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Contents

1 Adjoint equation 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Adjoint functors 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Spelling (or morphology) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Solutions to optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2.2 Symmetry of optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Formal definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.2 Universal morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.3 Counit-unit adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3.4 Hom-set adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Adjunctions in full . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4.1 Universal morphisms induce hom-set adjunction . . . . . . . . . . . . . . . . . . . . . . . 72.4.2 Counit-unit adjunction induces hom-set adjunction . . . . . . . . . . . . . . . . . . . . . . 72.4.3 Hom-set adjunction induces all of the above . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.1 Ubiquity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.2 Problems formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5.3 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6.1 Free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6.2 Free constructions and forgetful functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6.3 Diagonal functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6.4 Colimits and diagonal functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.7.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.7.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7.3 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

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2.7.4 Limit preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7.5 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.8.1 Universal constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.8.2 Equivalences of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.8.3 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Adjoint representation 183.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.1 Adjoint representation of a Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Roots of a semisimple Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4.1 Example SL(2, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Variants and analogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Adjoint representation of a Lie algebra 224.1 Adjoint representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Relation to Ad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Hermitian adjoint 255.1 Definition for bounded operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.3 Adjoint of densely defined operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4 Hermitian operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.5 Adjoints of antilinear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.6 Other adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.10 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.10.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.10.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.10.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Chapter 1

Adjoint equation

An adjoint equation is a linear differential equation, usually derived from its primal equation using integrationby parts. Gradient values with respect to a particular quantity of interest can be efficiently calculated by solvingthe adjoint equation. Methods based on solution of adjoint equations are used in wing shape optimization, fluidflow control and uncertainty quantification. For example dXt = a(Xt)dt + b(Xt)dW this is an Itō stochasticdifferential equation. Now by using Euler scheme, we integrate the parts of this equation and get another equation,Xn+1 = Xn + a∆t+ ζb

√∆t , here ζ is a random variable, later one is an adjoint equation.

1.1 See also

Adjoint state method

1.2 References• Jameson, Antony (1988). “Aerodynamic Design via Control Theory”. Journal of Scientific Computing 3 (3).

1

Chapter 2

Adjoint functors

For the construction in field theory, see Adjunction (field theory). For the construction in topology, see Adjunctionspace.

In mathematics, specifically category theory, adjunction is a possible relationship between two functors.Adjunction is ubiquitous in mathematics, as it specifies intuitive notions of optimization and efficiency.In the most concise symmetric definition, an adjunction between categories C and D is a pair of functors,

F : D → C and G : C → D

and a family of bijections

homC(FY,X) ∼= homD(Y,GX)

which is natural in the variables X and Y. The functor F is called a left adjoint functor, while G is called a rightadjoint functor. The relationship “F is left adjoint to G” (or equivalently, “G is right adjoint to F”) is sometimeswritten

F ⊣ G.

This definition and others are made precise below.

2.1 Introduction

“The slogan is ‘Adjoint functors arise everywhere’.” (Saunders Mac Lane, Categories for the working mathematician)The long list of examples in this article is only a partial indication of how often an interesting mathematical construc-tion is an adjoint functor. As a result, general theorems about left/right adjoint functors, such as the equivalence oftheir various definitions or the fact that they respectively preserve colimits/limits (which are also found in every areaof mathematics), can encode the details of many useful and otherwise non-trivial results.

2.1.1 Spelling (or morphology)

One can observe (e.g. in this article), two different roots are used: “adjunct” and “adjoint”. From Oxford shorterEnglish dictionary, “adjunct” is from Latin, “adjoint” is from French.In Mac Lane, Categories for the working mathematician, chap. 4, “Adjoints”, one can verify the following usage.φ : homC(FY,X) ∼= homD(Y,GX)

2

2.2. MOTIVATION 3

The hom-set bijection φ is an “adjunction”.If f an arrow in homC(FY,X) , φf is the right “adjunct” of f (p. 81).The functor F is left “adjoint” for G .

2.2 Motivation

2.2.1 Solutions to optimization problems

It can be said that an adjoint functor is a way of giving themost efficient solution to some problem via a method whichis formulaic. For example, an elementary problem in ring theory is how to turn a rng (which is like a ring that mightnot have a multiplicative identity) into a ring. The most efficient way is to adjoin an element '1' to the rng, adjoinall (and only) the elements which are necessary for satisfying the ring axioms (e.g. r+1 for each r in the ring), andimpose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is formulaicin the sense that it works in essentially the same way for any rng.This is rather vague, though suggestive, and can be made precise in the language of category theory: a constructionis most efficient if it satisfies a universal property, and is formulaic if it defines a functor. Universal properties comein two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessaryto discuss one of them.The idea of using an initial property is to set up the problem in terms of some auxiliary category E, and then identifythat what we want is to find an initial object of E. This has an advantage that the optimization— the sense that weare finding the most efficient solution — means something rigorous and is recognisable, rather like the attainment ofa supremum. The category E is also formulaic in this construction, since it is always the category of elements of thefunctor to which one is constructing an adjoint. In fact, this latter category is precisely the comma category over thefunctor in question.As an example, take the given rng R, and make a category E whose objects are rng homomorphisms R→ S, with S aring having a multiplicative identity. The morphisms in E between R→ S1 and R→ S2 are commutative triangles ofthe form (R → S1,R → S2, S1 → S2) where S1 → S2 is a ring map (which preserves the identity). Note that this isprecisely the definition of the comma category of R over the inclusion of unitary rings into rng. The existence of amorphism between R→ S1 and R→ S2 implies that S1 is at least as efficient a solution as S2 to our problem: S2 canhave more adjoined elements and/or more relations not imposed by axioms than S1. Therefore, the assertion that anobject R→ R* is initial in E, that is, that there is a morphism from it to any other element of E, means that the ringR* is a most efficient solution to our problem.The two facts that this method of turning rngs into rings ismost efficient and formulaic can be expressed simultaneouslyby saying that it defines an adjoint functor.

2.2.2 Symmetry of optimization problems

Continuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is therea problem to which F is the most efficient solution?The notion that F is the most efficient solution to the problem posed by G is, in a certain rigorous sense, equivalent tothe notion that G poses the most difficult problem that F solves.This has the intuitive meaning that adjoint functors should occur in pairs, and in fact they do, but this is not trivial fromthe universal morphism definitions. The equivalent symmetric definitions involving adjunctions and the symmetriclanguage of adjoint functors (we can say either F is left adjoint to G or G is right adjoint to F) have the advantage ofmaking this fact explicit.

2.3 Formal definitions

There are various definitions for adjoint functors. Their equivalence is elementary but not at all trivial and in facthighly useful. This article provides several such definitions:

4 CHAPTER 2. ADJOINT FUNCTORS

• The definitions via universal morphisms are easy to state, and require minimal verifications when constructingan adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuitioninvolving optimizations.

• The definition via counit-unit adjunction is convenient for proofs about functors which are known to be adjoint,because they provide formulas that can be directly manipulated.

• The definition via hom-sets makes symmetry the most apparent, and is the reason for using the word adjoint.

Adjoint functors arise everywhere, in all areas of mathematics. Their full usefulness lies in that the structure in anyof these definitions gives rise to the structures in the others via a long but trivial series of deductions. Thus, switchingbetween themmakes implicit use of a great deal of tedious details that would otherwise have to be repeated separatelyin every subject area. For example, naturality and terminality of the counit can be used to prove that any right adjointfunctor preserves limits.

2.3.1 Conventions

The theory of adjoints has the terms left and right at its foundation, and there are many components which live in oneof two categories C and D which are under consideration. It can therefore be extremely helpful to choose letters inalphabetical order according to whether they live in the “lefthand” category C or the “righthand” category D, and alsoto write them down in this order whenever possible.In this article for example, the letters X, F, f, ε will consistently denote things which live in the category C, the lettersY, G, g, η will consistently denote things which live in the category D, and whenever possible such things will bereferred to in order from left to right (a functor F:C←D can be thought of as “living” where its outputs are, in C).

2.3.2 Universal morphisms

A functor F : C ← D is a left adjoint functor if for each object X in C, there exists a terminal morphism from F toX. If, for each object X in C, we choose an object G0X of D for which there is a terminal morphism εX : F(G0X) →X from F to X, then there is a unique functor G : C → D such that GX = G0X and εXʹ ∘ FG(f) = f ∘ εX for f : X→Xʹ a morphism in C; F is then called a left adjoint to G.A functor G : C → D is a right adjoint functor if for each object Y in D, there exists an initial morphism from Y toG. If, for each object Y in D, we choose an object F0Y of C and an initial morphism ηY : Y → G(F0Y) from Y to G,then there is a unique functor F : C ← D such that FY = F0Y and GF(g) ∘ ηY = ηYʹ ∘ g for g : Y → Yʹ a morphismin D; G is then called a right adjoint to F.Remarks:

It is true, as the terminology implies, that F is left adjoint to G if and only ifG is right adjoint to F. This is apparent fromthe symmetric definitions given below. The definitions via universal morphisms are often useful for establishing thata given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitivelymeaningful in that finding a universal morphism is like solving an optimization problem.

2.3.3 Counit-unit adjunction

A counit-unit adjunction between two categories C and D consists of two functors F : C ← D and G : C → D andtwo natural transformations

ε : FG→ 1C

η : 1D → GF

respectively called the counit and the unit of the adjunction (terminology from universal algebra), such that thecompositions

FFη−−−→FGF εF−−→F

2.4. ADJUNCTIONS IN FULL 5

GηG−−−→GFG Gε−−→G

are the identity transformations 1F and 1G on F and G respectively.In this situation we say that F is left adjoint to G and G is right adjoint to F , and may indicate this relationshipby writing (ε, η) : F ⊣ G , or simply F ⊣ G .In equation form, the above conditions on (ε,η) are the counit-unit equations

1F = εF ◦ Fη1G = Gε ◦ ηG

which mean that for each X in C and each Y in D,

1FY = εFY ◦ F (ηY )1GX = G(εX) ◦ ηGX

Note that here 1 denotes identity functors, while above the same symbol was used for identity natural transformations.These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimescalled the zig-zag equations because of the appearance of the corresponding string diagrams. A way to rememberthem is to first write down the nonsensical equation 1 = ε ◦ η and then fill in either F or G in one of the two simpleways which make the compositions defined.Note: The use of the prefix “co” in counit here is not consistent with the terminology of limits and colimits, becausea colimit satisfies an initial property whereas the counit morphisms will satisfy terminal properties, and dually. Theterm unit here is borrowed from the theory of monads where it looks like the insertion of the identity 1 into a monoid.

2.3.4 Hom-set adjunction

A hom-set adjunction between two categories C and D consists of two functors F : C ← D and G : C → D and anatural isomorphism

Φ : homC(F−,−)→ homD(−, G−)

This specifies a family of bijections

ΦY,X : homC(FY,X)→ homD(Y,GX)

for all objects X in C and Y in D.In this situation we say that F is left adjoint to G and G is right adjoint to F , and may indicate this relationshipby writing Φ : F ⊣ G , or simply F ⊣ G .This definition is a logical compromise in that it is somewhat more difficult to satisfy than the universal morphismdefinitions, and has fewer immediate implications than the counit-unit definition. It is useful because of its obvioussymmetry, and as a stepping-stone between the other definitions.In order to interpret Φ as a natural isomorphism, one must recognize homC(F–, –) and homD(–, G–) as functors. Infact, they are both bifunctors from Dop × C to Set (the category of sets). For details, see the article on hom functors.Explicitly, the naturality of Φ means that for all morphisms f : X→ X′ in C and all morphisms g : Y′ → Y in D thefollowing diagram commutes:The vertical arrows in this diagram are those induced by composition with f and g. Formally, Hom(Fg, f) : HomC(FY,X) → HomC(FY′, X′ ) is given by h→ f o h o Fg for each h in HomC(FY, X). Hom(g, Gf) is similar.

2.4 Adjunctions in full

There are hence numerous functors and natural transformations associated with every adjunction, and only a smallportion is sufficient to determine the rest.

6 CHAPTER 2. ADJOINT FUNCTORS

Naturality of Φ

An adjunction between categories C and D consists of

• A functor F : C ← D called the left adjoint

• A functor G : C → D called the right adjoint

• A natural isomorphism Φ : homC(F–,–) → homD(–,G–)

• A natural transformation ε : FG → 1C called the counit

• A natural transformation η : 1D→ GF called the unit

An equivalent formulation, where X denotes any object of C and Y denotes any object of D:For every C-morphism f : FY→ X, there is a unique D-morphism ΦY, X(f) = g : Y→ GX such that the diagramsbelow commute, and for every D-morphism g : Y→ GX, there is a unique C-morphism Φ−1Y, X(g) = f : FY→ X inC such that the diagrams below commute:

From this assertion, one can recover that:

2.4. ADJUNCTIONS IN FULL 7

• The transformations ε, η, and Φ are related by the equations

f = Φ−1Y,X(g) = εX ◦ F (g) ∈ homC(F (Y ), X)

g = ΦY,X(f) = G(f) ◦ ηY ∈ homD(Y,G(X))

Φ−1GX,X(1GX) = εX ∈ homC(FG(X), X)

ΦY,FY (1FY ) = ηY ∈ homD(Y,GF (Y ))

• The transformations ε, η satisfy the counit-unit equations

1FY = εFY ◦ F (ηY )1GX = G(εX) ◦ ηGX

• Each pair (GX, εX) is a terminal morphism from F to X in C

• Each pair (FY, ηY) is an initial morphism from Y to G in D

In particular, the equations above allow one to define Φ, ε, and η in terms of any one of the three. However, theadjoint functors F and G alone are in general not sufficient to determine the adjunction. We will demonstrate theequivalence of these situations below.

2.4.1 Universal morphisms induce hom-set adjunction

Given a right adjoint functor G : C → D; in the sense of initial morphisms, one may construct the induced hom-setadjunction by doing the following steps.

• Construct a functor F : C ← D and a natural transformation η.

• For each object Y in D, choose an initial morphism (F(Y), ηY) from Y to G, so we have ηY : Y →G(F(Y)). We have the map of F on objects and the family of morphisms η.• For each f : Y0 → Y1, as (F(Y0), ηY0) is an initial morphism, then factorize ηY1 o f with ηY0 and getF(f) : F(Y0) → F(Y1). This is the map of F on morphisms.• The commuting diagram of that factorization implies the commuting diagram of natural transformations,so η : 1D→ G o F is a natural transformation.• Uniqueness of that factorization and thatG is a functor implies that the map of F on morphisms preservescompositions and identities.

• Construct a natural isomorphism Φ : homC(F-,-) → homD(-,G-).

• For each objectX in C, each object Y inD, as (F(Y), ηY) is an initial morphism, then ΦY,X is a bijection,where ΦY, X(f : F(Y) → X) = G(f) o ηY .• η is a natural transformation, G is a functor, then for any objects X0, X1 in C, any objects Y0, Y1 in D,any x : X0 → X1, any y : Y1 → Y0, we have ΦY1, X1(x o f o F(y)) = G(x) o G(f) o G(F(y)) o ηY1 =G(x) o G(f) o ηY0 o y = G(x) o ΦY0, X0(f) o y, and then Φ is natural in both arguments.

A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor.(The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairsis a trivially defined inclusion or forgetful functor.)

2.4.2 Counit-unit adjunction induces hom-set adjunction

Given functors F : C ← D, G : C → D, and a counit-unit adjunction (ε, η) : F ⊣ G, we can construct a hom-setadjunction by finding the natural transformation Φ : homC(F-,-) → homD(-,G-) in the following steps:

• For each f : FY → X and each g : Y → GX, define

8 CHAPTER 2. ADJOINT FUNCTORS

ΦY,X(f) = G(f) ◦ ηYΨY,X(g) = εX ◦ F (g)The transformations Φ and Ψ are natural because η and ε are natural.

• Using, in order, that F is a functor, that ε is natural, and the counit-unit equation 1FY = εFY o F(ηY), we obtain

ΨΦf = εX ◦ FG(f) ◦ F (ηY )= f ◦ εFY ◦ F (ηY )= f ◦ 1FY = f

hence ΨΦ is the identity transformation.

• Dually, using that G is a functor, that η is natural, and the counit-unit equation 1GX = G(εX) o ηGX, we obtain

ΦΨg = G(εX) ◦GF (g) ◦ ηY= G(εX) ◦ ηGX ◦ g= 1GX ◦ g = g

hence ΦΨ is the identity transformation. Thus Φ is a natural isomorphism with inverse Φ−1 = Ψ.

2.4.3 Hom-set adjunction induces all of the above

Given functors F : C ← D, G : C → D, and a hom-set adjunction Φ : homC(F-,-) → homD(-,G-), we can construct acounit-unit adjunction

(ε, η) : F ⊣ G ,

which defines families of initial and terminal morphisms, in the following steps:

• Let εX = Φ−1GX,X(1GX) ∈ homC(FGX,X) for each X in C, where 1GX ∈ homD(GX,GX) is the identity

morphism.

• Let ηY = ΦY,FY (1FY ) ∈ homD(Y,GFY ) for each Y in D, where 1FY ∈ homC(FY, FY ) is the identitymorphism.

• The bijectivity and naturality of Φ imply that each (GX, εX) is a terminal morphism from F to X in C, andeach (FY, ηY) is an initial morphism from Y to G in D.

• The naturality of Φ implies the naturality of ε and η, and the two formulas

ΦY,X(f) = G(f) ◦ ηYΦ−1

Y,X(g) = εX ◦ F (g)

for each f: FY → X and g: Y → GX (which completely determine Φ).

• Substituting FY for X and ηY = ΦY, FY(1FY) for g in the second formula gives the first counit-unit equation

1FY = εFY ◦ F (ηY ) ,and substitutingGX for Y and εX =Φ−1GX, X(1GX) for f in the first formula gives the second counit-unitequation1GX = G(εX) ◦ ηGX .

2.5. HISTORY 9

2.5 History

2.5.1 Ubiquity

The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the concepts in category theory,it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those facedwith giving tidy, systematic presentations of the subject would have noticed relations such as

hom(F(X), Y) = hom(X, G(Y))

in the category of abelian groups, where F was the functor − ⊗ A (i.e. take the tensor product with A), and G wasthe functor hom(A,–). The use of the equals sign is an abuse of notation; those two groups are not really identicalbut there is a way of identifying them that is natural. It can be seen to be natural on the basis, firstly, that these aretwo alternative descriptions of the bilinear mappings from X × A to Y. That is, however, something particular to thecase of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a naturalisomorphism.The terminology comes from the Hilbert space idea of adjoint operators T, U with ⟨Tx, y⟩ = ⟨x, Uy⟩ , which isformally similar to the above relation between hom-sets. We say that F is left adjoint to G, and G is right adjoint toF. Note that G may have itself a right adjoint that is quite different from F (see below for an example). The analogyto adjoint maps of Hilbert spaces can be made precise in certain contexts.[1]

If one starts looking for these adjoint pairs of functors, they turn out to be very common in abstract algebra, andelsewhere as well. The example section below provides evidence of this; furthermore, universal constructions, whichmay be more familiar to some, give rise to numerous adjoint pairs of functors.In accordance with the thinking of Saunders Mac Lane, any idea, such as adjoint functors, that occurs widely enoughin mathematics should be studied for its own sake.

2.5.2 Problems formulations

Mathematicians do not generally need the full adjoint functor concept. Concepts can be judged according to theiruse in solving problems, as well as for their use in building theories. The tension between these two motivations wasespecially great during the 1950s when category theory was initially developed. Enter Alexander Grothendieck, whoused category theory to take compass bearings in other work— in functional analysis, homological algebra and finallyalgebraic geometry.It is probably wrong to say that he promoted the adjoint functor concept in isolation: but recognition of the role ofadjunction was inherent in Grothendieck’s approach. For example, one of his major achievements was the formulationof Serre duality in relative form — loosely, in a continuous family of algebraic varieties. The entire proof turned onthe existence of a right adjoint to a certain functor. This is something undeniably abstract, and non-constructive, butalso powerful in its own way.

2.5.3 Posets

Every partially ordered set can be viewed as a category (with a single morphism between x and y if and only if x ≤ y).A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant,an antitone Galois connection). See that article for a number of examples: the case of Galois theory of course is aleading one. Any Galois connection gives rise to closure operators and to inverse order-preserving bijections betweenthe corresponding closed elements.As is the case for Galois groups, the real interest lies often in refining a correspondence to a duality (i.e. antitoneorder isomorphism). A treatment of Galois theory along these lines by Kaplansky was influential in the recognitionof the general structure here.The partial order case collapses the adjunction definitions quite noticeably, but can provide several themes:

• adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status

10 CHAPTER 2. ADJOINT FUNCTORS

• closure operators may indicate the presence of adjunctions, as corresponding monads (cf. the Kuratowskiclosure axioms)

• a very general comment of William Lawvere[2] is that syntax and semantics are adjoint: take C to be the setof all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For atheory T in C, let F(T) be the set of all structures that satisfy the axioms T ; for a set of mathematical structuresS, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if Tlogically implies G(S): the “semantics functor” F is left adjoint to the “syntax functor” G.

• division is (in general) the attempt to invert multiplication, but many examples, such as the introduction ofimplication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as theattempt to provide an adjoint.

Together these observations provide explanatory value all over mathematics.

2.6 Examples

2.6.1 Free groups

The construction of free groups is a common and illuminating example.Suppose that F :Grp← Set is the functor assigning to each set Y the free group generated by the elements of Y, andthat G : Grp→ Set is the forgetful functor, which assigns to each group X its underlying set. Then F is left adjointto G:Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements ofX. Let εX : FGX → X be the group homomorphism which sends the generators of FGX to the elements of X theycorrespond to, which exists by the universal property of free groups. Then each (GX, εX) is a terminal morphismfrom F to X, because any group homomorphism from a free group FZ to X will factor through εX : FGX → X viaa unique set map from Z to GX. This means that (F,G) is an adjoint pair.Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. LetηY : Y → GFY be the set map given by “inclusion of generators”. Then each (FY, ηY ) is an initial morphism fromY to G, because any set map from Y to the underlying set GW of a group will factor through ηY : Y → GFY via aunique group homomorphism from FY toW. This also means that (F,G) is an adjoint pair.Hom-set adjunction. Maps from the free group FY to a group X correspond precisely to maps from the set Y to theset GX: each homomorphism from FY to X is fully determined by its action on generators. One can verify directlythat this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair (F,G).Counit-unit adjunction. One can also verify directly that ε and η are natural. Then, a direct verification that theyform a counit-unit adjunction (ε, η) : F ⊣ G is as follows:The first counit-unit equation 1F = εF ◦ Fη says that for each set Y the composition

FYF (ηY )−−−−−→FGFY εFY−−−→FY

should be the identity. The intermediate group FGFY is the free group generated freely by the words of the freegroup FY. (Think of these words as placed in parentheses to indicate that they are independent generators.) Thearrow F (ηY ) is the group homomorphism from FY into FGFY sending each generator y of FY to the correspondingword of length one (y) as a generator of FGFY. The arrow εFY is the group homomorphism from FGFY to FY sendingeach generator to the word of FY it corresponds to (so this map is “dropping parentheses”). The composition of thesemaps is indeed the identity on FY.The second counit-unit equation 1G = Gε ◦ ηG says that for each group X the composition

GXηGX−−−−→GFGX G(εX)−−−−−→GX

should be the identity. The intermediate set GFGX is just the underlying set of FGX. The arrow ηGX is the “inclusionof generators” set map from the set GX to the set GFGX. The arrow G(εX) is the set map from GFGX to GX which

2.6. EXAMPLES 11

underlies the group homomorphism sending each generator of FGX to the element of X it corresponds to (“droppingparentheses”). The composition of these maps is indeed the identity on GX.

2.6.2 Free constructions and forgetful functors

Free objects are all examples of a left adjoint to a forgetful functor which assigns to an algebraic object its underlyingset. These algebraic free functors have generally the same description as in the detailed description of the free groupsituation above.

2.6.3 Diagonal functors and limits

Products, fibred products, equalizers, and kernels are all examples of the categorical notion of a limit. Any limitfunctor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question),and the counit of the adjunction provides the defining maps from the limit object (i.e. from the diagonal functor onthe limit, in the functor category). Below are some specific examples.

• Products Let Π : Grp2 → Grp the functor which assigns to each pair (X1, X2) the product group X1×X2,and let Δ : Grp2 ← Grp be the diagonal functor which assigns to every group X the pair (X, X) in the productcategory Grp2. The universal property of the product group shows that Π is right-adjoint to Δ. The counit ofthis adjunction is the defining pair of projection maps from X1×X2 to X1 and X2 which define the limit, andthe unit is the diagonal inclusion of a group X into X1×X2 (mapping x to (x,x)).

The cartesian product of sets, the product of rings, the product of topological spaces etc. follow thesame pattern; it can also be extended in a straightforward manner to more than just two factors. Moregenerally, any type of limit is right adjoint to a diagonal functor.

• Kernels. Consider the category D of homomorphisms of abelian groups. If f1 : A1 → B1 and f2 : A2 → B2

are two objects of D, then a morphism from f1 to f2 is a pair (gA, gB) of morphisms such that gBf1 = f2gA.Let G : D → Ab be the functor which assigns to each homomorphism its kernel and let F : D ← Ab be thefunctor which maps the groupA to the homomorphismA→0. ThenG is right adjoint to F, which expresses theuniversal property of kernels. The counit of this adjunction is the defining embedding of a homomorphism’skernel into the homomorphism’s domain, and the unit is the morphism identifying a group A with the kernelof the homomorphism A→ 0.

A suitable variation of this example also shows that the kernel functors for vector spaces and for modulesare right adjoints. Analogously, one can show that the cokernel functors for abelian groups, vector spacesand modules are left adjoints.

2.6.4 Colimits and diagonal functors

Coproducts, fibred coproducts, coequalizers, and cokernels are all examples of the categorical notion of a colimit.Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimitsin question), and the unit of the adjunction provides the defining maps into the colimit object. Below are some specificexamples.

• Coproducts. If F : Ab ← Ab2 assigns to every pair (X1, X2) of abelian groups their direct sum, and if G :Ab → Ab2 is the functor which assigns to every abelian group Y the pair (Y, Y), then F is left adjoint to G,again a consequence of the universal property of direct sums. The unit of this adjoint pair is the defining pairof inclusion maps from X1 and X2 into the direct sum, and the counit is the additive map from the direct sumof (X,X) to back to X (sending an element (a,b) of the direct sum to the element a+b of X).

Analogous examples are given by the direct sum of vector spaces and modules, by the free product ofgroups and by the disjoint union of sets.

12 CHAPTER 2. ADJOINT FUNCTORS

2.6.5 Further examples

Algebra

• Adjoining an identity to a rng. This example was discussed in the motivation section above. Given a rng R,a multiplicative identity element can be added by taking RxZ and defining a Z-bilinear product with (r,0)(0,1)= (0,1)(r,0) = (r,0), (r,0)(s,0) = (rs,0), (0,1)(0,1) = (0,1). This constructs a left adjoint to the functor taking aring to the underlying rng.

• Ring extensions. Suppose R and S are rings, and ρ : R→ S is a ring homomorphism. Then S can be seen asa (left) R-module, and the tensor product with S yields a functor F : R-Mod→ S-Mod. Then F is left adjointto the forgetful functor G : S-Mod→ R-Mod.

• Tensor products. If R is a ring and M is a right R module, then the tensor product with M yields a functor F: R-Mod→ Ab. The functor G : Ab→ R-Mod, defined by G(A) = homZ(M,A) for every abelian group A, isa right adjoint to F.

• From monoids and groups to rings The integral monoid ring construction gives a functor from monoidsto rings. This functor is left adjoint to the functor that associates to a given ring its underlying multiplicativemonoid. Similarly, the integral group ring construction yields a functor from groups to rings, left adjoint to thefunctor that assigns to a given ring its group of units. One can also start with a field K and consider the categoryof K-algebras instead of the category of rings, to get the monoid and group rings over K.

• Field of fractions. Consider the category Dom of integral domains with injective morphisms. The forgetfulfunctor Field→ Dom from fields has a left adjoint - it assigns to every integral domain its field of fractions.

• Polynomial rings. Let Ring* be the category of pointed commutative rings with unity (pairs (A,a) where Ais a ring, a ∈ A and morphisms preserve the distinguished elements). The forgetful functor G:Ring* → Ringhas a left adjoint - it assigns to every ring R the pair (R[x],x) where R[x] is the polynomial ring with coefficientsfrom R.

• Abelianization. Consider the inclusion functor G :Ab→Grp from the category of abelian groups to categoryof groups. It has a left adjoint called abelianization which assigns to every group G the quotient groupGab=G/[G,G].

• TheGrothendieck group. In K-theory, the point of departure is to observe that the category of vector bundleson a topological space has a commutative monoid structure under direct sum. One may make an abeliangroup out of this monoid, the Grothendieck group, by formally adding an additive inverse for each bundle(or equivalence class). Alternatively one can observe that the functor that for each group takes the underlyingmonoid (ignoring inverses) has a left adjoint. This is a once-for-all construction, in line with the third sectiondiscussion above. That is, one can imitate the construction of negative numbers; but there is the other optionof an existence theorem. For the case of finitary algebraic structures, the existence by itself can be referred touniversal algebra, or model theory; naturally there is also a proof adapted to category theory, too.

• Frobenius reciprocity in the representation theory of groups: see induced representation. This exampleforeshadowed the general theory by about half a century.

Topology

• A functor with a left and a right adjoint. Let G be the functor from topological spaces to sets that associatesto every topological space its underlying set (forgetting the topology, that is). G has a left adjoint F, creatingthe discrete space on a set Y, and a right adjoint H creating the trivial topology on Y.

• Suspensions and loop spaces Given topological spaces X and Y, the space [SX, Y] of homotopy classes ofmaps from the suspension SX of X to Y is naturally isomorphic to the space [X, ΩY] of homotopy classes ofmaps from X to the loop space ΩY of Y. This is an important fact in homotopy theory.

2.6. EXAMPLES 13

• Stone-Čech compactification. Let KHaus be the category of compact Hausdorff spaces and G : KHaus→Top be the inclusion functor to the category of topological spaces. ThenG has a left adjoint F : Top→KHaus,the Stone–Čech compactification. The unit of this adjoint pair yields a continuous map from every topologicalspace X into its Stone-Čech compactification. This map is an embedding (i.e. injective, continuous and open)if and only if X is a Tychonoff space.

• Direct and inverse images of sheaves Every continuous map f : X→ Y between topological spaces inducesa functor f∗ from the category of sheaves (of sets, or abelian groups, or rings...) on X to the correspondingcategory of sheaves on Y, the direct image functor. It also induces a functor f −1 from the category of sheavesof abelian groups on Y to the category of sheaves of abelian groups on X, the inverse image functor. f −1 isleft adjoint to f∗. Here a more subtle point is that the left adjoint for coherent sheaves will differ from that forsheaves (of sets).

• Soberification. The article on Stone duality describes an adjunction between the category of topological spacesand the category of sober spaces that is known as soberification. Notably, the article also contains a detaileddescription of another adjunction that prepares the way for the famous duality of sober spaces and spatiallocales, exploited in pointless topology.

Category theory

• A series of adjunctions. The functor π0 which assigns to a category its set of connected components is left-adjoint to the functor D which assigns to a set the discrete category on that set. Moreover, D is left-adjoint tothe object functor U which assigns to each category its set of objects, and finally U is left-adjoint to A whichassigns to each set the indiscrete category on that set.

• Exponential object. In a cartesian closed category the endofunctor C → C given by –×A has a right adjoint–A.

Categorical logic

• Quantification. If ϕY is a unary predicate expressing some property, then a sufficiently strong set theory mayprove the existence of the set Y = {y | ϕY (y)} of terms that fulfill the property. A proper subset T ⊂ Yand the associated injection of T into Y is characterized by a predicate ϕT (y) = ϕY (y) ∧ φ(y) expressing astrictly more restrictive property.

The role of quantifiers in predicate logics is in forming propositions and also in expressing sophisticatedpredicates by closing formulas with possibly more variables. For example, consider a predicate ψf withtwo open variables of sort X and Y . Using a quantifier to close X , we can form the set

{y ∈ Y | ∃x. ψf (x, y) ∧ ϕS(x)}

of all elements y of Y for which there is an x to which it is ψf -related, and which itself is characterizedby the property ϕS . Set theoretic operations like the intersection ∩ of two sets directly corresponds tothe conjunction ∧ of predicates. In categorical logic, a subfield of topos theory, quantifiers are identifiedwith adjoints to the pullback functor. Such a realization can be seen in analogy to the discussion ofpropositional logic using set theory but, interestingly, the general definition make for a richer range oflogics.

So consider an object Y in a category with pullbacks. Any morphism f : X → Y induces a functor

f∗ : Sub(Y ) −→ Sub(X)

on the category that is the preorder of subobjects. It maps subobjects T of Y (technically: monomor-phism classes of T → Y ) to the pullback X ×Y T . If this functor has a left- or right adjoint, they arecalled ∃f and ∀f , respectively.[3] They both map from Sub(X) back to Sub(Y ) . Very roughly, given adomain S ⊂ X to quantify a relation expressed via f over, the functor/quantifier closes X in X ×Y Tand returns the thereby specified subset of Y .

14 CHAPTER 2. ADJOINT FUNCTORS

Example: In Set , the category of sets and functions, the canonical subobjects are the subset (or rathertheir canonical injections). The pullback f∗T = X ×Y T of an injection of a subset T into Y along fis characterized as the largest set which knows all about f and the injection of T into Y . It thereforeturns out to be (in bijection with) the inverse image f−1[T ] ⊆ X .For S ⊆ X , let us figure out the left adjoinet, which is defined via

Hom(∃fS, T ) ∼= Hom(S, f∗T ),

which here just means

∃fS ⊆ T ↔ S ⊆ f−1[T ]

Consider f [S] ⊆ T . We see S ⊆ f−1[f [S]] ⊆ f−1[T ] . Conversely, If for an x ∈ S we also havex ∈ f−1[T ] , then clearly f(x) ∈ T . So S ⊆ f−1[T ] implies f [S] ⊆ T . We concude that left adjointto the inverse image functor f∗ is given by the direct image. Here is a characterization of this result,which matches more the logical interpretation: The image of S under ∃f is the full set of y 's, suchthat f−1[{y}] ∩ S is non-empty. This works because it neglects exactly those y ∈ Y which are in thecomplement of f [S] . So

∃fS = {y ∈ Y | ∃(x ∈ f−1[{y}]). x ∈ S } = f [S].

Put this in analogy to our motivation {y ∈ Y | ∃x. ψf (x, y) ∧ ϕS(x)} .The right adjoint to the inverse image functor is given (without doing the computation here) by

∀fS = {y ∈ Y | ∀(x ∈ f−1[{y}]). x ∈ S }.

The subset ∀fS of Y is characterized as the full set of y 's with the property that the inverse image of{y} with respect to f is fully contained within S . Note how the predicate determining the set is thesame as above, except that ∃ is replaced by ∀ .

See also powerset.

2.7 Properties

2.7.1 Existence

Not every functor G : C→ D admits a left adjoint. If C is a complete category, then the functors with left adjoints canbe characterized by the adjoint functor theorem of Peter J. Freyd: G has a left adjoint if and only if it is continuousand a certain smallness condition is satisfied: for every object Y of D there exists a family of morphisms

fi : Y → G(Xi)

where the indices i come from a set I, not a proper class, such that every morphism

h : Y → G(X)

can be written as

h = G(t) o fi

for some i in I and some morphism

t : Xi→ X in C.

An analogous statement characterizes those functors with a right adjoint.

2.7. PROPERTIES 15

2.7.2 Uniqueness

If the functor F : C ← D has two right adjoints G and G′, then G and G′ are naturally isomorphic. The same is truefor left adjoints.Conversely, if F is left adjoint to G, and G is naturally isomorphic to G′ then F is also left adjoint to G′. Moregenerally, if F, G, ε, η is an adjunction (with counit-unit (ε,η)) and

σ : F → F′τ : G → G′

are natural isomorphisms then F′, G′, ε′, η′ is an adjunction where

η′ = (τ ∗ σ) ◦ ηε′ = ε ◦ (σ−1 ∗ τ−1).

Here ◦ denotes vertical composition of natural transformations, and ∗ denotes horizontal composition.

2.7.3 Composition

Adjunctions can be composed in a natural fashion. Specifically, if F, G, ε, η is an adjunction between C and Dand F′, G′, ε′, η′ is an adjunction between D and E then the functor

F ′ ◦ F : C ← E

is left adjoint to

G ◦G′ : C → E .

More precisely, there is an adjunction between F′ F and G G′ with unit and counit given by the compositions:

1Eη−→GF Gη′F−−−→GG′F ′F

F ′FGG′ F ′εG′

−−−−→F ′G′ ε′−→1C .

This new adjunction is called the composition of the two given adjunctions.One can then form a category whose objects are all small categories and whose morphisms are adjunctions.

2.7.4 Limit preservation

The most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore is aright adjoint) is continuous (i.e. commutes with limits in the category theoretical sense); every functor that has a rightadjoint (and therefore is a left adjoint) is cocontinuous (i.e. commutes with colimits).Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. Forexample:

• applying a right adjoint functor to a product of objects yields the product of the images;

• applying a left adjoint functor to a coproduct of objects yields the coproduct of the images;

• every right adjoint functor is left exact;

• every left adjoint functor is right exact.

16 CHAPTER 2. ADJOINT FUNCTORS

2.7.5 Additivity

If C and D are preadditive categories and F : C ← D is an additive functor with a right adjoint G : C → D, then G isalso an additive functor and the hom-set bijections

ΦY,X : homC(FY,X) ∼= homD(Y,GX)

are, in fact, isomorphisms of abelian groups. Dually, if G is additive with a left adjoint F, then F is also additive.Moreover, if both C and D are additive categories (i.e. preadditive categories with all finite biproducts), then any pairof adjoint functors between them are automatically additive.

2.8 Relationships

2.8.1 Universal constructions

As stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one foreach object in C and one for each object in D. Conversely, if there exists a universal morphism to a functor G : C →D from every object of D, then G has a left adjoint.However, universal constructions are more general than adjoint functors: a universal construction is like an opti-mization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of D(equivalently, every object of C).

2.8.2 Equivalences of categories

If a functor F: C←D is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence ofcategories, i.e. an adjunction whose unit and counit are isomorphisms.Every adjunction F, G, ε, η extends an equivalence of certain subcategories. Define C1 as the full subcategory ofC consisting of those objects X of C for which εX is an isomorphism, and define D1 as the full subcategory of Dconsisting of those objects Y of D for which ηY is an isomorphism. Then F and G can be restricted to D1 and C1

and yield inverse equivalences of these subcategories.In a sense, then, adjoints are “generalized” inverses. Note however that a right inverse of F (i.e. a functor G such thatFG is naturally isomorphic to 1D) need not be a right (or left) adjoint of F. Adjoints generalize two-sided inverses.

2.8.3 Monads

Every adjunction F, G, ε, η gives rise to an associated monad T, η, μ in the category D. The functor

T : D → D

is given by T = GF. The unit of the monad

η : 1D → T

is just the unit η of the adjunction and the multiplication transformation

µ : T 2 → T

is given by μ = GεF. Dually, the triple FG, ε, FηG defines a comonad in C.Every monad arises from some adjunction—in fact, typically from many adjunctions—in the above fashion. Twoconstructions, called the category of Eilenberg–Moore algebras and the Kleisli category are two extremal solutions tothe problem of constructing an adjunction that gives rise to a given monad.

2.9. REFERENCES 17

2.9 References[1] arXiv.org: John C. Baez Higher-Dimensional Algebra II: 2-Hilbert Spaces.

[2] William Lawvere, Adjointness in foundations, Dialectica, 1969, available here. The notation is different nowadays; aneasier introduction by Peter Smith in these lecture notes, which also attribute the concept to the article cited.

[3] Saunders Mac Lane, Ieke Moerdijk, (1992) Sheaves in Geometry and Logic Springer-Verlag. ISBN 0-387-97710-4 Seepage 58

• Adámek, Jiří; Herrlich, Horst; Strecker, George E. (1990). Abstract and Concrete Categories. The joy of cats(PDF). John Wiley & Sons. ISBN 0-471-60922-6. Zbl 0695.18001.

• Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5(2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.

2.10 External links• Adjunctions Seven short lectures on adjunctions.

Chapter 3

Adjoint representation

Inmathematics, the adjoint representation (or adjoint action) of a Lie groupG is a way of representing the elementsof the group as linear transformations of the group’s Lie algebra, considered as a vector space. For example, in thecase where G is the Lie group of invertible matrices of size n, GL(n), the Lie algebra is the vector space of all (notnecessarily invertible) n-by-nmatrices. So in this case the adjoint representation is the vector space of n-by-nmatricesx , and any element g in GL(n) acts as a linear transformation of this vector space given by conjugation: x 7→ gxg−1

.For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action ofG on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields.

3.1 Definition

See also: Representation theory and Lie_group § The Lie algebra associated with a Lie group

LetG be a Lie group and let g be its Lie algebra (which we identify with TeG, the tangent space to the identity elementin G). Define the map

Ψ : G→ Aut(G), g 7→ Ψg

where Aut(G) is the automorphism group of G and the automorphism Ψg is defined by

Ψg(h) = ghg−1

for all h in G. The differential of Ψg at the identity is an automorphism of the Lie algebra g . We denote this map byAdg:

d(Ψg)e = Adg : g→ g.

To say that Adg is a Lie algebra automorphism is to say that Adg is a linear transformation of g that preserves the Liebracket. The map

Ad : G→ Aut(g), g 7→ Adgis called the adjoint representation ofG. This is indeed a representation ofG since Aut(g) is a closed[1] Lie subgroupof GL(g) and the above adjoint map is a Lie group homomorphism. Note Ad is a trivial map if G is abelian.If G is an (immersed) Lie subgroup of the general linear group GLn(C) , then, since the exponential map is thematrix exponential: exp(X) = eX , taking the derivative of Ψg(exp(tX)) = etgXg−1 at t = 0, one gets: for g in Gand X in g ,

18

3.2. EXAMPLES 19

Adg(X) = gXg−1

where on the right we have the products of matrices.

3.1.1 Adjoint representation of a Lie algebra

Main article: Adjoint representation of a Lie algebra

One may always pass from a representation of a Lie group G to a representation of its Lie algebra by taking thederivative at the identity.Taking the derivative of the adjoint map

Ad : G→ Aut(g)

gives the adjoint representation of the Lie algebra g :

d(Ad)x : Tx(G)→ TAd(x)(Aut(g))

ad : g→ Der(g).Here Der(g) is the Lie algebra of Aut(g) which may be identified with the derivation algebra of g . The adjointrepresentation of a Lie algebra is related in a fundamental way to the structure of that algebra. In particular, one canshow that

adx(y) = [x, y]

for all x, y ∈ g .[2]

3.2 Examples• If G is abelian of dimension n, the adjoint representation of G is the trivial n-dimensional representation.

• IfG is a matrix Lie group (i.e. a closed subgroup ofGL(n,C)), then its Lie algebra is an algebra of n×nmatriceswith the commutator for a Lie bracket (i.e. a subalgebra of gln(C) ). In this case, the adjoint map is given byAdg(x) = gxg−1.

• If G is SL(2, R) (real 2×2 matrices with determinant 1), the Lie algebra of G consists of real 2×2 matrices withtrace 0. The representation is equivalent to that given by the action of G by linear substitution on the space ofbinary (i.e., 2 variable) quadratic forms.

3.3 Properties

The following table summarizes the properties of the various maps mentioned in the definitionThe image of G under the adjoint representation is denoted by Ad(G). If G is connected, the kernel of the adjointrepresentation coincides with the kernel of Ψ which is just the center of G. Therefore the adjoint representation of aconnected Lie group G is faithful if and only if G is centerless. More generally, if G is not connected, then the kernelof the adjoint map is the centralizer of the identity component G0 of G. By the first isomorphism theorem we have

Ad(G) ∼= G/ZG(G0).

20 CHAPTER 3. ADJOINT REPRESENTATION

Given a finite-dimensional real Lie algebra g , by Lie’s third theorem, there is a connected Lie group Int(g) whoseLie algebra is the image of the adjoint representation of g (i.e., Lie(Int(g)) = ad(g) .) It is called the adjoint groupof g .Now, if g is the Lie algebra of a connected Lie group G, then Int(g) is the image of the adjoint representation of G:Int(g) = Ad(G) .

3.4 Roots of a semisimple Lie group

If G is semisimple, the non-zero weights of the adjoint representation form a root system. To see how this works,consider the case G = SL(n, R). We can take the group of diagonal matrices diag(t1, ..., tn) as our maximal torus T.Conjugation by an element of T sends

a11 a12 · · · a1na21 a22 · · · a2n...

... . . . ...an1 an2 · · · ann

7→

a11 t1t−12 a12 · · · t1t

−1n a1n

t2t−11 a21 a22 · · · t2t

−1n a2n

...... . . . ...

tnt−11 an1 tnt

−12 an2 · · · ann

.Thus, T acts trivially on the diagonal part of the Lie algebra of G and with eigenvectors titj−1 on the various off-diagonal entries. The roots of G are the weights diag(t1, ..., tn) → titj−1. This accounts for the standard descriptionof the root system of G = SLn(R) as the set of vectors of the form ei−ej.

3.4.1 Example SL(2, R)

Let us compute the root system for one of the simplest cases of Lie Groups. Let us consider the group SL(2, R) oftwo dimensional matrices with determinant 1. This consists of the set of matrices of the form:

[a bc d

]with a, b, c, d real and ad − bc = 1.A maximal compact connected abelian Lie subgroup, or maximal torus T, is given by the subset of all matrices of theform

[t1 00 t2

]=

[t1 00 1/t1

]=

[exp(θ) 0

0 exp(−θ)

]with t1t2 = 1 . The Lie algebra of the maximal torus is the Cartan subalgebra consisting of the matrices

[θ 00 −θ

]= θ

[1 00 0

]− θ

[0 00 1

]= θ(e1 − e2).

If we conjugate an element of SL(2, R) by an element of the maximal torus we obtain

[t1 00 1/t1

][a bc d

][1/t1 00 t1

]=

[at1 bt1c/t1 d/t1

][1/t1 00 t1

]=

[a bt21

ct−21 d

]The matrices

[1 00 0

][0 00 1

][0 10 0

][0 01 0

]

3.5. VARIANTS AND ANALOGUES 21

are then 'eigenvectors’ of the conjugation operation with eigenvalues 1, 1, t21, t−21 . The function Λ which gives t21 is

a multiplicative character, or homomorphism from the group’s torus to the underlying field R. The function λ givingθ is a weight of the Lie Algebra with weight space given by the span of the matrices.It is satisfying to show the multiplicativity of the character and the linearity of the weight. It can further be provedthat the differential of Λ can be used to create a weight. It is also educational to consider the case of SL(3, R).

3.5 Variants and analogues

The adjoint representation can also be defined for algebraic groups over any field.The co-adjoint representation is the contragredient representation of the adjoint representation. Alexandre Kirillovobserved that the orbit of any vector in a co-adjoint representation is a symplectic manifold. According to the philos-ophy in representation theory known as the orbit method (see also the Kirillov character formula), the irreduciblerepresentations of a Lie group G should be indexed in some way by its co-adjoint orbits. This relationship is closestin the case of nilpotent Lie groups.

3.6 Notes[1] The condition that a linear map is a Lie algebra homomorphism is a closed condition.

[2]

adx(y) = d(Adx)e(y)

= limε→0

(I + εx)y(I + εx)−1 − y

ε

= limε→0

(I + εx)y(I − εx+ (εx)2 +O(ε3))− y

ε

= limε→0

((I + εx)yI − (I + εx)yεx+ (I + εx)y(εx)2 +O(ε3))− y

ε

= limε→0

(IyI + εxyI − Iyεx− εxyεx+ Iy(εx)2 + εxy(εx)2 +O(ε3))− y

ε

= limε→0

y + xyε− yxε− xyxε2 + yx2ε2 + xyx2ε2 +O(ε3)− y

ε

= limε→0

xy − yx− xyxε+ yx2ε+ xyx2ε+O(ε2)

= [x, y]

3.7 References• Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics,Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN978-0-387-97527-6

Chapter 4

Adjoint representation of a Lie algebra

In mathematics, the adjoint endomorphism or adjoint action is a homomorphism of Lie algebras that plays afundamental role in the development of the theory of Lie algebras.Given an element x of a Lie algebra g , one defines the adjoint action of x on g as the map

adx : g→ g with adx(y) = [x, y]

for all y in g .The concept generates the adjoint representation of a Lie group Ad. In fact, ad is the differential of Ad at the identityelement of the group.

4.1 Adjoint representation

Let g be a Lie algebra over a field k. Then the linear mapping

ad : g→ End(g)

given by x↦ adx is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Itsimage actually lies in Der (g) . See below.)Within End (g) , the Lie bracket is, by definition, given by the commutator of the two operators:

[adx, ady] = adx ◦ ady − ady ◦ adx

where ○ denotes composition of linear maps.If g is finite-dimensional, then End (g) is isomorphic to gl(g) , the Lie algebra of the general linear group over thevector space g and if a basis for it is chosen, the composition corresponds to matrix multiplication.Using the above definition of the Lie bracket, the Jacobi identity

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0

takes the form

([adx, ady]) (z) =(ad[x,y]

)(z)

where x, y, and z are arbitrary elements of g .

22

4.2. STRUCTURE CONSTANTS 23

This last identity says that ad really is a Lie algebra homomorphism; i.e., a linear mapping that takes brackets tobrackets.In a more module-theoretic language, the construction simply says that g is a module over itself.The kernel of ad is, by definition, the center of g . Next, we consider the image of ad. Recall that a derivation on aLie algebra is a linear map δ : g→ g that obeys the Leibniz' law, that is,

δ([x, y]) = [δ(x), y] + [x, δ(y)]

for all x and y in the algebra.That adx is a derivation is a consequence of the Jacobi identity. This implies that the image of g under ad is asubalgebra of Der (g) , the space of all derivations of g .

4.2 Structure constants

The explicit matrix elements of the adjoint representation are given by the structure constants of the algebra. That is,let {ei} be a set of basis vectors for the algebra, with

[ei, ej ] =∑k

cijkek.

Then the matrix elements for adₑᵢ are given by

[adei ]kj= cijk .

Thus, for example, the adjoint representation of su(2) is the defining rep of so(3).

4.3 Relation to Ad

Ad and ad are related through the exponential map: crudely, Ad = exp ad, where Ad is the adjoint representation fora Lie group.To be more precise, let G be a Lie group, and let Ψ: G → Aut(G) be the mapping g↦Ψg, with Ψg: G → G givenby the inner automorphism

Ψg(h) = ghg−1 .

It is an example of a Lie group map. Define Adg to be the derivative of Ψg at the origin:

Adg = (dΨg)e : TeG→ TeG

where d is the differential and TeG is the tangent space at the origin e (e being the identity element of the group G).The Lie algebra of G is g = Te G. Since Adg ∈ Aut (g) , Ad: g↦Adg is a map from G to Aut(TₑG) which will havea derivative from TₑG to End(TₑG) (the Lie algebra of Aut(V) being End(V)).Then we have

ad = d(Ad)e : TeG→ End(TeG).The use of upper-case/lower-case notation is used extensively in the literature. Thus, for example, a vector x inthe algebra g generates a vector field X in the group G. Similarly, the adjoint map adₓy = [x,y] of vectors in g ishomomorphic to the Lie derivative LXY = [X,Y] of vector fields on the group G considered as a manifold.Further see the derivative of the exponential map.

24 CHAPTER 4. ADJOINT REPRESENTATION OF A LIE ALGEBRA

4.4 References• Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics,Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN978-0-387-97527-6

Chapter 5

Hermitian adjoint

In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has acorresponding adjoint operator. Adjoints of operators generalize conjugate transposes of square matrices to (pos-sibly) infinite-dimensional situations. If one thinks of operators on a complex Hilbert space as “generalized complexnumbers”, then the adjoint of an operator plays the role of the complex conjugate of a complex number.The adjoint of an operator A may also be called theHermitian adjoint,Hermitian conjugate orHermitian trans-pose[1] (after Charles Hermite) of A and is denoted by A* or A† (the latter especially when used in conjunction withthe bra–ket notation).

5.1 Definition for bounded operators

Suppose H is a complex Hilbert space, with inner product ⟨·, ·⟩ . Consider a continuous linear operator A : H → H(for linear operators, continuity is equivalent to being a bounded operator). Then the adjoint of A is the continuouslinear operator A* : H → H satisfying

⟨Ax, y⟩ = ⟨x,A∗y⟩ for all x, y ∈ H.Existence and uniqueness of this operator follows from the Riesz representation theorem.[2]

This can be seen as a generalization of the adjoint matrix of a square matrix which has a similar property involvingthe standard complex inner product.

5.2 Properties

The following properties of the Hermitian adjoint of bounded operators are immediate:[2]

1. A** = A – involutiveness2. If A is invertible, then so is A*, with (A*)−1 = (A−1)*3. (A + B)* = A* + B*4. (λA)* = λA*, where λ denotes the complex conjugate of the complex number λ – antilinearity (together with

3.)5. (AB)* = B* A*

If we define the operator norm of A by

∥A∥op := sup{∥Ax∥ : ∥x∥ ≤ 1}

then

25

26 CHAPTER 5. HERMITIAN ADJOINT

∥A∗∥op = ∥A∥op. [2]

Moreover,

∥A∗A∥op = ∥A∥2op. [2]

One says that a norm that satisfies this condition behaves like a “largest value”, extrapolating from the case of self-adjoint operators.The set of bounded linear operators on a complex Hilbert space H together with the adjoint operation and the operatornorm form the prototype of a C*-algebra.

5.3 Adjoint of densely defined operators

A densely defined operator A from a complex Hilbert space H to itself is a linear operator whose domain D(A) is adense linear subspace of H and whose values lies in H.[3] By definition, the domain D(A*) of its adjoint A* is the setof all y ∈ H for which there is a z ∈ H satisfying

⟨Ax, y⟩ = ⟨x, z⟩ for all x ∈ D(A),

and A*(y) is defined to be the z thus found.[4]

Properties 1.–5. hold with appropriate clauses about domains and codomains. For instance, the last property nowstates that (AB)* is an extension of B*A* if A, B and AB are densely defined operators.[5]

The relationship between the image of A and the kernel of its adjoint is given by:

kerA∗ = (im A)⊥

(kerA∗)⊥= im A

These statements are equivalent. See orthogonal complement for the proof of this and for the definition of ⊥ .Proof of the first equation:[6]

A∗x = 0 ⇐⇒ ⟨A∗x, y⟩ = 0 ∀y ∈ H⇐⇒ ⟨x,Ay⟩ = 0 ∀y ∈ H⇐⇒ x ⊥ im A

The second equation follows from the first by taking the orthogonal complement on both sides. Note that in general,the image need not be closed, but the kernel of a continuous operator[7] always is.

5.4 Hermitian operators

A bounded operator A : H → H is called Hermitian or self-adjoint if

A = A∗

which is equivalent to

⟨Ax, y⟩ = ⟨x,Ay⟩ for all x, y ∈ H. [8]

In some sense, these operators play the role of the real numbers (being equal to their own “complex conjugate”) andform a real vector space. They serve as the model of real-valued observables in quantum mechanics. See the articleon self-adjoint operators for a full treatment.

5.5. ADJOINTS OF ANTILINEAR OPERATORS 27

5.5 Adjoints of antilinear operators

For an antilinear operator the definition of adjoint needs to be adjusted in order to compensate for the complexconjugation. An adjoint operator of the antilinear operator A on a complex Hilbert space H is an antilinear operatorA* : H → H with the property:

⟨Ax, y⟩ = ⟨x,A∗y⟩ all forx, y ∈ H.

5.6 Other adjoints

The equation

⟨Ax, y⟩ = ⟨x,A∗y⟩

is formally similar to the defining properties of pairs of adjoint functors in category theory, and this is where adjointfunctors got their name from.

5.7 See also• Mathematical concepts

• Hermitian operator• Norm (mathematics)• Transpose of linear maps

• Physical applications

• Operator (physics)• †-algebra

5.8 Footnotes[1] David A. B. Miller (2008). Quantum Mechanics for Scientists and Engineers. Cambridge University Press. pp. 262, 280.

[2] Reed & Simon 2003, pp. 186–187; Rudin 1991, §12.9

[3] See unbounded operator for details.

[4] Reed & Simon 2003, pp. 252; Rudin 1991, §13.1

[5] Rudin 1991, Thm 13.2

[6] See Rudin 1991, Thm 12.10 for the case of bounded operators

[7] The same as a bounded operator.

[8] Reed & Simon 2003, pp. 187; Rudin 1991, §12.11

5.9 References• Reed, Michael; Simon, Barry (2003), Functional Analysis, Elsevier, ISBN 981-4141-65-8.

• Rudin, Walter (1991), Functional Analysis (second ed.), McGraw-Hill, ISBN 0-07-054236-8.

28 CHAPTER 5. HERMITIAN ADJOINT

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