0
Definition0.7
The wedge product is the product in an exterior algebra. If α, β are differential k-forms of degree p, g respectively, then
 α∧β=(-1)pq β∧α, is not in general commutative, but is associative,
(α∧β)∧u= α∧(β∧u) and bilinear
(c1 α1+c2 α2)∧ β= c1( α1∧ β) + c2( α2∧ β)
α∧( c1 β1+c2 β2)= c1( α∧ β1) + c2( α∧ β2).    (Becca 2014)
Chapter 1
Definition1.1
“Let (A, m, η) be an algebra over k and write mop (ab) = ab ꓯ a, bϵ A where mop=mτΑ,Α. Thus ab=ba ꓯa, b ϵA. The (A, mop, η) is the opposite algebra.”
Definition1.2
“A co-algebra C is

A vector space over K
A map Δ: C→C âŠ-  C which is coassociative in the sense of ∑ (c(1)(1) âŠ-   c(1)(2) âŠ-  c(2))= ∑ (c(1) âŠ-   c(2)(1) âŠ-  c(2)c(2) )   ꓯ cϵC (Δ called the co-product)
A map ε: C→ k obeying ∑[ε((c(1))c(2))]=c= ∑[(c(1)) εc(2))] ꓯ cϵC ( ε called the counit)”

Co-associativity and co-unit element can be expressed as commutative diagrams as follow:

Figure 1: Co-associativity map Δ

Figure 2: co-unit element map ε
Definition1.3
“A bi-algebra H is

An algebra (H, m ,η)
A co-algebra (H, Δ, ε)
Δ,ε are algebra maps, where HâŠ-  H has the tensor product algebra structure (hâŠ- g)(h‘âŠ-  g‘)= hh‘âŠ-   gg‘ ꓯh, h‘, g, g‘ ϵH. “

A representation of Hopf algebras as diagrams is the following:

Definition1.4
“A Hopf Algebra H is

A bi-algebra H, Δ, ε, m, η
A map S : H→ H such that ∑ [(Sh(1))h(2) ]= ε(h)= ∑ [h(1)Sh(2) ]ꓯ hϵH

The axioms that make a simultaneous algebra and co-algebra into Hopf algebra is
Ï„:  HâŠ- H→HâŠ-H
Is the map Ï„(hâŠ-g)=gâŠ-h called the flip map ꓯ h, g ϵ H.”
Definition1.5
“Hopf Algebra is commutative if it’s commutative as algebra. It is co-commutative if it’s co-commutative as a co-algebra, τΔ=Δ. It can be defined as S2=id.
A commutative algebra over K is an algebra (A, m, η) over k such that m=mop.”
Definition1.6
“Two Hopf algebras H,H‘ are dually paired by a map : H’ H →k if,
=ψ,Δh>, =ε(h)
g  >=, ε(φ)=
=
ꓯ φ, ψϵ H’ and h, g ϵH.
Let (C, Δ,ε) be a co-algebra over k. The co-algebra (C, Δcop, ε) is the opposite co-algebra.
A co-commutative co-algebra over k is a co-algebra (C, Δ, ε) over k such that Δ= Δcop.”
Definition1.7
“A bi-algebra or Hopf algebra H acts on algebra A (called H-module algebra) if:

H acts on A as a vector space.
The product map m: AA→A commutes with the action of H
The unit map η: k→ A commutes with the action of H.

From b,c we come to the next action
h⊳(ab)=∑(h(1)⊳a)(h(2)⊳b), h⊳1= ε(h)1, ꓯa, b ϵ A, h ϵ H
This is the left action.”
Definition1.8
“Let (A, m, η) be algebra over k and is a left H- module along with a linear map m: AâŠ-A→A and a scalar multiplication η: k âŠ- A→A if the following diagrams commute.”

Figure 3: Left Module map
Definition1.9
“Co-algebra (C, Δ, ε) is H-module co-algebra if:

C is an H-module
Δ: C→CC and ε: C→ k commutes with the action of H. (Is a right C- co-module).

Explicitly,
Δ(h⊳c)=∑h(1)⊳c(1)⨂h(2)⊳c(2), ε(h⊳c)= ε(h)ε(c), ꓯh ϵ H, c ϵ C.”

 Definition1.10
“A co-action of a co-algebra C on a vector space V is a map β: V→C⨂V such that,

(id⨂β) ∘β=(Δ⨂ id )β;
 id =(ε⨂id )∘β.”

Definition1.11
“A bi-algebra or Hopf algebra H co-acts on an algebra A (an H- co-module algebra) if:

A is an H- co-module
The co-action β: A→ H⨂A is an algebra homomorphism, where H⨂A has the tensor product algebra structure.”

Definition1.12
“Let C be co- algebra (C, Δ, ε), map β: A→ H⨂A is a right C- co- module if the following diagrams commute.”

Figure 6:Co-algebra of a right co-module
“Sub-algebras, left ideals and right ideals of algebra have dual counter-parts in co-algebras. Let (A, m, η) be algebra over k and suppose that V is a left ideal of A. Then m(A⨂V)⊆V. Thus the restriction of m to A⨂V determines a map A⨂V→V. Left co-ideal of a co-algebra C is a subspace V of C such that the co-product Δ restricts to a map V→C⨂V.”
Definition1.13
“Let V be a subspace of a co-algebra C over k. Then V is a sub-co-algebra of C if Δ(V)⊆V⨂V, for left co-ideal Δ(V)⊆C⨂V and for right co-ideal Δ(V)⊆V⨂C.”
Definition1.14
“Let V be a subspace of a co-algebra C over k. The unique minimal sub-co-algebra of C which contains V is the sub-co-algebra of C generated by V.”
Definition1.15
“A simple co-algebra is a co-algebra which has two sub-co-algebras.”
Definition1.16
“Let C be co-algebra over k. A group-like element of C is c ϵC with satisfies, Δ(s)=s⨂s  and ε(s)=1 ꓯ s ϵS. The set of group-like elements of C is denoted G(C).”
Definition1.17
“Let S be a set. The co-algebra k[S] has a co-algebra structure determined by
Δ(s)=s⨂s  and ε(s)=1
ꓯ s ϵS. If S=∅ we set C=k[∅]=0.
Is the group-like co-algebra of S over k.”
Definition1.18
“The co-algebra C over k with basis {co, c1, c2,…..} whose co-product and co-unit is satisfy by Δ(cn)= ∑cn-l⨂cl and ε(cn)=δn,0 for l=1,….,n and for all n≥0. Is denoted by P∞(k). The sub-co-algebra which is the span of co, c1, c2,…,cn is denoted Pn(k).”
Definition1.19
“A co-matrix co-algebra over k is a co-algebra over k isomorphic to Cs(k) for some finite set S. The co-matrix identities are:

Δ(ei, j)= ∑ei, l⨂el, j
ε(ei, j)=δi, j

∀ i, j ϵS. Set C∅(k)=(0).”
Definition1.20
“Let S be a non-empty finite set. A standard basis for Cs(k) is a basis {c i ,j}I, j ϵS for Cs(k) which satisfies the co-matrix identities.”
Definition1.21
“Let (C, Δc, εc) and (D, ΔD, εD) be co-algebras over the field k. A co-algebra map f: C→D is a linear map of underlying vector spaces such that ΔD∘f=(f⨂f)∘ Δc and εD∘f= εc. An isomorphism of co-algebras is a co-algebra map which is a linear isomorphism.”
Definition1.22
“Let C be co-algebra over the field k. A co-ideal of C is a subspace I of C such that ε (I) = (0) and Δ (Ι) ⊆ I⨂C+C⨂I.”
Definition1.23
“The co-ideal Ker (ε) of a co-algebra C over k is denoted by C+.”
Definition1.24
“Let I be a co-ideal of co-algebra C over k. The unique co-algebra structure on C /I such that the projection Ï€: C→ C/I is a co-algebra map, is the quotient co-algebra structure on C/I.”
Definition1.25
“The tensor product of co-algebra has a natural co-algebra structure as the tensor product of vector space CâŠ-D is a co-algebra over k where Δ(c(1)⨂d(1))⨂( c(2)⨂d(2)) and ε(c⨂d)=ε(c)ε(d) ∀ c in C and d in D.”
Definition1.26
“Let C be co-algebra over k. A skew-primitive element of C is a cϵC which satisfies Δ(c)= g⨂c +c⨂h, where c, h ϵG(c). The set of g:h-skew primitive elements of C is denoted  by
Pg,h (C).”
Definition1.27
“Let C be co-algebra over a field k. A co-commutative element of C is cϵC such that Δ(c) = Δcop(c). The set of co-commutative elements of C is denoted by Cc(C).
Cc(C) ⊆C.”
Definition1.28
“The category whose objects are co-algebras over k and whose morphisms are co-algebra maps under function composition is denoted by k-Coalg.”
Definition1.29
“The category whose objects are algebras over k and whose morphisms are co-algebra maps under function composition is denoted by k-Alg.”
Definition1.30
“Let (C, Δ, ε) be co-algebra over k. The algebra (Câˆ-, m, η) where m= Δâˆ-| Câˆ-⨂Câˆ-, η (1) =ε, is the dual algebra of (C, Δ, ε).”
Definition1.31
“Let A be algebra over the field k. A locally finite A-module is an A-module M whose finitely generated sub-modules are finite-dimensional. The left and right C∖module actions on C are locally finite.”
Definition1.32
“Let A be algebra over the field k. A derivation of A is a linear endomorphism F of A such that F (ab) =F (a) b-aF(b) for all a, b ϵA.
For fixed b ϵA note that F: A→A defined by F(a)=[a, b]= ab- ba  for all a ϵA is a derivation of A.”
Definition1.33
“Let C be co-algebra over the field k. A co-derivation of C is a linear endomorphism f of C such that Δ∘f= (f⨂IC + IC ⨂f) ∘Δ.”
Definition1.34
“Let A and B ne algebra over the field k. The tensor product algebra structure on A⨂B is determined by (a⨂b)(a’⨂b’)= aa’⨂bb’ ꓯ a, a’ϵA and b, b’ϵB.”
Definition1.35
“Let X, Y be non-empty subsets of an algebra A over the field k. The centralizer of Y in X is
ZX(Y) = {xϵX|yx=xy ꓯyϵY}
For y ϵA the centralizer of y in X is ZX(y) = ZX({y}).”
Definition1.36
“The centre of an algebra A over the field Z (A) = ZA(A).”
Definition1.37
“Let (S, ≤) be a partially ordered set which is locally finite, meaning that ꓯ, I, jϵS which satisfy i≤j the interval [i, j] = {lϵS|i≤l≤j} is a finite set. Let S= {[i, j] |I, jϵS, i≤j} and let A be the algebra which is the vector space of functions f: S→k under point wise operations whose product is given by
(f⋆g)([i, j])=f([i, l])g([l, j])  i≤l≤j
For all f, g ϵA and [i, j]ϵS and whose unit is given by 1([I,j])= δi,j ꓯ[I,j]ϵS.”
Definition1.38
“The algebra of A over the k described above is the incidence algebra of the locally finite partially ordered set (S, ≤).”
Definition1.39
“Lie co-algebra over k is a pair (C, δ), where C is a vector space over k and δ: C→C⨂C is a linear map, which satisfies:
τ∘δ=0 and (Ι+(τ⨂Ι)∘(Ι⨂τ)+(Ι⨂τ)∘ (τ⨂Ι))∘(Ι⨂δ)∘δ=0
Ï„=Ï„C,C and I is the appropriate identity map.”
Definition1.40
“Suppose that C is co-algebra over the field k. The wedge product of subspaces U and V is U∧V = Δ-1(U⨂C+ C⨂V).”
Definition1.41
“Let C be co-algebra over the field k. A saturated sub-co-algebra of C is a sub-co-algebra D of C such that U∧V⊆D, ꓯ U, V of D.”
Definition1.42
“Let C be co-algebra over k and (N, ρ) be a left co-module. Then U∧X= ρ-1(U⨂N+ C⨂X) is the wedge product of subspaces U of C and X of N.”
Definition1.43
“Let C be co-algebra over k and U be a subspace of C. The unique minimal saturated sub-co-algebra of C containing U is the saturated closure of U in C.”
Definition1.44
“Let (A, m, η) be algebra over k. Then,

A∘=m∖1(Aâˆ-⨂Aâˆ- )
(A∘, Δ, ε) is a co-algebra over k, where Δ= mâˆ-| A∘ and ε=ηâˆ-.

Τhe co-algebra (A∘, Δ, ε) is the dual co-algebra of (A, m, η).
Also we denote A∘ by a∘ and Δ∘= a∘(1)⨂ a∘(2), ꓯ a∘ ϵ A∘.”
Definition1.45
“Let A be algebra over k. An η:ξ- derivation of A is a linear map f: A→k which satisfies f(ab)= η(a)f(b)+f(a) ξ(b), ꓯ a, bϵ A and η, ξ ϵ Alg(A, k).”
Definition1.46
“The full subcategory of k-Alg (respectively of k-Co-alg) whose objects are finite dimensional algebras (respectively co-algebras) over k is denoted k-Alg fd (respectively     k-Co-alg fd).”
Definition1.47
“A proper algebra over k is an algebra over k such that the intersection of the co-finite ideals of A is (0), or equivalently the algebra map jA:A→(A∘)*, be linear map defined by jA(a)(a∘)=a∘(a), a ϵA and a∘ϵA∘. Then:

jA:A→(A∘)* is an algebra map
Ker(jA) is the intersection of the co-finite ideals of A
Im(jA) is a dense subspace of (A∘)*.

Is one-to-one.”
Definition1.48
“Let A (respectively C) be an algebra (respectively co-algebra ) over k. Then A (respectively C) is reflexive if jA:A→(A∘)*, as defined before and jC:C→(C*)∘, defined as:
jC(c)(c*)=c*(c), ꓯ c*ϵC* and cϵC. Then:

Im(jC)⊆(C*)∘ and jC:C→(C*)∘ is a co-algebra map.
jC is one-to-one.
Im(jC) is the set of all aϵ(C*)* which vanish on a closed co-finite ideal of C*.

Is an isomorphism.”
Definition1.49
“Almost left noetherian algebra over k is an algebra over k whose co-finite left ideal are finitely generated. (M is called almost noetherian if every co-finite submodule of M is finitely generated).”
Definition1.50
“Let f:U→V be a map of vector spaces over k. Then f is an almost one-to-one linear map if ker(f) is finite-dimensional, f is an almost onto linear map if Im(f) is co-finite subspace of V and f is an almost isomorphism if f is an almost one-to-one and an almost linear map.”
Definition1.51
“Let A be algebra over k and C be co-algebra over k. A pairing of A and C is a bilinear map
 β: AÃ-C→k which satisfies, β(ab,c)= β (a, c(1))β (b, c(2)) and β(1, c) = ε(c), ꓯ a, b ϵ A and        c ϵC.”
Definition1.52
“Let V be a vector space over k. A co-free co-algebra on V is a pair (Ï€, Tco(V)) such that:

Tco(V) is a co-algebra over k and π: Tco(V)→T is a linear map.
If C is a co-algebra over k and f:C→V is a linear map,∃ a co-algebra map F: C→ Tco(V) determined by π∘F=f.”

Definition1.53
“Let V be a vector space over k. A co-free co-commutative co-algebra on V is any pair (Ï€, C(V)) which satisfies:

C(V) is a co-commutative co-algebra over k and π:C(V)→V is a linear map.
If C is a co-commutative co-algebra over k and f: C→V is linear map, ∃ co-algebra map F:C →C(V) determined by π∘F=f. “                  (Majid 2002, Radford David E)

Chapter 2
Proposition (Anti-homomorphism property of antipodes) 2.1
“The antipode of a Hopf algebra is unique and obey S(hg)=S(g)S(h), S(1)=1 and (S⨂S)∘Δh=τ∘Δ∘Sh, εSh=εh, ∀h,g ∈ H. “             (Majid 2002, Radford David E)
Proof
Let S and S1 be two antipodes for H. Then using properties of antipode, associativity of τ and co-associativity of Δ we get
S= τ∘(SâŠ-[ τ∘(IdâŠ-S1)∘Δ])∘Δ= τ∘(IdâŠ- Ï„)∘(S⨂IdâŠ-S1)∘(Id âŠ-Δ)∘Δ=
τ∘(τ⨂Id)∘(S⨂IdâŠ-S1)∘(Δ âŠ-Id)∘Δ = τ∘( [τ∘(S⨂Id)∘Δ]⨂S1)∘ Δ=S1.
So the antipode is unique.
Let Sâˆ-id=εs idâˆ-S=εt
To check that S is an algebra anti-homomorphism, we compute
S(1)= S(1(1))1(2)S(1(3))= S(1(1)) εt (1(2))= εs(1)=1,
S(hg)=S(h(1)g(1)) εt(h(2)g(2))= S(h(1)g(1))h(2) εt(g(2))S(h(3))=εs (h(1)g(1))S(g(2))S(h(2))=
S(g(1)) εs(h(1)) εt (g(2))S(h(2))=S(g)S(h), ∀h,g ∈H and we used εt(hg)= εt(h εt(g)) and εs(hg)= εt(εs(h)g).
Dualizing the above we can show that S is also a co-algebra anti-homomorphism:
ε(S(h))= ε(S(h(1) εt(h(2)))= ε(S(h(1)h(2))= ε(εt(h))= ε(h),
Δ(S(h))= Δ(S(h(1) εt(h(2)))= Δ(S(h(1) εt(h(2))⨂1)= Δ(S(h(1) ))(h(2)S(h(4))⨂ εt (h(3))=
Δ(εs(h(1))(S(h(3))⨂S(h(2)))=S(h(3))⨂ εs(h(1))S(h(2))=S(h(2))⨂ S(h(1)). (New directions)
Example2.2
“The Hopf Algebra H=Uq(b+) is generated by 1 and the elements X,g,g-1 with relations
gg-1=1=g-1g and g X=q X g, where q  is a fixed invertible element of the field k. Here
ΔX= X⨂1 +g ⨂ X, Δg=g ⨂ g, Δg-1=g-1⨂g-1,
εX=0, εg=1=ε g-1, SX=- g-1X, Sg= g-1, S g-1=g.
S2X=q-1X.”
Proof
We have Δ, ε on the generators and extended them multiplicatively to products of the generators.
ΔgX=(Δg)( ΔX)=( g⨂g)( X⨂1 +gâ