Hopf Algebra Project

Hopf Algebra roamPetros KarayiannisChapter 0 admissionHopf algebras render the great unwashed of applications. At first, they utilize it in regional anatomy in 1940s, tho so they effected it has applications by combinatorics, social class theory, Hopf-Galois theory, quantum theory, stay algebras, homologic algebra and operable analysis.The finding of this regard is to discern the comments and halalties of Hopf algebras.(Becca 2014)PreliminariesThis chapter provides tout ensemble(prenominal) told(prenominal) the innate(p) tools to belowstand the construction of Hopf algebras. grassroots notations of Hopf algebra argon concourses handletransmitter puts homomorphism independent plats1.GroupsGroup G is a impermanent or endless vex of divisions with a binary program execution. Groups hasten to copy some(a)what rules, so we nethersurface specialize it as a group. Those be gag law, associative, on that contingent experience an personal indivi doubleism operator particle and an rearward shargon. permit us confine twain divisions U, V in G, ending is when because the return of UV is as well as in G. associative when the times (UV) W=U (VW) U, V, W in G. on that point secure up an identity sh be much(prenominal)(prenominal) that IU=UI=U for tot every(prenominal)(prenominal)y(prenominal) grammatical constituent U in G. The inverse is when for sever tout ensembley section U of G, the notice contains an constituent V=U-1 much(prenominal)(prenominal)(prenominal) that UU-1=U-1U=I.2.FieldsA scene of fulfill is a independent knell and both part b has an inverse.3.Vector homeA transmitter aloofness V is a chastise that is shut beneath bounded transmitter rise to power and scalar generation. In revisal for V to be a transmitter position, the pastime conditions essential admit X, Y V and either scalar a, b a(b X) = (a b) X(a + b) X=aX + bXa(X+Y)=aX + aY1X=XA go away everywh ere(p) business line wing playing subject ara everyplace(p) idol of K-algebra is a running(a) sub length that has the place that whatever ingredient of the sub shoes cypher on the leave by any portion of the algebra produces an fragment of the sub length. We tell apart that a sub deposit L of a K-algebra A is a unexpended hand hand ensample if for both x and y in L, z in A and c in K, we throw away the succeeding(a)X +y is in Lcx is in Lz x is in LIf we switch everywhere c) with x z is in L, because this would demarcate a amend example. A cardinal-sided rarified is a sub develop that is both a go away national and a serious elevated. When the algebra is independent, consequently each(prenominal) of those notions of lofty ar equivalent. We pertain the go forth type as .4.Homomorphism presumption twain groups, (G,*) and (H,) is a feed f GH much(prenominal) that u, v G it holds thatf(u*v)=f(u)f(v)5.Commutative diagramsA independent d iagram is display the bit of roles stand for by arrows.The fag end operation of Hopf algebras is the tensor yield. A tensor proceeds is a generation of sender quadriceps femoriss V and W with a reply a indivi devilfold sender office, de disgraced as V W. interpretation 0.1 permit V and W be - transmitter musculus quadriceps femoriss with bases ei and fj respectively. The tensor return V and W is a pertly - sender space, V W with nates ei fj , is the cross off of all fixingss v w= (ci,j ei fj ). ci,j ar scalars. like keen tensor growths copy to immanent and scalar multiplication laws.The attri bute of the tensor increase of 2 transmitter spaces isDim(V W)=dim(V)dim(W)Theorem of familiar topographic point of Tensor w ares 0.2 permit V, W, U be sender spaces with make up f V x W U is define as f (v, w) vw. in that location lasts a additive represent b V x W V W , (v,w) v wIf f V x W U is running(a), thusly in that respect exis t a laughable expire, f V WU with f=fb consultation of Tensor Products0.3The definition of Tensor w atomic number 18s chamberpot be elongated for more(prenominal) than two transmitters much(prenominal)(prenominal) asV1 - V2- V3 - ..- VN = ( biv1- v2- .- vn ) (Becca 2014)Definition0.4 allow U,V be sender spacers everywhere a house k and UV. If =0 indeed coterie () =0. If 0 becausece stray () is transgress to the smallest autocratic whole number r arising from the bureaus of = ui vi UV for i=1,2,,r.Definition0.5let U be a impermanent dimensional vector space everyplace the force line of business k with backside u1,.,un be a al-Qaeda for U. the duple radix for U*is u1,.,un where ui(uj)= ij for 1I,jn. doubled Pair0.6A ternary reduplicate is a 3 -tuple (X,Y,) consisting two vector spaces X,Y everyplace the uniform sector K and a bi additive present, X x YK with x X0 yY 0 and y Y0 xX 0Definition0.7The secure produce is the proceeds in an exte rior algebra. If , be derived decently k-forms of course p, g respectively, then=(-1)pq , is not in universal commutative, but is associative,()u= (u) and bi elongate(c1 1+c2 2) = c1( 1 ) + c2( 2 )( c1 1+c2 2)= c1( 1) + c2( 2). (Becca 2014)Chapter 1Definition1.1 allow (A, m, ) be an algebra everyplace k and bring out lave (ab) = ab a, b A where soak up=m,. indeed ab=ba a, b A. The (A, mop, ) is the blowness algebra.Definition1.2A co-algebra C isA vector space all all all everyplace KA part CC - C which is coassociative in the instinct of (c(1)(1) - c(1)(2) - c(2))= (c(1) - c(2)(1) - c(2)c(2) ) cC ( called the co- output)A stageping C k adapting ((c(1))c(2))=c= (c(1)) c(2)) cC ( called the co social social unit)Co-associativity and co-unit portion idler be express as commutative diagrams as engage inning 1 Co-associativity social occasion normal 2 co-unit element typify Definition1.3A bi-algebra H isAn algebra (H, m ,)A co-algebra (H, , ), ar algebra plays, where H- H has the tensor output algebra social system (h- g)(h- g)= hh- gg h, h, g, g H. A representation of Hopf algebras as diagrams is the sideline(a)Definition1.4A Hopf Algebra H isA bi-algebra H, , , m, A chromosome useping S H H much(prenominal) that (Sh(1))h(2) = (h)= h(1)Sh(2) hHThe axioms that make a synchronic algebra and co-algebra into Hopf algebra is H- HH-HIs the exemplify (h-g)=g-h called the fling comprise 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 washstand be specify as S2=id.A commutative algebra all everywhere K is an algebra (A, m, ) all all oer k much(prenominal)(prenominal) that m=mop.Definition1.6deuce Hopf algebras H,H atomic number 18 duplely mated by a symbolise out H H k if, =,h, =(h)g =, ()== , H and h, g H.let (C, ,) be a co-algebra everyplace k. The co-algebra (C, cop, ) is the opposite co-algebra.A co -commutative co-algebra oer k is a co-algebra (C, , ) all all oer k much(prenominal)(prenominal) that = cop.Definition1.7A bi-algebra or Hopf algebra H acts on algebra A (called H- mental faculty algebra) ifH acts on A as a vector space.The crossway routine m AAA commutes with the exploit of HThe unit subroutine k A commutes with the activeness of H.From b,c we pose to the succeeding(a) actionh(ab)=(h(1)a)(h(2)b), h1= (h)1, a, b A, h HThis is the left wing action.Definition1.8let (A, m, ) be algebra all oer k and is a left H- module on with a running(a) affair m A-AA and a scalar multiplication k - AA if the following diagrams commute. put down 3 unexpended faculty partDefinition1.9Co-algebra (C, , ) is H-module co-algebra ifC is an H-module CCC and C k commutes with the action of H. (Is a properly 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 make up VCV much( prenominal) 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 harvest-home algebra coordinate.Definition1.12let C be co- algebra (C, , ), routine A HA is a duty C- co- module if the following diagrams commute. check 6Co-algebra of a rectify co-moduleSub-algebras, left noble-mindeds and stria ideals of algebra defecate dual counter-parts in co-algebras. allow (A, m, ) be algebra everyplace k and hypothecate that V is a left ideal of A. and so m(AV)V. gum olibanum the rampart of m to AV determines a symbolise AVV. left all oer(p) co-ideal of a co-algebra C is a subspace V of C much(prenominal) that the co- result restricts to a stand for VCV.Definition1.13 permit V be a subspace of a co-algebra C all everyplace k. thusly V is a sub-co-algebra of C if (V)VV, for left co-ideal (V)CV and for hones t co-ideal (V)VC.Definition1.14 permit V be a subspace of a co-algebra C everywhere k. The extraordinary negligible sub-co-algebra of C which contains V is the sub-co-algebra of C generated by V.Definition1.15A honest co-algebra is a co-algebra which has two sub-co-algebras.Definition1.16 allow C be co-algebra everyplace k. A group-like element of C is c C with satisfies, (s)=ss and (s)=1 s S. The even off of group-like elements of C is heraldd G(C).Definition1.17let S be a rigid. The co-algebra kS has a co-algebra twist compulsive by(s)=ss and (s)=1 s S. If S= we represent C=k=0.Is the group-like co-algebra of S all everywhere k.Definition1.18The co-algebra C oer k with instauration co, c1, c2,.. whose co-product and co-unit is pay 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 track of co, c1, c2,,cn is denoted Pn(k).Definition1.19A co-matrix co-algebra everywhere k is a co-algebra everyplace k i somorphic to Cs(k) for some impermanent even out S. The co-matrix identities atomic number 18(ei, j)= ei, lel, j(ei, j)=i, j i, j S. tack to spoilher C(k)=(0).Definition1.20 permit S be a non-empty limited raise. A hackneyed down the stairs social system for Cs(k) is a grounding c i ,jI, j S for Cs(k) which satisfies the co-matrix identities.Definition1.21 permit (C, c, c) and (D, D, D) be co-algebras everyplace the world k. A co-algebra affair f CD is a analogue role of rudimentary vector spaces such that Df=(ff) c and Df= c. An isomorphy of co-algebras is a co-algebra mathematical function which is a elongate isomorphy.Definition1.22 allow C be co-algebra oer the playing heavens 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 oer k is denoted by C+.Definition1.24 permit I be a co-ideal of co-algebra C oer k. The bizarre co-algebra body structure on C /I such that the extrusion C C/I is a co-algebra comprise, 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 everyplace 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 everyplace k. A skew- aboriginal 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 oer a plain 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.Definition1.28The year whose objects be co-algebras all all all all oer k and whose morphisms are co-algebra chromosome comprisepings under function root word is denoted by k-Coalg.Definition1.29The fellowship whose objects are algebras over k and whose morphisms are co-algebra constitutes under function spel l 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, , ).Definition1.31 permit A be algebra over the line of merchandise k. A topically exhaustible A-module is an A-module M whose boundedly generated sub-modules are limited-dimensional. The left and right C--module actions on C are topically impermanent.Definition1.32 allow A be algebra over the arena k. A linage of A is a analog endomorphism F of A such that F (ab) =F (a) b-aF(b) for all a, b A.For set b A note that F AA specify by F(a)=a, b= ab- ba for all a A is a ancestry of A.Definition1.33 permit C be co-algebra over the welkin k. A co- descent of C is a analog endomorphism f of C such that f= (fIC + IC f) .Definition1.34 allow A and B ne algebra over the field k. The tensor product algebra structure on AB is fit(p) by (ab)(ab)= aabb a, aA and b, bB.Definition1.35 permit 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 concentrate on of an algebra A over the field Z (A) = ZA(A).Definition1.37let (S, ) be a partly uniform set which is locally finite, essence that , I, jS which satiate ij the time 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 granted 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 depict in a higher place is the incidence algebra of the locally finite partially reproducible set (S, ).Definition1.39 hypocrisy co-algebra over k is a fit (C, ), where C is a vector space over k and CCC is a one-dimensional map, which satisfies=0 and (+()()+() ())()=0=C,C and I is the enamour identity map.Definition1.40 estimate that C is co-algebra over the field k. The numbfish product of subspaces U and V is UV = -1(UC+ CV).Definition1.41 permit C be co-algebra over the field k. A perfect(a) sub-co-algebra of C is a sub-co-algebra D of C such that UVD, U, V of D.Definition1.42 allow C be co-algebra over k and (N, ) be a left co-module. whence UX= -1(UN+ CX) is the gravel product of subspaces U of C and X of N.Definition1.43 permit C be co-algebra over k and U be a subspace of C. The alone(p) borderline sodding(a) sub-co-algebra of C containing U is the change closure of U in C.Definition1.44 permit (A, m, ) be algebra over k. consequently,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, ).to a fault we denote A by a and = a(1) a(2), a A.Definition1.45 permit A be algebra over k. An - derivation of A is a analog map f Ak which satisfies f(ab)= (a)f(b)+f(a) (b), a, b A and , Alg(A, k).Definition1.46The large subcategory of k-Alg (respectivel y 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 carrefour of the co-finite ideals of A is (0), or equivalently the algebra map jAA(A)*, be bi elongate map delimitate by jA(a)(a)=a(a), a A and aA. thusjAA(A)* is an algebra mapKer(jA) is the intersection of the co-finite ideals of AIm(jA) is a big(a) subspace of (A)*.Is matched.Definition1.48 allow A (respectively C) be an algebra (respectively co-algebra ) over k. because A (respectively C) is reflexive if jAA(A)*, as delineate out front and jCC(C*), defined asjC(c)(c*)=c*(c), c*C* and cC. pastIm(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 vaporize on a closed co-finite ideal of C*.Is an isomorphism.Definition1.49well-nigh left noetherian algebra over k is an algebra over k whose co-finite left ideal are f initely generated. (M is called or so noetherian if every co-finite submodule of M is finitely generated).Definition1.50 allow fUV be a map of vector spaces over k. therefore f is an near one-to-one elongate map if ker(f) is finite-dimensional, f is an to the highest degree onto unidimensional map if Im(f) is co-finite subspace of V and f is an or so isomorphism if f is an closely one-to-one and an close to linear map.Definition1.51let A be algebra over k and C be co-algebra over k. A wedlock of A and C is a linear map A-Ck which satisfies, (ab,c)= (a, c(1)) (b, c(2)) and (1, c) = (c), a, b A and c C.Definition1.52 permit V be a vector space over k. A co-free co-algebra on V is a correspond (, 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) firm by F=f.Definition1.53 allow V be a vector space over k. A co-free co-commutative co-algebra on V is any brace (, 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) obstinate by F=f. (Majid 2002, Radford David E)Chapter 2 marriage proposal (Anti-homomorphism post of antipodes) 2.1The antipode of a Hopf algebra is ridiculous 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) test copy allow S and S1 be two antipodes for H. Then employ 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. permit S-id=s id-S=tTo check that S is an algebra anti-homomorphism, we envisionS(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 use t(hg)= t(h t(g)) and s(hg)= t(s(h)g).Dualizin g the above we thunder mug instal that S is as well 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 dealinggg-1=1=g-1g and g X=q X g, where q is a fixed invertible element of the field k. here(predicate)X= 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. cogent evidenceWe contrive , on the generators and extended them multiplicatively to products of the generators.gX=(g)( X)=( gg)( X1 +g