. This file cannot be posted on any website not belonging to the authors. We learnt the concept of adjoint functors and showed that the construction quiver algebra can be interpreted as a pair of adjoint functors. Jan 10, 2017 yes, but the shared name is an accident. Such functors give representations of an abstract category in terms of known structures i. The rest follows from the general theory of mackey functors, the particular role the burnside ring plays for mackey functors, and explicit formulae for the. Adjoint functors, projectivization, and differentiation. This section provides the lecture notes from the course. Most of the proofs in this section are standard, so we leave the details to the reader.
At the cost of some repetition compare adjoint functor, we outline how one gets from the homfunctor formulation of adjunction in cat to the elementary definition in terms of units and counits. Frobenius reciprocity in the representation theory of groups. A basic result of category theory is that right adjoint functor s preserve all limit s that exist in their domain, and, dually, left adjoints preserve all. Category theory in context my second book, category theory in context, originated as lecture notes written for an undergraduate topics course taught at harvard in the spring of 2015, and taught against at johns hopkins at the graduate level in the fall of 2015. The present lecture notes arose from a representation theory course given by prof.
In mathematics, specifically category theory, adjunction is a relationship that two functors may have. This idea gets the name frobenius reciprocity in representation theory. Pdf fcategories and ffunctors in the representation. Representation theory authorstitles recent submissions. Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Adjoint functors can be thought of as being conceptual inverses. Conceptual explanations are given for the combinatorial construction of. Climent, notre dame journal of formal logic, 2008 symmetric bundles and representations of lie triple systems bertram, wolfgang and didry, manon, journal of generalized lie theory and applications, 2009. Are representable functors related to the representation theory. This will motivate the definition in the section that follows, which is. The adjoint representation can also be defined for algebraic groups over any field. Jun 20, 2017 adjoint functors and monads june 20, 2017 june 20, 2017 anton hilado in category theory we introduced the language of categories, and in many posts in this blog we have seen how useful it is in describing concepts in modern mathematics, for example in the two most recent posts, the theory of motives and algebraic spaces and stacks. Mackey functors and related structures in representation. In chapter ii we deal with direct and inverse limits.
Etingof in march 2004 within the framework of the clay mathematics institute research academy for high school students. We learnt the concept of adjoint functors and showed that the construction quiver algebra can be interpreted as a pair of adjoint functors between certain categories. There is a second group of functors also discussed in. A theory of adjoint functors with some thoughts about their. Alexandre kirillov observed that the orbit of any vector in a co adjoint representation is a symplectic manifold. Our focus here is to present a theory of adjoint functors. Heteromorphisms and adjoint functors david ellerman philosophy department university of california at riverside abstract category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematicss with adjoint functors being the primary lense.
In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. We will study the concept of adjoint functors and show the correspondence. In category theory we have a weaker condition for saying. Mostly the lectures do not contain the proofs and the theory is accom.
Frobenius is a cornerstone in the representation theory of finite groups. I want to know about the functors between this categories, i can decide for myself what functor is adjoint to some of the others. The amazing aspect is that number theory is only used for the almost obvious fact that the class groups form a mackey functor. Adjoint functors and heteromorphisms david ellerman.
We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. The general theory of adjoint functors constitutes chapter i. The fields of algebra and representation theory contain abundant examples of functors on categories of modules over a ring. Representation theory was born in 1896 in the work of the german mathematician f. Introduction in this paper we revisit some now classical constructions of modern representation theory. Notice that i didnt say on which side the adjointness holds. Adjoint functors and heteromorphisms david ellerman abstract category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. A theory of adjoint functors with some thoughts about. Category theory organizes mathematics using morphisms that transmitstructure anddetermination.
On adjoint functors in representation theory springerlink. It is a beautiful mathematical subject which has many applications, ranging from number theory and combinatorics to geometry, probability theory, quantum mechanics, and quantum eld theory. This is nothing but the statement of the frobenius reciprocity. We usually use isomorphisms to say two objects are the same, but this is a very strong condition. These lecture notes arose from a representation theory course given by prof. There is a second group of functors also discussed in 3 which includes some specialized versions of tor. How to determine the matrix of adjoint representation of. Conceptual explanations are given for the combinatorial construction of the derived set and for the differentiation functor. In recent decades, the notion of adjoint functors has moved. Oscillator realisations associated to the dtype yangian and extremal qoperators for orthogonal spin chains. Adjoint functors in algebra, topology and mathematical logic keegan smith august 8, 2008 1 introduction in mathematics we enjoy di. Chapter 4 develops the abstract theory of the twisted inverse image functor f.
Among some of the more interesting examples are enrights completion functors and zuckermans derived functors of the tfinite functor. Nowadays, every user of category theory agrees that adjunction is the concept which. On adjoint and brain functors david ellerman philosophy department university of california at riverside abstract there is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. Very roughlyspeaking, representation theory studies symmetryin linear spaces. Another fundamental notion in category theory is the notion of adjoint functors. We will see the truth of this, meeting examples of adjoint functors from diverse parts of. It is a beautiful mathematical subject which has many applications, ranging from numbertheory and combinatorics to geometry, probability theory, quantum mechanics and quantum eld theory. An associative algebra over a eld kis a vector space aover k equipped with an associative bilinear multiplication a. An adjunction in cat is a pair of adjoint functors. Among some of the more interesting examples are enrights completion functors and zuck ermans derived functors of the tfinite functor. Using the cohom and cotensor functors we state takeuchimorita equivalence and use it to sketch the proof of existence of basic coalgebras, due to.
Representation theory was born in 1896 in the work of the ger. Adjoint functors drew armstrong contents 1 from posets to categories 2 2 galois connections 17. Pdf induction and restriction as adjoint functors on. Lecture notes introduction to representation theory. Category theory in context krieger school of arts and. Green we present the block theory of coalgebras using indecomposable injectives comodules. In particular, we show that if an artin algebra is switched from the other, then they have the same representation dimension. Adjoint functors in algebra, topology and mathematical logic. Then the left and right adjoint to the functor v on the category of representations of g is the functor v. As the link of zhen lin says adjoint functors deal with universals. In mathematics, specifically category theory, adjunction is a relationship that two functors may. Induction and restriction as adjoint functors on representations of locally compact groups article pdf available in international journal of mathematics and mathematical sciences 161.
Typical problems of representation theory are as follows. Abstract the question what is category theory is approached by focusing on universal mapping properties and adjoint functors. Its general content can be very brie y summarized as follows. These include of course hom, ext, and tor as well as the more specialized examples of comple. Obviously, irreducible implies indecomposable, but not vice versa. This paper aims to establish general representation theorems for. The students in that course oleg golberg, sebastian hensel, tiankai liu, alex schwendner, elena yudovina, and dmitry vaintrob co. Adjoint functor theorems are theorems stating that under certain conditions a functor that preserves limits is a right adjoint, and that a functor that preserves colimits is a left adjoint. We will study the concept of adjoint functors and show the correspondence quiver. A nonzero representation v of ais said to be irreducible if its only subrepresentations are 0 and v itself, and indecomposable if it cannot be written as a direct sum of two nonzero subrepresentations. Functors of lindenbaumtarski, schematic interpretations, and adjoint cylinders between sentential logics tur, j. Adjoint functors and projectivization in representation theory of partially ordered sets are used to generalize the algorithms of differentiation by a maximal and by a minimal point.
The present lecture notes arose from a representation theory course given by. Adjoint functors and representation dimensions springerlink. Others have been lead to the focus on adjoints by di. The question what is category theory is approached by focusing on universal mapping properties and adjoint functors. Apr 08, 20 adjoint functors in graph theory 5 with h k n and u n. Our interest in adjoint functor techniques for unitary representations arose from.
The book introduction to representation theory based on these notes was published by the american mathematical society in 2016. An example of adjoint functors in two variables are the functors and horn mentioned above. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Essential for building up the theory in this general setting is the existence of a static generator b2bor dually a static cogenerator a2a. For example, the group of all permutations of a set of three elements can act linearly on a vector space i. The notion of adjoint functor applies everything that weve learned up to now to unify and subsume all the di.
Introduction to representation theory mit opencourseware. A representation in the sense of representation theory is a linear action of an algebraic object. The adjoint representation 1 the adjoint representation besides the left and right actions of g on itself, there is the conjugation action cg. This idea is then exploited to compare the representation dimensions of two algebras. This theorem provides a nice result allowing us to classify exactly which functors have left. Usually i think to adjointness situations in two different ways. My aim was to provide a short introduction to basic notions of category theory and representation theory of finitedimensional algebras. Marco grandis has been working to collect and formalize the ideas of directed homotopy theory his main work on the subject has been listed in the references at the nlab page on the subject. In the 1920s, the focus of investigations was representation theory of compact lie groups and its relations to invariant theory. Wojtowicz department of mathematics, university of dallas, 1845 e.
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