Hopf Algebra Project

Hopf Algebra ProjectPetros KarayiannisChapter 0IntroductionHopf algebras have lot of applications. At first, they used it in topology in 1940s, but because 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)PreliminariesThis chapter provides all the essential tools to chthonianstand the structure of Hopf algebras. Basic notations of Hopf algebra beGroupsFieldsVector outer distancesHomomorphismCommutative diagrams1.GroupsGroup G is a finite or infinite embed of elements with a binary operation. Groups have to pursue some rules, so we can learn it as a group. Those be closure, associative, there exist an identity element and an inverse element. allow us define two elements U, V in G, closure is when then the intersection point of UV is also in G. Associative when t he multiplication (UV) W=U (VW) U, V, W in G. There exist an identity element much(prenominal) 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.FieldsA issue is a commutative ring and every element b has an inverse.3.Vector SpaceA transmitter space V is a set that is closed under finite sender addition and scalar multiplication. In order for V to be a vector space, the following conditions must hold X, Y V and any scalar a, b a(b X) = (a b) X(a + b) X=aX + bXa(X+Y)=aX + aY1X=XA unexpended ideal of K-algebra is a bilinear 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 followingX +y is in Lcx is in Lz x is in LIf we replace c) with x z is in L, then this would defi ne a just 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 .4.HomomorphismGiven two groups, (G,*) and (H,) is a function f GH such that u, v G it holds thatf(u*v)=f(u)f(v)5.Commutative diagramsA commutative diagram is showing the composition of roles 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 allow V and W be -vector spaces with bases ei and fj respectively. The tensor product V and W is a new -vector space, V W with basis ei fj , is the set of all elements v w= (ci,j ei fj ). ci,j are scalars. oerly tensor products obey to distributive and scalar multiplication laws.The dimension of the tensor product of two vector spaces isDim(V W)=dim(V)dim(W)Theorem of Universa l Property of Tensor products 0.2Let V, W, U be vector spaces with procedure f V x W U is defined as f (v, w) vw.There exists a bilinear mapping b V x W V W , (v,w) v wIf f V x W U is bilinear, then there exist a funny function, f V WU with f=fbExtension of Tensor Products0.3The definition of Tensor products can be extended for more than two vectors such asV1 - V2- V3 - ..- VN = ( biv1- v2- .- vn ) (Becca 2014)Definition0.4Let U,V be vector spacers everyplace a landing field k and UV. If =0 then Rank () =0. If 0 then tell () is equal to the smallest positive integer r arising from the representations of = ui vi UV for i=1,2,,r.Definition0.5Let U be a finite dimensional vector space everywhere the field k with basis u1,.,un be a basis for U. the doubled basis for U*is u1,.,un where ui(uj)= ij for 1I,jn.Dual Pair0.6A dual pair is a 3 -tuple (X,Y,) consisting two vector spaces X,Y everyplace the same field K and a bilinear map, X x YK with x X0 yY 0 and y Y0 xX 0Defi nition0.7The wedge product is the product in an exterior algebra. If , are differential k-forms of degree p, g respectively, then=(-1)pq , is not in general commutative, but is associative,()u= (u) and bilinear(c1 1+c2 2) = c1( 1 ) + c2( 2 )( c1 1+c2 2)= c1( 1) + c2( 2). (Becca 2014)Chapter 1Definition1.1Let (A, m, ) be an algebra all all everyplace k and write mop (ab) = ab a, b A where mop=m,. then ab=ba a, b A. The (A, mop, ) is the opposite algebra.Definition1.2A co-algebra C isA vector space over KA map CC - C which is coassociative in the sense of (c(1)(1) - c(1)(2) - c(2))= (c(1) - c(2)(1) - c(2)c(2) ) cC ( called the co-product)A map C k obeying ((c(1))c(2))=c= (c(1)) c(2)) cC ( called the counit)Co-associativity and co-unit element can be expressed as commutative diagrams as follow go out 1 Co-associativity map Figure 2 co-unit element map Definition1.3A bi-algebra H isAn algebra (H, m ,)A co-algebra (H, , ), are algebra maps, where H- H has the tensor produc t algebra structure (h- g)(h- g)= hh- gg h, h, g, g H. A representation of Hopf algebras as diagrams is the followingDefinition1.4A Hopf Algebra H isA bi-algebra H, , , m, A map S H H such that (Sh(1))h(2) = (h)= h(1)Sh(2) hHThe axioms that make a simultaneous algebra and co-algebra into Hopf algebra is H- HH-HIs the map (h-g)=g-h called the flip map h, g H.Definition1.5Hopf Algebra is commutative if its commutative as algebra. It is co-commutative if its co-commutative as a co-algebra, =. It can be defined as S2=id.A commutative algebra over K is an algebra (A, m, ) over k such that m=mop.Definition1.6Two Hopf algebras H,H are dually paired by a map H H k if, =,h, =(h)g =, ()== , H and h, g H.Let (C, ,) be a co-algebra over k. The co-algebra (C, cop, ) is the opposite co-algebra.A co-commutative co-algebra over k is a co-algebra (C, , ) over k such that = cop.Definition1.7A bi-algebra or Hopf algebra H acts on algebra A (called H-module algebra) ifH acts on A as a vector s pace.The product map m AAA commutes with the transaction of HThe unit map k A commutes with the action of H.From b,c we come to the next actionh(ab)=(h(1)a)(h(2)b), h1= (h)1, a, b A, h HThis is the left action.Definition1.8Let (A, m, ) be algebra over k and is a left H- module along with a linear map m A-AA and a scalar multiplication k - AA if the following diagrams commute.Figure 3 Left Module mapDefinition1.9Co-algebra (C, , ) is H-module co-algebra ifC is an H-module CCC and C k commutes with the action of H. (Is a right C- co-module).Explicitly,(hc)=h(1)c(1)h(2)c(2), (hc)= (h)(c), h H, c C.Definition1.10A co-action of a co-algebra C on a vector space V is a map VCV such that,(id) =( id )id =(id ).Definition1.11A bi-algebra or Hopf algebra H co-acts on an algebra A (an H- co-module algebra) ifA is an H- co-moduleThe co-action A HA is an algebra homomorphism, where HA has the tensor product algebra structure.Definition1.12Let C be co- algebra (C, , ), map A HA is a rig ht C- co- module if the following diagrams commute.Figure 6Co-algebra of a right co-moduleSub-algebras, left ideals and right ideals of algebra have dual counter-parts in co-algebras. Let (A, m, ) be algebra over k and suppose that V is a left ideal of A. Then m(AV)V. Thus the restriction of m to AV determines a map AVV. Left co-ideal of a co-algebra C is a subspace V of C such that the co-product restricts to a map VCV.Definition1.13Let V be a subspace of a co-algebra C over k. Then V is a sub-co-algebra of C if (V)VV, for left co-ideal (V)CV and for right co-ideal (V)VC.Definition1.14Let 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.Definition1.15A simple co-algebra is a co-algebra which has two sub-co-algebras.Definition1.16Let C be co-algebra over k. A group-like element of C is c C with satisfies, (s)=ss and (s)=1 s S. The set of group-like elements of C is denoted G(C).Definition1.1 7Let S be a set. The co-algebra kS has a co-algebra structure determined by(s)=ss and (s)=1 s S. If S= we set C=k=0.Is the group-like co-algebra of S over k.Definition1.18The co-algebra C over k with basis co, c1, c2,.. whose co-product and co-unit is live up to by (cn)= cn-lcl and (cn)=n,0 for l=1,.,n and for all n0. Is denoted by P(k). The sub-co-algebra which is the span of co, c1, c2,,cn is denoted Pn(k).Definition1.19A 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(ei, j)= ei, lel, j(ei, j)=i, j i, j S. Set C(k)=(0).Definition1.20Let S be a non-empty finite set. A standard basis for Cs(k) is a basis c i ,jI, j S for Cs(k) which satisfies the co-matrix identities.Definition1.21Let (C, c, c) and (D, D, D) be co-algebras over the field k. A co-algebra map f CD is a linear map of underlying vector spaces such that Df=(ff) c and Df= c. An isomorphism of co-algebras is a co-algebra map which is a linear iso morphism.Definition1.22Let C be co-algebra over the field k. A co-ideal of C is a subspace I of C such that (I) = (0) and () IC+CI.Definition1.23The co-ideal Ker () of a co-algebra C over k is denoted by C+.Definition1.24Let I be a co-ideal of co-algebra C over k. The unique co-algebra structure on C /I such that the projection C C/I is a co-algebra map, is the quotient co-algebra structure on C/I.Definition1.25The tensor product of co-algebra has a natural co-algebra structure as the tensor product of vector space C-D is a co-algebra over k where (c(1)d(1))( c(2)d(2)) and (cd)=(c)(d) c in C and d in D.Definition1.26Let C be co-algebra over k. A skew-primitive element of C is a cC which satisfies (c)= gc +ch, where c, h G(c). The set of gh-skew primitive elements of C is denoted byPg,h (C).Definition1.27Let C be co-algebra over a field k. A co-commutative element of C is cC such that (c) = cop(c). The set of co-commutative elements of C is denoted by Cc(C).Cc(C) C.Definition1.2 8The category whose objects are co-algebras over k and whose morphisms are co-algebra maps under function composition is denoted by k-Coalg.Definition1.29The category whose objects are algebras over k and whose morphisms are co-algebra maps under function composition is denoted by k-Alg.Definition1.30Let (C, , ) be co-algebra over k. The algebra (C-, m, ) where m= - C-C-, (1) =, is the dual algebra of (C, , ).Definition1.31Let 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.Definition1.32Let 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 A.For fixed b A note that F AA defined by F(a)=a, b= ab- ba for all a A is a derivation of A.Definition1.33Let C be co-algebra over the field k. A co-derivation of C is a linear endomorphism f of C such that f= (fIC + IC f) . Definition1.34Let A and B ne algebra over the field k. The tensor product algebra structure on AB is determined by (ab)(ab)= aabb a, aA and b, bB.Definition1.35Let X, Y be non-empty subsets of an algebra A over the field k. The centralizer of Y in X isZX(Y) = xXyx=xy yYFor y A the centralizer of y in X is ZX(y) = ZX(y).Definition1.36The summation of an algebra A over the field Z (A) = ZA(A).Definition1.37Let (S, ) be a partially ordered set which is locally finite, meaning that , I, jS which satisfy ij the interval i, j = lSilj is a finite set. Let S= i, j I, jS, ij and let A be the algebra which is the vector space of functions f Sk under point wise operations whose product is given by(fg)(i, j)=f(i, l)g(l, j) iljFor all f, g A and i, jS and whose unit is given by 1(I,j)= i,j I,jS.Definition1.38The algebra of A over the k described supra is the incidence algebra of the locally finite partially ordered set (S, ).Definition1.39Lie co-algebra over k is a pair (C, ), where C is a vec tor space over k and CCC is a linear map, which satisfies=0 and (+()()+() ())()=0=C,C and I is the appropriate identity map.Definition1.40Suppose that C is co-algebra over the field k. The wedge product of subspaces U and V is UV = -1(UC+ CV).Definition1.41Let 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 UVD, U, V of D.Definition1.42Let C be co-algebra over k and (N, ) be a left co-module. Then UX= -1(UN+ CX) is the wedge product of subspaces U of C and X of N.Definition1.43Let 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.Definition1.44Let (A, m, ) be algebra over k. Then,A=m1(A-A- )(A, , ) is a co-algebra over k, where = m- A and =-.he co-algebra (A, , ) is the dual co-algebra of (A, m, ).Also we denote A by a and = a(1) a(2), a A.Definition1.45Let A be algebra over k. An - derivation of A is a linear map f Ak which satis fies f(ab)= (a)f(b)+f(a) (b), a, b A and , Alg(A, k).Definition1.46The 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).Definition1.47A 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 jAA(A)*, be linear map defined by jA(a)(a)=a(a), a A and aA. ThenjAA(A)* is an algebra mapKer(jA) is the intersection of the co-finite ideals of AIm(jA) is a dense subspace of (A)*.Is one-to-one.Definition1.48Let A (respectively C) be an algebra (respectively co-algebra ) over k. Then A (respectively C) is reflexive if jAA(A)*, as defined before and jCC(C*), defined asjC(c)(c*)=c*(c), c*C* and cC. ThenIm(jC)(C*) and jCC(C*) is a co-algebra map.jC is one-to-one.Im(jC) is the set of all a(C*)* which vanish on a closed co-finite ideal of C*.Is an isomorphism.Definition1.49Almost left noetherian algebra over k is an algebra over k whose co-finite left ideal are finitely generated. (M is called close noetherian if every co-finite submodule of M is finitely generated).Definition1.50Let fUV 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.Definition1.51Let A be algebra over k and C be co-algebra over k. A pairing of A and C is a bilinear map A-Ck which satisfies, (ab,c)= (a, c(1)) (b, c(2)) and (1, c) = (c), a, b A and c C.Definition1.52Let V be a vector space over k. A co-free co-algebra on V is a pair (, Tco(V)) such thatTco(V) is a co-algebra over k and Tco(V)T is a linear map.If C is a co-algebra over k and fCV is a linear map, a co-algebra map F C Tco(V) determined by F=f.Definition1.53Let V be a vector space over k. A co-free co-commutative co-al gebra on V is any pair (, C(V)) which satisfiesC(V) is a co-commutative co-algebra over k and C(V)V is a linear map.If C is a co-commutative co-algebra over k and f CV is linear map, co-algebra map FC C(V) determined by F=f. (Majid 2002, Radford David E)Chapter 2Proposition (Anti-homomorphism property of antipodes) 2.1The antipode of a Hopf algebra is unique and obey S(hg)=S(g)S(h), S(1)=1 and (SS)h=Sh, Sh=h, h,g H. (Majid 2002, Radford David E)ProofLet S and S1 be two antipodes for H. Then using properties of antipode, associativity of and co-associativity of we getS= (S- (Id-S1))= (Id- )(SId-S1)(Id -)=(Id)(SId-S1)( -Id) = ( (SId)S1) =S1.So the antipode is unique.Let S-id=s id-S=tTo check that S is an algebra anti-homomorphism, we computeS(1)= S(1(1))1(2)S(1(3))= S(1(1)) t (1(2))= s(1)=1,S(hg)=S(h(1)g(1)) t(h(2)g(2))= S(h(1)g(1))h(2) t(g(2))S(h(3))=s (h(1)g(1))S(g(2))S(h(2))=S(g(1)) s(h(1)) t (g(2))S(h(2))=S(g)S(h), h,g H and we used t(hg)= t(h t(g)) and s(hg)= t(s(h)g).Duali zing the above we can show that S is also a co-algebra anti-homomorphism(S(h))= (S(h(1) t(h(2)))= (S(h(1)h(2))= (t(h))= (h),(S(h))= (S(h(1) t(h(2)))= (S(h(1) t(h(2))1)= (S(h(1) ))(h(2)S(h(4)) t (h(3))=(s(h(1))(S(h(3))S(h(2)))=S(h(3)) s(h(1))S(h(2))=S(h(2)) S(h(1)). (New directions)Example2.2The Hopf Algebra H=Uq(b+) is generated by 1 and the elements X,g,g-1 with relationsgg-1=1=g-1g and g X=q X g, where q is a fixed invertible element of the field k. HereX= X1 +g X, g=g g, g-1=g-1g-1,X=0, g=1= g-1, SX=- g-1X, Sg= g-1, S g-1=g.S2X=q-1X.ProofWe have , on the generators and extended them multiplicatively to products of the generators.gX=(g)( X)=( gg)( X1 +g