Tuesday, January 28, 2020
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 ¢
Sunday, January 19, 2020
A Comparison of Satire in Voltaires Candide and Gullivers Travels Ess
A Comparison of the Satire of Candide and Gulliver's Travels An impartial observer has the ability to make the most critical and objective observation on society and the behavior of man. This impartial observer would see the truth as it is. This same premise may be applied to literary works. A naive character or narrator may be used as an impartial observer, who reveals social truths to the audience through his or her naivete. As Maurois has noted, in writing about Candide, by Voltaire," It was novel of apprenticeship, that is, the shaping of an adolescent's ideas by rude contact with the universe" (101). Jonathan Swift also takes this approach in his work Gulliver's Travels, where Gulliver, the main character, provides a impartial point of reference. The satires Gulliver's Travels, by Jonathan Swift, and Candide, by Voltaire, both make use of naivete to convey satirical attacks on society. In both works, litotes [understatements] are made of extremely absurd situations, which further illuminates the ridiculous nature of a situation. Characters in each novel are made vulnerable by their overly trusting natures. This is taken advantage of, and these characters are left exploited by corrupt people in society. Attacks are also made on authority figures of the world. This can be seen in the characters' reaction to authority. Finally, both works are travel tales, which expose the main characters to many perspectives. This allows the authors to satirize many aspects of society. These two satirical works make litotes of preposterous situations, thus shedding light on the absurdity at hand. This is an especially effective technique, because a character or narrator is involved in a ridiculous situation. The reader, from an... ... French Novelist Manners and Ideas. New York: D Appleton and Company, 1929. "Introduction to Gulliver's Travels." Norton Anthology of English Literature, The Major Authors. Ed. M.H. Abrhams et al. Sixth ed. New York: W. W. Norton and Company, 1995. Lawler, John. "The Evolution of Gulliver's Character." Norton Critical Editions. Maurois, Andre'. Voltaire. New York: D. Appleton and Company, 1932. Mylne, Vivienne. The Eighteenth-Century French Novel. Manchester: University of Manchester Press, 1965. Pasco, Allan H. Novel Configurations A Study of French Fiction. Birmingham: Summa Publications, 1987. Quintana, Ricardo "Situation as Satirical Method." Norton Critical Editions: Jonathan Swift Gulliver's Travels. Ed. Robert A Greenberg. New York: W. W. Norton and Company Inc., 1961. Van Doren, Carl. Swift .New York: The Viking Press, 1930.
Saturday, January 11, 2020
Jimmy Carter and Political Maxim Essay
Hardball by Chris Matthews is an interpretation of what many know as â€Å"hard-politics†. The book describes the skill of playing the game in Washington and how to be successful at it. The book is a guide that teaches a series of maxims that would be in favor of politicians to learn in order to be successful. The different tactics provided in the book hold a lot of knowledge that would make the life of anyone following these strategies much easier when trying to get ahead in life. Matthews describes a countless number of examples of successful politicians that rose to the top. Those politicians are the ones who learned how to play hardball in Washington. Matthews provides many of his observations over the years and describes them with quotes from various sources. Quoted by Chris Matthews, he states â€Å"JFK would call 5 or 6 †¦ LBJ would take 19 names and call them all. †The quote lies deep in the heart of the political maxim retail politics. Retail politics is the management of one-to-one communication among voters or other politicians. Lyndon B. Johnson was a mastermind of retail politics and embraced every moment of it. Johnson’s success came from his unique instinct to work at a man’s ego. While JFK was more of a wholesaler, Johnson worked retail politics like it was candy. That is exactly what the quote is symbolizing, the difference between a wholesaler and a retail politician. JFK was more widespread and focused more on voters than other politicians. Johnson had the patience and humility to work with every legislator one at a time and get that connection that he needed. This type of networking is what made LBJ successful in politics and it’s what made him stand out from an average politician. Another quote provided in the book Hardball was in Chapter 3. It stated, â€Å"He’s not going to win. It’s a Republican district. He’d be better for us is he loses. He’ll work for me. He’ll bring his organization with him. †This was a strategy that President Jimmy Carter used to benefit his campaign. Jimmy Carter played smart politics in this quote that is connected to the â€Å"it’s better to receive than to give maxim. †The maxim means to let others give to you because it makes them feel involved. Carter played this strategy to a tee. Carter knew that his best line of defense would come from those who had lost their elections and were looking for a job. The quote symbolizes the tactic Carter used and how his intelligence allowed him to take advantage of the situation. He knew that those individuals faced a tough race and when they lost, they would become his support because people like to be used and not ignored. Another quotes that is related to the â€Å"it’s better to give then to receive,†maxim is a quote I found to be very witty. The quotes states, â€Å"I’ve lived across the street from you for 18 years †¦ I shoveled your walk in winter. I cut your grass in summer†¦ I didn’t think I had to ask you for your vote. He never forgot her response. ‘Tom, I want you to know something: people like to be asked. †The lady knew Tip O’Neil and all he had done for her over the years, but out of respect she found that all he needed to do was ask. She wanted to make sure she was considered and thought about, so she wouldn’t be taken for granted. The main point from this quote is that people don’t mind being used, but they do mind being taken for granted. It’s important to know that asking favors only brings in more supporters. People want to invest in others and are often not critical of that person because they also have a lot invested in them. An important maxim that a politician can learn is â€Å"don’t get mad, don’t get even; get ahead. †The following quotes states, â€Å"Cry Baby†, screamed the headline of the New York Daily News above a picture of Newt in diapers. â€Å"Newt’s Tantrum: He closed down the government because Clinton made him sit at the back of the plane. I find this quote to be the funniest of the rest of the quotes. It clearly demonstrates the fallout of trying to get even with someone. It doesn’t work so ignore the revenge part and surpass your expectations. The political maxim â€Å"don’t get mad, don’t get even; get ahead,†means to maintain calm and concentrate all your power on progressing and becoming a success. Newt Gingrich didn’t apply the political maxim to his career which resulted in the quote above. His ego allowed a government shutdown to occur and the pressed slammed him in all newspapers. Gingrich forgot to concentrate on moving past the government shutdown, but instead played a game and lost the political game. My favorite political maxim in the book is â€Å"Leave no shot unanswered. †The following quote corresponds with the political maxim as it states, â€Å"The purpose of the war room was not just to respond to Republican attacks †¦ It was to respond to them fast, even before they were broadcasted or published, when the lead of the story was still rolling around in the reporter’s mind†¦ †Bill Clinton knew of that certain political maxim and he knew it well. Attacks to a candidate are harmful if left alone. They stick to you and become a reality if they are not disputed. Aided by George Stephanopoulos, Bill Clinton was alerted of the lethal combination of an unanswered shot. It was an important strategy because if a shot is unanswered, people start to believe that it is true which can sometimes ruin an entire campaign. The quote also makes you realize the necessity of responding to an attack. Clinton had to create an entire new room named, â€Å"the War Room†in order to fire back. That shows how much of an importance it is to leave no shot unanswered.
Friday, January 3, 2020
Diversity at Coca Cola - 1061 Words
Diversity at Coca Cola Karrie McHugh Managing Global Diversity/Keller Coca Cola thought they had it all. Their product was selling rather well when they first started. They found through studies that one in every four people were drinking a soft drink and most of this early on was Coca Cola. In 2002, they were still doing great with over $3.9 billion in netted profits. They not only sold in the United States but in over 200 countries all together. They even diversified their product to where it was not just coke. They sold sprite, Diet Coke, and many others. The biggest hardship that they ran across was when they started their marketing trying to become†¦show more content†¦Ã… ¸ Established a mentoring program Ã… ¸ Initiated executive briefings for senior management concerning diversity strategy. Ã… ¸ Implemented a â€Å"Solutions†program that included an ombudsman and a hotline to resolve employees disputes. (Herman et al., 2002) Even with all the changes a survey implemented by the task force still says that minority employees still perceive that their career opportunities are not comparable to the whites. Now in a way I can see this being perceived early on. There was a lot of restructuring going on and Coca Cola had to move the employees around giving everyone an equal opportunity. The minorities were still saying that they were not being paid equally. With that being said, there are a lot of factors that needs to be considered that the report did not discuss. Were people equally qualified for the positions and their pay. Did they have the education necessary to be at the top, regardless of their race? That is a lot of what the pay is based off of. A lot of companies will pay an employee a bit more because they have an Associate, Bachelors or Master’ degree. For me to comment fairly on this, I would need to know the background of the employees and their education, not the race. 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