Wednesday, August 21, 2019
Hopf Algebra Project
Hopf Algebra Project Petros Karayiannis Chapter 0 Introduction Hopf algebras have lot of applications. At first, they used it in topology in 1940s, but then they realized it has applications through combinatorics, category theory, Hopf-Galois theory, quantum theory, Lie algebras, Homological algebra and functional analysis. The purpose of this project is to see the definitions and properties of Hopf algebras.(Becca 2014) Preliminaries This chapter provides all the essential tools to understand the structure of Hopf algebras. Basic notations of Hopf algebra are: Groups Fields Vector spaces Homomorphism Commutative diagrams 1.Groups Group G is a finite or infinite set of elements with a binary operation. Groups have to obey some rules, so we can define it as a group. Those are: closure, associative, there exist an identity element and an inverse element. Let us define two elements U, V in G, closure is when then the product of UV is also in G. Associative when the multiplication (UV) W=U (VW) à ªÃ¢â‚¬Å" ¯ U, V, W in G. There exist an identity element such that IU=UI=U for every element U in G. The inverse is when for each element U of G, the set contains an element V=U-1 such that UU-1=U-1U=I. 2.Fields A field Ã’Å“ is a commutative ring and every element b à  µ Ã’Å“ has an inverse. 3.Vector Space A vector space V is a set that is closed under finite vector addition and scalar multiplication. In order for V to be a vector space, the following conditions must hold à ªÃ¢â‚¬Å" ¯ X, Y à  µ V and any scalar a, b à  µ Ã’Å“: a(b X) = (a b) X (a + b) X=aX + bX a(X+Y)=aX + aY 1X=X A left ideal of K-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. We say that a subset L of a K-algebra A is a left ideal if for every x and y in L, z in A and c in K, we have the following: X +y is in L cx is in L zà ¢Ã¢â‚¬ ¹Ã¢â‚¬ ¦ x is in L If we replace c) with xà ¢Ã¢â‚¬ ¹Ã¢â‚¬ ¦ z is in L, then this would define a right ideal. A two-sided ideal is a subset that is both a left and a right ideal. When the algebra is commutative, then all of those notions of ideal are equivalent. We denote the left ideal as à ¢Ã…  ³. 4.Homomorphism Given two groups, (G,*) and (H, °) is a function f: Gà ¢Ã¢â‚¬ ’H such that à ªÃ¢â‚¬Å" ¯ u, v à  µ G it holds that f(u*v)=f(u) °f(v) 5.Commutative diagrams A commutative diagram is showing the composition of maps represented by arrows. The fundament operation of Hopf algebras is the tensor product. A tensor product is a multiplication of vector spaces V and W with a result a single vector space, denoted as V  W. Definition 0.1 Let V and W be Ã’Å“-vector spaces with bases {ei } and {fj } respectively. The tensor product V and W is a new Ã’Å“-vector space,  V  W with basis { ei fj }, is the set of all elements v  w= à ¢Ã‹â€ ‘ (ci,j ei  fj ). ci,j à  µÃƒâ€™Ã…“ are scalars. Also tensor products obey to distributive and scalar multiplication laws. The dimension of the tensor product of two vector spaces is: Dim(V W)=dim(V)dim(W) Theorem of Universal Property of Tensor products 0.2 Let V, W, U be vector spaces with map f: V x W à ¢Ã¢â‚¬ ’ U is defined as f: (v, w) à ¢Ã¢â‚¬ ’vw. There exists a bilinear mapping b: V x W à ¢Ã¢â‚¬ ’ V W , (v,w) à ¢Ã¢â‚¬ ’ v   w If f: V x W à ¢Ã¢â‚¬ ’ U is bilinear, then there exist a unique function, f: V Wà ¢Ã¢â‚¬ ’U with f=f °b  Extension of Tensor Products0.3 The definition of Tensor products can be extended for more than two vectors such as; V1 à ¢Ã… - V2à ¢Ã… -  V3 à ¢Ã… -Â à ¢Ã¢â€š ¬Ã‚ ¦..à ¢Ã… - VN = à ¢Ã‹â€ ‘( biv1à ¢Ã… - v2à ¢Ã… -Â à ¢Ã¢â€š ¬Ã‚ ¦.à ¢Ã… - vn ) (Becca 2014) Definition0.4 Let U,V be vector spacers over a field k and ÃŽÂ ½ à  µ Uà ¢Ã‚ ¨Ã¢â‚¬Å¡V. If ÃŽÂ ½=0 then Rank (ÃŽÂ ½) =0. If ÃŽÂ ½Ãƒ ¢Ã¢â‚¬ °Ã‚ 0 then rank (ÃŽÂ ½) is equal to the smallest positive integer r arising from the representations of ÃŽÂ ½= à ¢Ã‹â€ ‘ui à ¢Ã‚ ¨Ã¢â‚¬Å¡ vi à  µUà ¢Ã‚ ¨Ã¢â‚¬Å¡V for i=1,2,à ¢Ã¢â€š ¬Ã‚ ¦,r. Definition0.5 Let U be a finite dimensional vector space over the field k with basis {u1,à ¢Ã¢â€š ¬Ã‚ ¦.,un} be a basis for U. the dual basis for U*is {u1,à ¢Ã¢â€š ¬Ã‚ ¦.,un} where ui(uj)= ÃŽÂ ´ij for 1à ¢Ã¢â‚¬ °Ã‚ ¤I,jà ¢Ã¢â‚¬ °Ã‚ ¤n. Dual Pair0.6 A dual pair is a 3 -tuple (X,Y,) consisting two vector spaces X,Y over the same field K and a bilinear map, : X x Yà ¢Ã¢â‚¬ ’K with à ªÃ¢â‚¬Å" ¯x à  µ X{0} yà  µY: 0 and à ªÃ¢â‚¬Å" ¯y à  µ Y{0} xà  µX: 0 Definition0.7 The wedge product is the product in an exterior algebra. If ÃŽÂ ±, ÃŽÂ ² are differential k-forms of degree p, g respectively, then  Π±Ãƒ ¢Ã‹â€  §ÃƒÅ½Ã‚ ²=(-1)pq ÃŽÂ ²Ãƒ ¢Ã‹â€  §ÃƒÅ½Ã‚ ±, is not in general commutative, but is associative, (ÃŽÂ ±Ãƒ ¢Ã‹â€  §ÃƒÅ½Ã‚ ²)à ¢Ã‹â€  §u= ÃŽÂ ±Ãƒ ¢Ã‹â€  §(ÃŽÂ ²Ãƒ ¢Ã‹â€  §u) and bilinear (c1 ÃŽÂ ±1+c2 ÃŽÂ ±2)à ¢Ã‹â€  § ÃŽÂ ²= c1( ÃŽÂ ±1à ¢Ã‹â€  § ÃŽÂ ²) + c2( ÃŽÂ ±2à ¢Ã‹â€  § ÃŽÂ ²) ÃŽÂ ±Ãƒ ¢Ã‹â€  §( c1 ÃŽÂ ²1+c2 ÃŽÂ ²2)= c1( ÃŽÂ ±Ãƒ ¢Ã‹â€  § ÃŽÂ ²1) + c2( ÃŽÂ ±Ãƒ ¢Ã‹â€  § ÃŽÂ ²2).   (Becca 2014) Chapter 1 Definition1.1 Let (A, m, ÃŽÂ ·) be an algebra over k and write mop (ab) = ab à ªÃ¢â‚¬Å" ¯ a, bà  µ A where mop=mà „Α,Α. Thus ab=ba à ªÃ¢â‚¬Å" ¯a, b à  µA. The (A, mop, ÃŽÂ ·) is the opposite algebra. Definition1.2 A co-algebra C is A vector space over K A map Ά: Cà ¢Ã¢â‚¬ ’C à ¢Ã… - C which is coassociative in the sense of à ¢Ã‹â€ ‘ (c(1)(1) à ¢Ã… -  c(1)(2) à ¢Ã… - c(2))= à ¢Ã‹â€ ‘ (c(1) à ¢Ã… -  c(2)(1) à ¢Ã… - c(2)c(2) )Â Â à ªÃ¢â‚¬Å" ¯ cà  µC (Άcalled the co-product) A map ÃŽÂ µ: Cà ¢Ã¢â‚¬ ’ k obeying à ¢Ã‹â€ ‘[ÃŽÂ µ((c(1))c(2))]=c= à ¢Ã‹â€ ‘[(c(1)) ÃŽÂ µc(2))] à ªÃ¢â‚¬Å" ¯ cà  µC ( ÃŽÂ µ called the counit) Co-associativity and co-unit element can be expressed as commutative diagrams as follow: Figure 1: Co-associativity map ΆFigure 2: co-unit element map ÃŽÂ µ Definition1.3 A bi-algebra H is An algebra (H, m ,ÃŽÂ ·) A co-algebra (H, Ά, ÃŽÂ µ) Ά,ÃŽÂ µ are algebra maps, where Hà ¢Ã… - H has the tensor product algebra structure (hà ¢Ã… - g)(hà ¢Ã… - g)= hhà ¢Ã… -  gg à ªÃ¢â‚¬Å" ¯h, h, g, g à  µH. A representation of Hopf algebras as diagrams is the following: Definition1.4 A Hopf Algebra H is A bi-algebra H, Ά, ÃŽÂ µ, m, ÃŽÂ · A map S : Hà ¢Ã¢â‚¬ ’ H such that à ¢Ã‹â€ ‘ [(Sh(1))h(2) ]= ÃŽÂ µ(h)= à ¢Ã‹â€ ‘ [h(1)Sh(2) ]à ªÃ¢â‚¬Å" ¯ hà  µH The axioms that make a simultaneous algebra and co-algebra into Hopf algebra is à „: Hà ¢Ã… - Hà ¢Ã¢â‚¬ ’Hà ¢Ã… -H Is the map à „(hà ¢Ã… -g)=gà ¢Ã… -h called the flip map à ªÃ¢â‚¬Å" ¯ h, g à  µ H. Definition1.5 Hopf Algebra is commutative if its commutative as algebra. It is co-commutative if its co-commutative as a co-algebra, à „Î†=Ά. It can be defined as S2=id. A commutative algebra over K is an algebra (A, m, ÃŽÂ ·) over k such that m=mop. Definition1.6 Two Hopf algebras H,H are dually paired by a map : H H à ¢Ã¢â‚¬ ’k if, =à ˆ,Άh>, =ÃŽÂ µ(h) g >=, ÃŽÂ µ(à †)= = à ªÃ¢â‚¬Å" ¯ à †, à ˆÃ  µ H and h, g à  µH. Let (C, Ά,ÃŽÂ µ) be a co-algebra over k. The co-algebra (C, Άcop, ÃŽÂ µ) is the opposite co-algebra. A co-commutative co-algebra over k is a co-algebra (C, Ά, ÃŽÂ µ) over k such that Ά= Άcop. 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: AAà ¢Ã¢â‚¬ ’A commutes with the action of H The unit map ÃŽÂ ·: kà ¢Ã¢â‚¬ ’ A commutes with the action of H. From b,c we come to the next action hà ¢Ã…  ³(ab)=à ¢Ã‹â€ ‘(h(1)à ¢Ã…  ³a)(h(2)à ¢Ã…  ³b), hà ¢Ã…  ³1= ÃŽÂ µ(h)1, à ªÃ¢â‚¬Å" ¯a, b à  µ A, h à  µ H This is the left action. Definition1.8 Let (A, m, ÃŽÂ ·) be algebra over k and is a left H- module along with a linear map m: Aà ¢Ã… -Aà ¢Ã¢â‚¬ ’A and a scalar multiplication ÃŽÂ ·: k à ¢Ã… - Aà ¢Ã¢â‚¬ ’A if the following diagrams commute. Figure 3: Left Module map Definition1.9 Co-algebra (C, Ά, ÃŽÂ µ) is H-module co-algebra if: C is an H-module Ά: Cà ¢Ã¢â‚¬ ’CC and ÃŽÂ µ: Cà ¢Ã¢â‚¬ ’ k commutes with the action of H. (Is a right C- co-module). Explicitly, Ά(hà ¢Ã…  ³c)=à ¢Ã‹â€ ‘h(1)à ¢Ã…  ³c(1)à ¢Ã‚ ¨Ã¢â‚¬Å¡h(2)à ¢Ã…  ³c(2), ÃŽÂ µ(hà ¢Ã…  ³c)= ÃŽÂ µ(h)ÃŽÂ µ(c), à ªÃ¢â‚¬Å" ¯h à  µ H, c à  µ C.  Definition1.10 A co-action of a co-algebra C on a vector space V is a map ÃŽÂ ²: Và ¢Ã¢â‚¬ ’Cà ¢Ã‚ ¨Ã¢â‚¬Å¡V such that, (idà ¢Ã‚ ¨Ã¢â‚¬Å¡ÃƒÅ½Ã‚ ²) à ¢Ã‹â€ ËÅ"ÃŽÂ ²=(레 ¢Ã‚ ¨Ã¢â‚¬Å¡ id )ÃŽÂ ²;  id =(ÃŽÂ µÃƒ ¢Ã‚ ¨Ã¢â‚¬Å¡id )à ¢Ã‹â€ ËÅ"ÃŽÂ ². 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 ÃŽÂ ²: Aà ¢Ã¢â‚¬ ’ Hà ¢Ã‚ ¨Ã¢â‚¬Å¡A is an algebra homomorphism, where Hà ¢Ã‚ ¨Ã¢â‚¬Å¡A has the tensor product algebra structure. Definition1.12 Let C be co- algebra (C, Ά, ÃŽÂ µ), map ÃŽÂ ²: Aà ¢Ã¢â‚¬ ’ Hà ¢Ã‚ ¨Ã¢â‚¬Å¡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, ÃŽÂ ·) be algebra over k and suppose that V is a left ideal of A. Then m(Aà ¢Ã‚ ¨Ã¢â‚¬Å¡V)à ¢Ã… †V. Thus the restriction of m to Aà ¢Ã‚ ¨Ã¢â‚¬Å¡V determines a map Aà ¢Ã‚ ¨Ã¢â‚¬Å¡Và ¢Ã¢â‚¬ ’V. Left co-ideal of a co-algebra C is a subspace V of C such that the co-product Άrestricts to a map Và ¢Ã¢â‚¬ ’Cà ¢Ã‚ ¨Ã¢â‚¬Å¡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 Ά(V)à ¢Ã… †Và ¢Ã‚ ¨Ã¢â‚¬Å¡V, for left co-ideal Ά(V)à ¢Ã… †Cà ¢Ã‚ ¨Ã¢â‚¬Å¡V and for right co-ideal Ά(V)à ¢Ã… †Và ¢Ã‚ ¨Ã¢â‚¬Å¡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 à  µC with satisfies, Ά(s)=sà ¢Ã‚ ¨Ã¢â‚¬Å¡s and ÃŽÂ µ(s)=1 à ªÃ¢â‚¬Å" ¯ s à  µS. 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 Ά(s)=sà ¢Ã‚ ¨Ã¢â‚¬Å¡s and ÃŽÂ µ(s)=1 à ªÃ¢â‚¬Å" ¯ s à  µS. If S=à ¢Ã‹â€ †¦ we set C=k[à ¢Ã‹â€ †¦]=0. Is the group-like co-algebra of S over k. Definition1.18 The co-algebra C over k with basis {co, c1, c2,à ¢Ã¢â€š ¬Ã‚ ¦..} whose co-product and co-unit is satisfy by Ά(cn)= à ¢Ã‹â€ ‘cn-là ¢Ã‚ ¨Ã¢â‚¬Å¡cl and ÃŽÂ µ(cn)=ÃŽÂ ´n,0 for l=1,à ¢Ã¢â€š ¬Ã‚ ¦.,n and for all nà ¢Ã¢â‚¬ °Ã‚ ¥0. Is denoted by Pà ¢Ã‹â€ Ã… ¾(k). The sub-co-algebra which is the span of co, c1, c2,à ¢Ã¢â€š ¬Ã‚ ¦,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: Ά(ei, j)= à ¢Ã‹â€ ‘ei, là ¢Ã‚ ¨Ã¢â‚¬Å¡el, j ÃŽÂ µ(ei, j)=ÃŽÂ ´i, j à ¢Ã‹â€ â‚ ¬ i, j à  µS. Set Cà ¢Ã‹â€ †¦(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 à  µS for Cs(k) which satisfies the co-matrix identities. Definition1.21 Let (C, Άc, ÃŽÂ µc) and (D, ΆD, ÃŽÂ µD) be co-algebras over the field k. A co-algebra map f: Cà ¢Ã¢â‚¬ ’D is a linear map of underlying vector spaces such that ΆDà ¢Ã‹â€ ËÅ"f=(fà ¢Ã‚ ¨Ã¢â‚¬Å¡f)à ¢Ã‹â€ ËÅ" Άc and ÃŽÂ µDà ¢Ã‹â€ ËÅ"f= ÃŽÂ µc. 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 ÃŽÂ µ (I) = (0) and Ά(ÃŽâ„ ¢) à ¢Ã… †Ià ¢Ã‚ ¨Ã¢â‚¬Å¡C+Cà ¢Ã‚ ¨Ã¢â‚¬Å¡I. Definition1.23 The co-ideal Ker (ÃŽÂ µ) 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 à ₠¬: Cà ¢Ã¢â‚¬ ’ 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 Cà ¢Ã… -D is a co-algebra over k where Ά(c(1)à ¢Ã‚ ¨Ã¢â‚¬Å¡d(1))à ¢Ã‚ ¨Ã¢â‚¬Å¡( c(2)à ¢Ã‚ ¨Ã¢â‚¬Å¡d(2)) and ÃŽÂ µ(cà ¢Ã‚ ¨Ã¢â‚¬Å¡d)=ÃŽÂ µ(c)ÃŽÂ µ(d) à ¢Ã‹â€ â‚ ¬ c in C and d in D. Definition1.26 Let C be co-algebra over k. A skew-primitive element of C is a cà  µC which satisfies Ά(c)= gà ¢Ã‚ ¨Ã¢â‚¬Å¡c +cà ¢Ã‚ ¨Ã¢â‚¬Å¡h, where c, h à  µG(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 cà  µC such that Ά(c) = Άcop(c). The set of co-commutative elements of C is denoted by Cc(C). Cc(C) à ¢Ã… †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, Ά, ÃŽÂ µ) be co-algebra over k. The algebra (Cà ¢Ã‹â€ -, m, ÃŽÂ ·) where m= 레 ¢Ã‹â€ -| Cà ¢Ã‹â€ -à ¢Ã‚ ¨Ã¢â‚¬Å¡Cà ¢Ã‹â€ -, ÃŽÂ · (1) =ÃŽÂ µ, is the dual algebra of (C, Ά, ÃŽÂ µ). 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 Cà ¢Ã‹â€ --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 à  µA. For fixed b à  µA note that F: Aà ¢Ã¢â‚¬ ’A defined by F(a)=[a, b]= ab- ba for all a à  µA 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 레 ¢Ã‹â€ ËÅ"f= (fà ¢Ã‚ ¨Ã¢â‚¬Å¡IC + IC à ¢Ã‚ ¨Ã¢â‚¬Å¡f) à ¢Ã‹â€ ËÅ"Ά. Definition1.34 Let A and B ne algebra over the field k. The tensor product algebra structure on Aà ¢Ã‚ ¨Ã¢â‚¬Å¡B is determined by (aà ¢Ã‚ ¨Ã¢â‚¬Å¡b)(aà ¢Ã‚ ¨Ã¢â‚¬Å¡b)= aaà ¢Ã‚ ¨Ã¢â‚¬Å¡bb à ªÃ¢â‚¬Å" ¯ a, aà  µA and b, bà  µB. 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) = {xà  µX|yx=xy à ªÃ¢â‚¬Å" ¯yà  µY} For y à  µA 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, à ¢Ã¢â‚¬ °Ã‚ ¤) be a partially ordered set which is locally finite, meaning that à ªÃ¢â‚¬Å" ¯, I, jà  µS which satisfy ià ¢Ã¢â‚¬ °Ã‚ ¤j the interval [i, j] = {là  µS|ià ¢Ã¢â‚¬ °Ã‚ ¤là ¢Ã¢â‚¬ °Ã‚ ¤j} is a finite set. Let S= {[i, j] |I, jà  µS, ià ¢Ã¢â‚¬ °Ã‚ ¤j} and let A be the algebra which is the vector space of functions f: Sà ¢Ã¢â‚¬ ’k under point wise operations whose product is given by (fà ¢Ã¢â‚¬ ¹Ã¢â‚¬ g)([i, j])=f([i, l])g([l, j]) ià ¢Ã¢â‚¬ °Ã‚ ¤là ¢Ã¢â‚¬ °Ã‚ ¤j For all f, g à  µA and [i, j]à  µS and whose unit is given by 1([I,j])= ÃŽÂ ´i,j à ªÃ¢â‚¬Å" ¯[I,j]à  µS. Definition1.38 The algebra of A over the k described above is the incidence algebra of the locally finite partially ordered set (S, à ¢Ã¢â‚¬ °Ã‚ ¤). Definition1.39 Lie co-algebra over k is a pair (C, ÃŽÂ ´), where C is a vector space over k and ÃŽÂ ´: Cà ¢Ã¢â‚¬ ’Cà ¢Ã‚ ¨Ã¢â‚¬Å¡C is a linear map, which satisfies: à „à ¢Ã‹â€ ËÅ"ÃŽÂ ´=0 and (ÃŽâ„ ¢+(à „à ¢Ã‚ ¨Ã¢â‚¬Å¡ÃƒÅ½Ã¢â€ž ¢)à ¢Ã‹â€ ËÅ"(ÃŽâ„ ¢Ãƒ ¢Ã‚ ¨Ã¢â‚¬Å¡Ãƒ „)+(ÃŽâ„ ¢Ãƒ ¢Ã‚ ¨Ã¢â‚¬Å¡Ãƒ „)à ¢Ã‹â€ ËÅ" (à „à ¢Ã‚ ¨Ã¢â‚¬Å¡ÃƒÅ½Ã¢â€ž ¢))à ¢Ã‹â€ ËÅ"(ÃŽâ„ ¢Ãƒ ¢Ã‚ ¨Ã¢â‚¬Å¡ÃƒÅ½Ã‚ ´)à ¢Ã‹â€ ËÅ"ÃŽÂ ´=0 à „=à „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 Uà ¢Ã‹â€  §V = Ά-1(Uà ¢Ã‚ ¨Ã¢â‚¬Å¡C+ Cà ¢Ã‚ ¨Ã¢â‚¬Å¡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 Uà ¢Ã‹â€  §Và ¢Ã… †D, à ªÃ¢â‚¬Å" ¯ U, V of D. Definition1.42 Let C be co-algebra over k and (N, à  ) be a left co-module. Then Uà ¢Ã‹â€  §X= à  -1(Uà ¢Ã‚ ¨Ã¢â‚¬Å¡N+ Cà ¢Ã‚ ¨Ã¢â‚¬Å¡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, ÃŽÂ ·) be algebra over k. Then, Aà ¢Ã‹â€ ËÅ"=mà ¢Ã‹â€ 1(Aà ¢Ã‹â€ -à ¢Ã‚ ¨Ã¢â‚¬Å¡Aà ¢Ã‹â€ - ) (Aà ¢Ã‹â€ ËÅ", Ά, ÃŽÂ µ) is a co-algebra over k, where Ά= mà ¢Ã‹â€ -| Aà ¢Ã‹â€ ËÅ" and ÃŽÂ µ=ÃŽÂ ·Ãƒ ¢Ã‹â€ -. ÃŽÂ ¤he co-algebra (Aà ¢Ã‹â€ ËÅ", Ά, ÃŽÂ µ) is the dual co-algebra of (A, m, ÃŽÂ ·). Also we denote Aà ¢Ã‹â€ ËÅ" by aà ¢Ã‹â€ ËÅ" and 레 ¢Ã‹â€ ËÅ"= aà ¢Ã‹â€ ËÅ"(1)à ¢Ã‚ ¨Ã¢â‚¬Å¡ aà ¢Ã‹â€ ËÅ"(2), à ªÃ¢â‚¬Å" ¯ aà ¢Ã‹â€ ËÅ" à  µ Aà ¢Ã‹â€ ËÅ". Definition1.45 Let A be algebra over k. An ÃŽÂ ·:ÃŽÂ ¾- derivation of A is a linear map f: Aà ¢Ã¢â‚¬ ’k which satisfies f(ab)= ÃŽÂ ·(a)f(b)+f(a) ÃŽÂ ¾(b), à ªÃ¢â‚¬Å" ¯ a, bà  µ A and ÃŽÂ ·, ÃŽÂ ¾ à  µ 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:Aà ¢Ã¢â‚¬ ’(Aà ¢Ã‹â€ ËÅ")*, be linear map defined by jA(a)(aà ¢Ã‹â€ ËÅ")=aà ¢Ã‹â€ ËÅ"(a), a à  µA and aà ¢Ã‹â€ ËÅ"à  µAà ¢Ã‹â€ ËÅ". Then: jA:Aà ¢Ã¢â‚¬ ’(Aà ¢Ã‹â€ ËÅ")* is an algebra map Ker(jA) is the intersection of the co-finite ideals of A Im(jA) is a dense subspace of (Aà ¢Ã‹â€ ËÅ")*. 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:Aà ¢Ã¢â‚¬ ’(Aà ¢Ã‹â€ ËÅ")*, as defined before and jC:Cà ¢Ã¢â‚¬ ’(C*)à ¢Ã‹â€ ËÅ", defined as: jC(c)(c*)=c*(c), à ªÃ¢â‚¬Å" ¯ c*à  µC* and cà  µC. Then: Im(jC)à ¢Ã… †(C*)à ¢Ã‹â€ ËÅ" and jC:Cà ¢Ã¢â‚¬ ’(C*)à ¢Ã‹â€ ËÅ" is a co-algebra map. jC is one-to-one. Im(jC) is the set of all aà  µ(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:Uà ¢Ã¢â‚¬ ’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  Π²: AÃÆ'-Cà ¢Ã¢â‚¬ ’k which satisfies, ÃŽÂ ²(ab,c)= ÃŽÂ ² (a, c(1))ÃŽÂ ² (b, c(2)) and ÃŽÂ ²(1, c) = ÃŽÂ µ(c), à ªÃ¢â‚¬Å" ¯ a, b à  µ A and       c à  µC. Definition1.52 Let V be a vector space over k. A co-free co-algebra on V is a pair (à ₠¬, Tco(V)) such that: Tco(V) is a co-algebra over k and à ₠¬: Tco(V)à ¢Ã¢â‚¬ ’T is a linear map. If C is a co-algebra over k and f:Cà ¢Ã¢â‚¬ ’V is a linear map,à ¢Ã‹â€ Æ’ a co-algebra map F: Cà ¢Ã¢â‚¬ ’ Tco(V) determined by à ₠¬Ãƒ ¢Ã‹â€ ËÅ"F=f. Definition1.53 Let V be a vector space over k. A co-free co-commutative co-algebra on V is any pair (à ₠¬, C(V)) which satisfies: C(V) is a co-commutative co-algebra over k and à ₠¬:C(V)à ¢Ã¢â‚¬ ’V is a linear map. If C is a co-commutative co-algebra over k and f: Cà ¢Ã¢â‚¬ ’V is linear map, à ¢Ã‹â€ Æ’ co-algebra map F:C à ¢Ã¢â‚¬ ’C(V) determined by à ₠¬Ãƒ ¢Ã‹â€ ËÅ"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 (Sà ¢Ã‚ ¨Ã¢â‚¬Å¡S)à ¢Ã‹â€ ËÅ"Άh=à „à ¢Ã‹â€ ËÅ"레 ¢Ã‹â€ ËÅ"Sh, ÃŽÂ µSh=ÃŽÂ µh, à ¢Ã‹â€ â‚ ¬h,g à ¢Ã‹â€ ˆH.             (Majid 2002, Radford David E) Proof Let S and S1 be two antipodes for H. Then using properties of antipode, associativity of à „ and co-associativity of Άwe get S= à „à ¢Ã‹â€ ËÅ"(Sà ¢Ã… -[ à „à ¢Ã‹â€ ËÅ"(Idà ¢Ã… -S1)à ¢Ã‹â€ ËÅ"Ά])à ¢Ã‹â€ ËÅ"Ά= à „à ¢Ã‹â€ ËÅ"(Idà ¢Ã… - à „)à ¢Ã‹â€ ËÅ"(Sà ¢Ã‚ ¨Ã¢â‚¬Å¡Idà ¢Ã… -S1)à ¢Ã‹â€ ËÅ"(Id à ¢Ã… -Ά)à ¢Ã‹â€ ËÅ"Ά= à „à ¢Ã‹â€ ËÅ"(à „à ¢Ã‚ ¨Ã¢â‚¬Å¡Id)à ¢Ã‹â€ ËÅ"(Sà ¢Ã‚ ¨Ã¢â‚¬Å¡Idà ¢Ã… -S1)à ¢Ã‹â€ ËÅ"(레 ¢Ã… -Id)à ¢Ã‹â€ ËÅ"Ά= à „à ¢Ã‹â€ ËÅ"( [à „à ¢Ã‹â€ ËÅ"(Sà ¢Ã‚ ¨Ã¢â‚¬Å¡Id)à ¢Ã‹â€ ËÅ"Ά]à ¢Ã‚ ¨Ã¢â‚¬Å¡S1)à ¢Ã‹â€ ËÅ" Ά=S1. So the antipode is unique. Let Sà ¢Ã‹â€ -id=ÃŽÂ µs idà ¢Ã‹â€ -S=ÃŽÂ µt To check that S is an algebra anti-homomorphism, we compute S(1)= S(1(1))1(2)S(1(3))= S(1(1)) ÃŽÂ µt (1(2))= ÃŽÂ µs(1)=1, S(hg)=S(h(1)g(1)) ÃŽÂ µt(h(2)g(2))= S(h(1)g(1))h(2) ÃŽÂ µt(g(2))S(h(3))=ÃŽÂ µs (h(1)g(1))S(g(2))S(h(2))= S(g(1)) ÃŽÂ µs(h(1)) ÃŽÂ µt (g(2))S(h(2))=S(g)S(h), à ¢Ã‹â€ â‚ ¬h,g à ¢Ã‹â€ ˆH and we used ÃŽÂ µt(hg)= ÃŽÂ µt(h ÃŽÂ µt(g)) and ÃŽÂ µs(hg)= ÃŽÂ µt(ÃŽÂ µs(h)g). Dualizing the above we can show that S is also a co-algebra anti-homomorphism: ÃŽÂ µ(S(h))= ÃŽÂ µ(S(h(1) ÃŽÂ µt(h(2)))= ÃŽÂ µ(S(h(1)h(2))= ÃŽÂ µ(ÃŽÂ µt(h))= ÃŽÂ µ(h), Ά(S(h))= Ά(S(h(1) ÃŽÂ µt(h(2)))= Ά(S(h(1) ÃŽÂ µt(h(2))à ¢Ã‚ ¨Ã¢â‚¬Å¡1)= Ά(S(h(1) ))(h(2)S(h(4))à ¢Ã‚ ¨Ã¢â‚¬Å¡ ÃŽÂ µt (h(3))= Ά(ÃŽÂ µs(h(1))(S(h(3))à ¢Ã‚ ¨Ã¢â‚¬Å¡S(h(2)))=S(h(3))à ¢Ã‚ ¨Ã¢â‚¬Å¡ ÃŽÂ µs(h(1))S(h(2))=S(h(2))à ¢Ã‚ ¨Ã¢â‚¬Å¡ 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 ΆX= Xà ¢Ã‚ ¨Ã¢â‚¬Å¡1 +g à ¢Ã‚ ¨Ã¢â‚¬Å¡ X, Άg=g à ¢Ã‚ ¨Ã¢â‚¬Å¡ g, Άg-1=g-1à ¢Ã‚ ¨Ã¢â‚¬Å¡g-1, ÃŽÂ µX=0, ÃŽÂ µg=1=ÃŽÂ µ g-1, SX=- g-1X, Sg= g-1, S g-1=g. S2X=q-1X. Proof We have Ά, ÃŽÂ µ on the generators and extended them multiplicatively to products of the generators. ΆgX=(Άg)( ΆX)=( gà ¢Ã‚ ¨Ã¢â‚¬Å¡g)( Xà ¢Ã‚ ¨Ã¢â‚¬Å¡1 +gà ¢
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