SAURAV DWIVEDI

CONSTRUCTIVE MATHEMATICSIn processIncepted. January 5, 2013

mwww.geocities.ws/dwivedi/data/cmt.pdf

November 4, 2015

Contents

Part I Algebras

1 Map, Relation & Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Algebraic Objects with One Binary Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1 Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Coset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 Normal (Invariant) Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 Quotient Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.6 Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.1 Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.1 Linear Representations of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Part II Geometry

8 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.1 Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 One Form/Dual Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.3 Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8.3.1 Homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3.2 Raising/Lowering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9 Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Part III Topology

10 Homeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

v

Lists of Abbreviations

A,B,C, . . . Arbitrary sets.a,b, c, . . . Elements of a set.F,#,,. . . Binary operations on a set A .eF, e#, . . . Identity elements corresponding to binary operations F,#, . . . .A,B,C, . . . Algebraic objects, such as group, ring, module.S(A;F) A Semigroup, with underlying set A , and binary operation F on it.M(A;F; eF) A Monoid, with underlying set A , binary operation F on it, and identity eF .G(A;F; eF) A Group, with underlying set A , binary operation F on it, and identity eF .R(A;F,#; eF, e#) A Ring, with underlying set A , binary operations F,# on it, and identities

eF, e# .< Subsemigroup, submonoid, subgroup, subring,. . .C Normal (invariant) subgroup.xixi = xixi = xixi = ∑

ixixi Einstein’s summation convention.

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Part IAlgebras

Notes inspired by Basic Algebra (N. Jacobson).

Chapter 1Map, Relation & Function

1.1 Function

Lemma 1.1 (Function). A special relation R on A×B , with unique ordered pairs (a, f(a)) for ∀ a ∈ A,forms a function f .

Proof. Let f be injective, with (a1, f(a1)) , (a1, f(a2)) =⇒ (a1, f(a2)) , (a2, f(a2)) =⇒ f is surjective =⇒f is bijective ⇐⇒ (a, f(a)) is unique. ut

3

Chapter 2Algebraic Structures

Abstract

2.1 Algebraic Objects with One Binary Operation

Definition 2.1.1 (Semigroup) The construct S(A;F) with underlying non-empty set A , and associativebinary operation F on A ,

F :A×A −→A , (2.1)

forms a semigroup.

Definition 2.1.2 (Monoid) A semigroup S(A;F) with two-sided identity eF ,

aFeF = eFFa = a , ∀ a ∈ A , (2.2)

forms a monoid M(A;F; eF) .

5

Chapter 3Group

Abstract Groups are categorical objects with one binary operation.

3.1 Group

Definition 3.1.1 (Group) A monoid M(A;F; eF) with two-sided inverse,

aFa−1 = a−1Fa = eF , ∀ a, a−1 ∈ A , (3.1)

forms a group G(A;F; eF) .

One-sided identity, and one-sided inverse suffice to form a group. A group G(A;F; eF) hasunique identity eF .

Corollary 3.1. A finite monoid forms a group.

Lemma 3.1. [Uniqueness] Let a,b, c be unique elements of a group G(A;F; eF) . The binary operation ofany two is unique,

aFb 6= aFc, ∀ a,b, c ∈ A .

Proof. [Contradiction] Let a,b, c,d ∈ A . aFb = c and aFd = c =⇒ b = d . ut

Corollary 3.2. Let a group G(A;F; eF) with order n = |A| , be represented by

A = {e, a2, a3, . . . , an} .

G(A;F; eF) can alternatively be represented by

A = {eFai, a2Fai, a3Fai, . . . , anFai} , or (3.2)A = {aiFe, aiFa2, aiFa3, . . . , aiFan} , (3.3)

∀ i ∈ {1,2,3, . . . ,n} , except for order.

Proof. Let ai ∈ A in a group G(A;F; eF) . From Theorem 3.1, aiFaj is unique ∀ j ∈ {1,2,3, . . . ,n} ,and |A|= n is definite, implying |{aiFaj| ∀ j∈ {1,2,3, . . . ,n}}|= n . aiFaj ∈A ∀ j∈ {1,2,3, . . . ,n} =⇒A = {aiFaj| ∀ j ∈ {1,2,3, . . . ,n}} . ut

3.2 Subgroup

Definition 3.2.1 (Subgroup) For H ⊂ A in G(A;F; eF) , if H forms a group under (the same) binaryoperation F , then G(H;F; eF) is termed subgroup of G(A;F; eF) , abbreviated

G(H;F; eF) < G(A;F; eF) .

7

8 3 Group

3.3 Coset

Definition 3.3.1 (Coset) A left [right] coset of a subgroup G(H;F; eF) < G(A;F; eF) in G(A;F; eF)is,

COSETL (G(H;F; eF)) = {aFH| ∃ a(“a ∈ A′′ ∧ “¬(a ∈ H)′′)} , (3.4)

COSETR (G(H;F; eF)) = {HFa| ∃ a(“a ∈ A′′ ∧ “¬(a ∈ H)′′)} . (3.5)

Definition 3.3.2 (Conjugate) The element bFaFb−1 is termed conjugate of a ∈ A, ∀ b ∈ A in a groupG(A;F; eF).

Definition 3.3.3 (Class) Conjugates of a ∈ A form its class in the group G(A;F; eF).

CLASS (a) = {bFaFb−1| ∀ b ∈ A} . (3.6)

3.4 Normal (Invariant) Subgroup

3.5 Quotient Group

3.6 Homomorphism

Definition 3.6.1 (Homomorphism) A map f : A −→ B between groups G(A;F; eF) and G(B;#; e#) ,with binary operations

F : A×A −→A , (3.7)# : B × B −→ B , (3.8)

is homomorphism of groups, with

f (aFb) = f (a)# f (b) ∀ a,b ∈ A . (3.9)

Chapter 4Ring

Abstract Rings are categorical objects with two associative binary operations.

4.1 Ring

Definition 4.1.1 (Ring) A construct R(A;F,#; eF, e#) with underlying non-empty set A , associative bi-nary operations F and # on A ,

F,# :A×A −→A , (4.1)

forms a ring, when

1. GA(A;F; eF) is a abelian group,2. M(A;#; e#) is a monoid, and3. # is two-sided distributive over F,

a#(bFc) = (a#b)F(a#c) , (4.2)(aFb)#c = (a#c)F(b#c) , (4.3)

∀ a,b, c ∈ A .

R(A;F,#; eF, e#) is abelian ring when ∃ abelian monoid MA(A;#; e#) . R(A;F,#; eF) is ringwithout identity e# , when S(A#;#) , A# = {A|e# /∈ A} , is a semigroup.

An observation in rings, of a prime relevance, is

eF#a = a#eF = eF ∀ a ∈ A . (4.4)

Definition 4.1.2 (Zero Divisor) a ∈ A is termed left [right] zero divisor of the ring R(A;F,#; eF, e#),when ∃ b ∈ A(b 6= eF) =⇒ a#b = eF [= b#a] . Both left and right zero divisor is termed zero divisor ofthe ring.

eF is a trivial zero divisor of all rings R(A;F,#; eF, e#) with |A| ≥ 2 .

Definition 4.1.3 (Domain) A ring with no non-trivial zero divisors is termed a domain. Let AF ={A|eF /∈A} . The ring R(A;F,#; eF, e#) forms a domain, when M(AF;#; e#)<M(A;#; e#) . This impliesthat, for a,b 6= eF =⇒ a#b 6= eF,∀ a,b ∈ A ; none is a zero divisor in A∗ .

eF is one and only zero divisor of a domain RD(A;F,#; eF, e#) . All rings R(A;F,#; eF, e#) withAF = {A|eF /∈ A} are trivially domains.

A domain RD(A;F,#; eF, e#) with #−invertible elements u∈ U ⊂A , is termed a domain of unitsRD(U ;F,#; eF, e#) with M(U ;#; e#) < M(A;#; e#) .

Definition 4.1.4 (Division Ring) A domain RD(A;F,#; eF, e#) is termed a division ring when ∃G(A;#; e#) .The ring R(A;F,#; eF, e#) forms a division ring, when G(AF;#; e#) < G(A;#; e#) , for AF = {A|eF /∈A} . This implies that, for a ∈ A, e# 6= eF, ∃ b ∈ A =⇒ a#b = e# = b#a,∀ a,b ∈ A .

9

10 4 Ring

All division rings are domain, but not conversely.

Definition 4.1.5 (Field) A abelian division ring is termed a field. The ring R(A;F,#; eF, e#) forms a field,when ∃ a abelian group GA(A;#; e#), such that GA(AF;#; e#) < GA(A;#; e#) , for AF = {A|eF /∈ A} .

4.2 Ideal

Analogous to groups, congruences exist in rings, and have relevant consequences. Groups have onecongruence relation each; rings have two. Let I ⊂A , and R(I ;F,#; eF, e#)< R(A;F,#; eF, e#) . Let≡F and ≡F be congruence relations on Amodulo GA(I ;F; eF) and M(I ;#; e#) . For a,b ∈ A ,

a ≡F b [mod GA(I ;F; eF)] , (4.5)a ≡# b [mod M(I ;#; e#)] . (4.6)

Theorem 4.1. F−congruence modulo GA(I ;F; eF) (≡F) is a equivalence relation on A .

Proof. Let a,b, c ∈ A . a ≡F a (reflexive), a ≡F b =⇒ b ≡F a (symmetric), and a ≡F b , b ≡F c =⇒a ≡F c (transitive). ut

Equivalence class a of a ∈A is coset IFa of I inA . As GA(A;F; eF) is abelian, GA(I ;F; eF)CGA(A;F; eF) . Cosets {IFa|a ∈ A} form F−quotient group G(A/I ;⊕; e⊕) with A/I ⊂ A andp,q ∈ A/I =⇒ p⊕ q = DO: . . . Heuristic!

It is observed that #−congruence modulo M(I ;#; e#) (≡#) is not equivalence relation on A , un-less ∃ abelian group GA(A;#; e#) .

Theorem 4.1 implies that, a ∼ a, i ∼ eF ∀ a ∈ A , i ∈ I . Thus a#i ∼ a#eF = eF ∼ i ∼ eF#a ∼i#a =⇒ a#i, i#a ∈ I ∀ a ∈ A , i ∈ I .

Definition 4.2.1 (Ideal) Let I ⊂ A , and R(I ;F,#; eF, e#) < R(A;F,#; eF, e#) . I is termed left [right]ideal of the ring R(A;F,#; eF, e#) , when a ∈A , i ∈ I =⇒ a#i [i#a]∈ I . Both left and right ideal is termedideal of the ring.

Lemma 4.1 (Ideal). Let I ⊂A . I is an ideal of the ring R(A;F,#; eF, e#) iff ∃ a subgroup G(I ;F; eF)<G(A;F; eF) .

Proof. R(A;F,#; eF, e#) =⇒ ∃GA(A;F; eF) and M(A;#; e#) . If ∃G(I ;F; eF)<G(A;F; eF) , then∃ a equivalence relation ≡F onA [cf. Theorem 4.1]. For a ∈ A, i ∈ I , a∼ a and i∼ eF =⇒ a#i andi#a ∈ I . ut

Corollary 4.1. eF and A are trivial ideals of the ring R(A;F,#; eF, e#) .

Chapter 5Modules

Abstract . . .

11

Chapter 6Representations

Abstract Based on Serre’s classic.

6.1 Linear Representations of Finite Groups

Let V : Mn −→K be n−dimensional vector space over a manifold Mn , with basis {~ei} . Linear

l : V −→ V , (6.1)

maps vector to vector,

l(~u) = ~v , l(ui~ei) = uil(~ei) = vj~ej , i, j ∈ [1,n] , (6.2)

Or

l(~ei) =vj

ui~ej = wji~ej , ui,vj,wj

i ∈K . (6.3)

Being a bijection, l entails unique {wj} for each~ei . Thus, l generates square n× n matrix {wji} for the

whole basis {~ei} .The bijections l ∈ L form a group GL(L;⊗;I) of square (non singular) matrices under matrix

multiplication, termed general linear group. The binary operation ⊗ : L×L→ L is defined as,

l(ei)⊗ l(ej) = wki wl

jek ⊗ el = wki wl

jδkl = wki wk

j = λij ∈K . (6.4)

Definition 6.1.1 (Linear Representation) Let G(A;F; eF) be an arbitrary group. The homomorphism

ρ :A −→L , ρ(a1Fa2) = ρ(a1)⊗ ρ(a2) = l1 ⊗ l2 ∈ L , (6.5)

turns elements of the group into square matrices. ρ is termed linear representation of G(A;F; eF) in V ofdegree n, and vector space V is termed representation space of G(A;F; eF) .

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Chapter 7Categories

Abstract . . .

15

Part IIGeometry

Based on Nakahara’s classic text.

Chapter 8Differential Geometry

Abstract Vectors are multilinear objects over a manifold Mn . Tensors are multilinear objects fromproduct vector spaces and product one forms to an arbitrary field K .

8.1 Vector Space

Definition 8.1.1 (Vector Space) A vector ~v ∈ ~V over a manifold Mn is a multilinear from Mn to an arbi-trary field K ,

~v : Mn −→K . (8.1)

Any vector can be expanded into linear combination of basis vectors {~eµ} ,

~v = vµ~eµ , (sum over µ) , (8.2)

where vµ ∈K are termed components of the vector ~v for the basis {~eµ} .

In the context of physics, each reference frame has its own basis.

8.2 One Form/Dual Vector Space

Theorem 8.1 (One Form). The set of linears from ~V to an arbitrary field K ,

V : ~V −→K , (8.3)

form a vector space with basis {ων} .

Proof. Let V : ~V −→K be a vector space with basis {ων} . An arbitrary vector v∈ V can be expandedto,

v(~u) = vνων(~u) , ∀ ~u ∈ ~V , vν ∈K . (8.4)

For a trivial case ~u =~eµ , we have v(~eµ) = vνων(~eµ) . Our choice

ων(~eµ) = δνµ , (8.5)

makes v(~eµ) = vνδνµ = vµ ∈ K . For vector space ~V with basis {~eµ} , our choice ων(~eµ) = δν

µ makesV : ~V −→K to be a vector space with basis {ων} . ut

The vector space V is termed one form, and is dual to vector space ~V . It turns out that V ∼= ~V anddim(V) = n = dim(~V) .

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20 8 Differential Geometry

8.3 Tensor

Definition 8.3.1 (Tensor of type (0,q)) A tensor t ∈ T 0q of type (0,q) is a multilinear from product vector

spaces to an arbitrary field K ,

T 0q :

q⊗~V −→K . (8.6)

Definition 8.3.2 (Tensor of type (p,0)) A tensor t ∈ T p0 of type (p,0) is a multilinear from product one

forms to an arbitrary field K ,

T p0 :

p⊗V −→K . (8.7)

Definition 8.3.3 (Tensor of type (p,q)) A tensor t∈ T pq of type (p,q) is a multilinear from product vector

spaces and product one forms to an arbitrary field K ,

T pq :

p⊗V

q⊗ ~V −→K . (8.8)

8.3.1 Homomorphism

Let a (0,2) type tensor

t02 : ~V ⊗ ~V −→K (8.9)

be a homomorphism,

t(~u,~v) = t(~u)⊗ t(~v) , ∀ ~u,~v ∈ ~V ,& t(~u), t(~v) ∈K . (8.10)

The homomorphism allows a (0,2) type tensor to decompose into two (0,1) type tensors,

t(0,2) ≡ t(0,1) ⊗ t(0,1) . (8.11)

Generalizing further, a (0,q) type tensor decomposes into q (0,1) type tensors,

t(0,q) =q⊗t(0,1) . (8.12)

A tensor of type (0,1) is a one form t : ~V −→K , expanded as

t(0,1) = tµωµ , (8.13)

for basis {ωµ} . A tensor of type (0,2) , t : ~V ⊗ ~V −→K , decomposes as

t(0,2) = t(0,1) ⊗ t(0,1) = tµωµ ⊗ tνων = tµνωµ ⊗ ων , (8.14)

where tµν = tµtν ,∀ tµ, tν ∈K . Generalizing further, A tensor of type (0,q) decomposes as

t(0,q) = tµ1µ2 ...µq

q⊗

i=1ωµi . (8.15)

where tµ1µ2 ...µq = tµ1 tµ2 . . . tµq ,∀ tµ1 , tµ2 . . . tµq ∈K .

8.3.2 Raising/Lowering

DO: . . . Heuristic!

Chapter 9Algebraic Geometry

DO: . . . Heuristic!

21

Part IIITopology

Based on General Topology, Bourbaki, N.

Chapter 10Homeomorphism

25