0

Definition0.7

The wedge product is the product in an exterior algebra. If Î±, Î² are differential k-forms of degree p, g respectively, then

Â Î±âˆ§Î²=(-1)pq Î²âˆ§Î±, is not in general commutative, but is associative,

(Î±âˆ§Î²)âˆ§u= Î±âˆ§(Î²âˆ§u) and bilinear

(c1 Î±1+c2 Î±2)âˆ§ Î²= c1( Î±1âˆ§ Î²) + c2( Î±2âˆ§ Î²)

Î±âˆ§( c1 Î²1+c2 Î²2)= c1( Î±âˆ§ Î²1) + c2( Î±âˆ§ Î²2).Â Â Â (Becca 2014)

Chapter 1

Definition1.1

“Let (A, m, Î·) be an algebra over k and write mop (ab) = ab ê“¯ a, bÏµ A where mop=mÏ„Î‘,Î‘. Thus ab=ba ê“¯a, b ÏµA. The (A, mop, Î·) is the opposite algebra.”

Definition1.2

“A co-algebra C is

A vector space over K

A map Î”: Câ†’C âŠ-Â C which is coassociative in the sense of âˆ‘ (c(1)(1) âŠ-Â Â c(1)(2) âŠ-Â c(2))= âˆ‘ (c(1) âŠ-Â Â c(2)(1) âŠ-Â c(2)c(2) )Â Â ê“¯ cÏµC (Î” called the co-product)

A map Îµ: Câ†’ k obeying âˆ‘[Îµ((c(1))c(2))]=c= âˆ‘[(c(1)) Îµc(2))] ê“¯ cÏµC ( Îµ called the counit)”

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

Figure 1: Co-associativity map Î”

Figure 2: co-unit element map Îµ

Definition1.3

“A bi-algebra H is

An algebra (H, m ,Î·)

A co-algebra (H, Î”, Îµ)

Î”,Îµ are algebra maps, where HâŠ-Â H has the tensor product algebra structure (hâŠ- g)(h‘âŠ-Â g‘)= hh‘âŠ-Â Â gg‘ ê“¯h, h‘, g, g‘ ÏµH. “

A representation of Hopf algebras as diagrams is the following:

Definition1.4

“A Hopf Algebra H is

A bi-algebra H, Î”, Îµ, m, Î·

A map S : Hâ†’ H such that âˆ‘ [(Sh(1))h(2) ]= Îµ(h)= âˆ‘ [h(1)Sh(2) ]ê“¯ hÏµH

The axioms that make a simultaneous algebra and co-algebra into Hopf algebra is

Ï„:Â HâŠ- Hâ†’HâŠ-H

Is the map Ï„(hâŠ-g)=gâŠ-h called the flip map ê“¯ h, g Ïµ 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, Ï„Î”=Î”. It can be defined as S2=id.

A commutative algebra over K is an algebra (A, m, Î·) over k such that m=mop.”

Definition1.6

“Two Hopf algebras H,H‘ are dually paired by a map : H’ H â†’k if,

=Ïˆ,Î”h>, =Îµ(h)

gÂ >=, Îµ(Ï†)=

=

ê“¯ Ï†, ÏˆÏµ H’ and h, g ÏµH.

Let (C, Î”,Îµ) be a co-algebra over k. The co-algebra (C, Î”cop, Îµ) is the opposite co-algebra.

A co-commutative co-algebra over k is a co-algebra (C, Î”, Îµ) over k such that Î”= Î”cop.”

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: AAâ†’A commutes with the action of H

The unit map Î·: kâ†’ A commutes with the action of H.

From b,c we come to the next action

hâŠ³(ab)=âˆ‘(h(1)âŠ³a)(h(2)âŠ³b), hâŠ³1= Îµ(h)1, ê“¯a, b Ïµ A, h Ïµ H

This is the left action.”

Definition1.8

“Let (A, m, Î·) be algebra over k and is a left H- module along with a linear map m: AâŠ-Aâ†’A and a scalar multiplication Î·: k âŠ- Aâ†’A if the following diagrams commute.”

Figure 3: Left Module map

Definition1.9

“Co-algebra (C, Î”, Îµ) is H-module co-algebra if:

C is an H-module

Î”: Câ†’CC and Îµ: Câ†’ k commutes with the action of H. (Is a right C- co-module).

Explicitly,

Î”(hâŠ³c)=âˆ‘h(1)âŠ³c(1)â¨‚h(2)âŠ³c(2), Îµ(hâŠ³c)= Îµ(h)Îµ(c), ê“¯h Ïµ H, c Ïµ C.”

Â Definition1.10

“A co-action of a co-algebra C on a vector space V is a map Î²: Vâ†’Câ¨‚V such that,

(idâ¨‚Î²) âˆ˜Î²=(Î”â¨‚ id )Î²;

Â id =(Îµâ¨‚id )âˆ˜Î².”

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 Î²: Aâ†’ Hâ¨‚A is an algebra homomorphism, where Hâ¨‚A has the tensor product algebra structure.”

Definition1.12

“Let C be co- algebra (C, Î”, Îµ), map Î²: Aâ†’ Hâ¨‚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, Î·) be algebra over k and suppose that V is a left ideal of A. Then m(Aâ¨‚V)âŠ†V. Thus the restriction of m to Aâ¨‚V determines a map Aâ¨‚Vâ†’V. Left co-ideal of a co-algebra C is a subspace V of C such that the co-product Î” restricts to a map Vâ†’Câ¨‚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 Î”(V)âŠ†Vâ¨‚V, for left co-ideal Î”(V)âŠ†Câ¨‚V and for right co-ideal Î”(V)âŠ†Vâ¨‚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 ÏµC with satisfies, Î”(s)=sâ¨‚sÂ and Îµ(s)=1 ê“¯ s ÏµS. 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

Î”(s)=sâ¨‚sÂ and Îµ(s)=1

ê“¯ s ÏµS. If S=âˆ… we set C=k[âˆ…]=0.

Is the group-like co-algebra of S over k.”

Definition1.18

“The co-algebra C over k with basis {co, c1, c2,â€¦..} whose co-product and co-unit is satisfy by Î”(cn)= âˆ‘cn-lâ¨‚cl and Îµ(cn)=Î´n,0 for l=1,â€¦.,n and for all nâ‰¥0. Is denoted by Pâˆž(k). The sub-co-algebra which is the span of co, c1, c2,â€¦,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:

Î”(ei, j)= âˆ‘ei, lâ¨‚el, j

Îµ(ei, j)=Î´i, j

âˆ€ i, j ÏµS. Set Câˆ…(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 ÏµS for Cs(k) which satisfies the co-matrix identities.”

Definition1.21

“Let (C, Î”c, Îµc) and (D, Î”D, ÎµD) be co-algebras over the field k. A co-algebra map f: Câ†’D is a linear map of underlying vector spaces such that Î”Dâˆ˜f=(fâ¨‚f)âˆ˜ Î”c and ÎµDâˆ˜f= Îµc. 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 Îµ (I) = (0) and Î” (Î™) âŠ† Iâ¨‚C+Câ¨‚I.”

Definition1.23

“The co-ideal Ker (Îµ) 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€: Câ†’ 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 Î”(c(1)â¨‚d(1))â¨‚( c(2)â¨‚d(2)) and Îµ(câ¨‚d)=Îµ(c)Îµ(d) âˆ€ c in C and d in D.”

Definition1.26

“Let C be co-algebra over k. A skew-primitive element of C is a cÏµC which satisfies Î”(c)= gâ¨‚c +câ¨‚h, where c, h ÏµG(c). The set of g:h-skew primitive elements of C is denotedÂ by

Pg,h (C).”

Definition1.27

“Let C be co-algebra over a field k. A co-commutative element of C is cÏµC such that Î”(c) = Î”cop(c). The set of co-commutative elements of C is denoted by Cc(C).

Cc(C) âŠ†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, Î”, Îµ) be co-algebra over k. The algebra (Câˆ-, m, Î·) where m= Î”âˆ-| Câˆ-â¨‚Câˆ-, Î· (1) =Îµ, is the dual algebra of (C, Î”, Îµ).”

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 Câˆ–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 ÏµA.

For fixed b ÏµA note that F: Aâ†’A defined by F(a)=[a, b]= ab- baÂ for all a ÏµA 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 Î”âˆ˜f= (fâ¨‚IC + IC â¨‚f) âˆ˜Î”.”

Definition1.34

“Let A and B ne algebra over the field k. The tensor product algebra structure on Aâ¨‚B is determined by (aâ¨‚b)(a’â¨‚b’)= aa’â¨‚bb’ ê“¯ a, a’ÏµA and b, b’ÏµB.”

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) = {xÏµX|yx=xy ê“¯yÏµY}

For y ÏµA 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, â‰¤) be a partially ordered set which is locally finite, meaning that ê“¯, I, jÏµS which satisfy iâ‰¤j the interval [i, j] = {lÏµS|iâ‰¤lâ‰¤j} is a finite set. Let S= {[i, j] |I, jÏµS, iâ‰¤j} and let A be the algebra which is the vector space of functions f: Sâ†’k under point wise operations whose product is given by

(fâ‹†g)([i, j])=f([i, l])g([l, j])Â iâ‰¤lâ‰¤j

For all f, g ÏµA and [i, j]ÏµS and whose unit is given by 1([I,j])= Î´i,j ê“¯[I,j]ÏµS.”

Definition1.38

“The algebra of A over the k described above is the incidence algebra of the locally finite partially ordered set (S, â‰¤).”

Definition1.39

“Lie co-algebra over k is a pair (C, Î´), where C is a vector space over k and Î´: Câ†’Câ¨‚C is a linear map, which satisfies:

Ï„âˆ˜Î´=0 and (Î™+(Ï„â¨‚Î™)âˆ˜(Î™â¨‚Ï„)+(Î™â¨‚Ï„)âˆ˜ (Ï„â¨‚Î™))âˆ˜(Î™â¨‚Î´)âˆ˜Î´=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 Uâˆ§V = Î”-1(Uâ¨‚C+ Câ¨‚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 Uâˆ§VâŠ†D, ê“¯ U, V of D.”

Definition1.42

“Let C be co-algebra over k and (N, Ï) be a left co-module. Then Uâˆ§X= Ï-1(Uâ¨‚N+ Câ¨‚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, Î·) be algebra over k. Then,

Aâˆ˜=mâˆ–1(Aâˆ-â¨‚Aâˆ- )

(Aâˆ˜, Î”, Îµ) is a co-algebra over k, where Î”= mâˆ-| Aâˆ˜ and Îµ=Î·âˆ-.

Î¤he co-algebra (Aâˆ˜, Î”, Îµ) is the dual co-algebra of (A, m, Î·).

Also we denote Aâˆ˜ by aâˆ˜ and Î”âˆ˜= aâˆ˜(1)â¨‚ aâˆ˜(2), ê“¯ aâˆ˜ Ïµ Aâˆ˜.”

Definition1.45

“Let A be algebra over k. An Î·:Î¾- derivation of A is a linear map f: Aâ†’k which satisfies f(ab)= Î·(a)f(b)+f(a) Î¾(b), ê“¯ a, bÏµ A and Î·, Î¾ Ïµ 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 (respectivelyÂ Â Â Â 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:Aâ†’(Aâˆ˜)*, be linear map defined by jA(a)(aâˆ˜)=aâˆ˜(a), a ÏµA and aâˆ˜ÏµAâˆ˜. Then:

jA:Aâ†’(Aâˆ˜)* is an algebra map

Ker(jA) is the intersection of the co-finite ideals of A

Im(jA) is a dense subspace of (Aâˆ˜)*.

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:Aâ†’(Aâˆ˜)*, as defined before and jC:Câ†’(C*)âˆ˜, defined as:

jC(c)(c*)=c*(c), ê“¯ c*ÏµC* and cÏµC. Then:

Im(jC)âŠ†(C*)âˆ˜ and jC:Câ†’(C*)âˆ˜ is a co-algebra map.

jC is one-to-one.

Im(jC) is the set of all aÏµ(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:Uâ†’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

Â Î²: AÃ-Câ†’k which satisfies, Î²(ab,c)= Î² (a, c(1))Î² (b, c(2)) and Î²(1, c) = Îµ(c), ê“¯ a, b Ïµ A andÂ Â Â Â Â Â Â c ÏµC.”

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 Ï€: Tco(V)â†’T is a linear map.

If C is a co-algebra over k and f:Câ†’V is a linear map,âˆƒ a co-algebra map F: Câ†’ Tco(V) determined by Ï€âˆ˜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 Ï€:C(V)â†’V is a linear map.

If C is a co-commutative co-algebra over k and f: Câ†’V is linear map, âˆƒ co-algebra map F:C â†’C(V) determined by Ï€âˆ˜F=f. “Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (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 (Sâ¨‚S)âˆ˜Î”h=Ï„âˆ˜Î”âˆ˜Sh, ÎµSh=Îµh, âˆ€h,g âˆˆ H. “Â Â Â Â Â Â Â Â Â Â Â Â (Majid 2002, Radford David E)

Proof

Let S and S1 be two antipodes for H. Then using properties of antipode, associativity of Ï„ and co-associativity of Î” we get

S= Ï„âˆ˜(SâŠ-[ Ï„âˆ˜(IdâŠ-S1)âˆ˜Î”])âˆ˜Î”= Ï„âˆ˜(IdâŠ- Ï„)âˆ˜(Sâ¨‚IdâŠ-S1)âˆ˜(Id âŠ-Î”)âˆ˜Î”=

Ï„âˆ˜(Ï„â¨‚Id)âˆ˜(Sâ¨‚IdâŠ-S1)âˆ˜(Î” âŠ-Id)âˆ˜Î” = Ï„âˆ˜( [Ï„âˆ˜(Sâ¨‚Id)âˆ˜Î”]â¨‚S1)âˆ˜ Î”=S1.

So the antipode is unique.

Let Sâˆ-id=Îµs idâˆ-S=Îµt

To check that S is an algebra anti-homomorphism, we compute

S(1)= S(1(1))1(2)S(1(3))= S(1(1)) Îµt (1(2))= Îµs(1)=1,

S(hg)=S(h(1)g(1)) Îµt(h(2)g(2))= S(h(1)g(1))h(2) Îµt(g(2))S(h(3))=Îµs (h(1)g(1))S(g(2))S(h(2))=

S(g(1)) Îµs(h(1)) Îµt (g(2))S(h(2))=S(g)S(h), âˆ€h,g âˆˆH and we used Îµt(hg)= Îµt(h Îµt(g)) and Îµs(hg)= Îµt(Îµs(h)g).

Dualizing the above we can show that S is also a co-algebra anti-homomorphism:

Îµ(S(h))= Îµ(S(h(1) Îµt(h(2)))= Îµ(S(h(1)h(2))= Îµ(Îµt(h))= Îµ(h),

Î”(S(h))= Î”(S(h(1) Îµt(h(2)))= Î”(S(h(1) Îµt(h(2))â¨‚1)= Î”(S(h(1) ))(h(2)S(h(4))â¨‚ Îµt (h(3))=

Î”(Îµs(h(1))(S(h(3))â¨‚S(h(2)))=S(h(3))â¨‚ Îµs(h(1))S(h(2))=S(h(2))â¨‚ 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 qÂ is a fixed invertible element of the field k. Here

Î”X= Xâ¨‚1 +g â¨‚ X, Î”g=g â¨‚ g, Î”g-1=g-1â¨‚g-1,

ÎµX=0, Îµg=1=Îµ g-1, SX=- g-1X, Sg= g-1, S g-1=g.

S2X=q-1X.”

Proof

We have Î”, Îµ on the generators and extended them multiplicatively to products of the generators.

Î”gX=(Î”g)( Î”X)=( gâ¨‚g)( Xâ¨‚1 +gâ