Polytopes and Hopf algebras

Transcription

1 Polytopes and Hopf algebras Victor M. Buchstaber Steklov Institute, RAS, Moscow School of Mathematics, University of Manchester Shanghai 02 May 2010

2 Part I Abstract We show that the differential ring of convex polytopes possesses additional rich algebraic structures, in particular a Hopf comodule structure. This opens ways to applying technique and results of Hopf algebras to studying combinatorial polytopes. We shall discuss applications to toric topology. The talk is based on the works: 1. V. M. Buchstaber, N. Yu. Erokhovets, Ring of polytopes, quasi-symmetric functions and Fibonacci numbers., arxiv: v1 [math CO] 3 Feb V. M. Buchstaber, N. Yu. Erokhovets, Operator algebra on the ring of polytopes and quasisymmetric functions., UMN, v. 65, Issue 2(392), 2010,

3 Contents Basic notions Group of combinatorial polytopes -ring of combinatorial polytopes The Rota-Hopf Algebra Hopf -algebra of combinatorial polytopes Leibnitz Hopf algebras Universal Leibnitz Hopf algebras Operator Hopf algebra References 2

4 Basic notions Let us consider n-dimensional Euclidean space R n. A convex hull of a finite set {v 1,..., v N } R n is conv(v 1,..., v N ) = { x R n : x = N t i v i, t i 0, i=1 } N t i = 1. Definition. A convex polytope P is conv(v 1,..., v N ). We will speak of convex polytopes without the word convex. Definition. An n-dim convex polyhedron P is an intersection of finitely many half-spaces in R n : i=1 {x R n : a i, x + 1 0, i = 1,..., m}. where, is the canonical scalar product in R n. A polytope is a bounded convex polyhedron. A polytope P n is simple if there are exactly n facets meeting at each vertex of P n. Note that each face of a simple polytope is again a simple polytope. A polytope P n is simplicial if all proper faces of P n are simplices. 3

5 Group of combinatorial polytopes Definition. Two polytopes P and Q of the same dimension are said to be combinatorially equivalent if there is a bijection between their sets of faces that preserves the inclusion relation. Definition. A combinatorial polytope is a class of combinatorially equivalent polytopes. Denote by P 2n the free abelian group generated by all n-dimensional combinatorial polytopes. For n 1 we have the direct sum P 2n = m n+1 P 2n,2(m n), where P 2n,2(m n) is generated by all n-dimensional polytopes with m facets. Note that rank P 2n,2(m n) < for any fixed n and m. Denote by P 0 the group Z generated by point. Set P = P 2n = P 0 + n 0 m 2 m 1 n=1 P 2n,2(m n). 4

6 -ring of combinatorial polytopes Definition. The direct product P Q of polytopes P and Q turns the bigraded group P into a bigraded commutative associative ring, the ring of polytopes (P, ). The unit is P 0, a point. Let P s be the abelian subgroup in P generated by simple polytopes. The direct product P Q of simple polytopes P and Q is a simple polytope as well. Thus (P s, ) is a subring in (P, ). A polytope is indecomposable if it can not be represented as a direct product of two other polytopes of positive dimensions. Theorem. The ring (P, ) is a polynomial ring generated by indecomposable combinatorial polytopes. 5

7 The Rota-Hopf Algebra Let P be a finite poset with a minimal element ˆ0 and a maximal element ˆ1. An element y in P covers another element x in P, if x < y and there is no z in P such that x < z < y. A rank function ρ is a map P Z such that ρ(ˆ0) = 0 and ρ(y) = ρ(x) + 1 if y covers x. If a rank function ρ exists for a given P then it is defined in a unique way. A poset P with a rank function ρ is called graded. Set ρ(p) = ρ(ˆ1) and deg P = 2ρ(P). Two finite graded posets are isomorphic if there exists an order preserving bijection between them. Denote by R the graded free abelian group with basis the set of all isomorphism classes of finite graded posets. 6

8 The graded connected Hopf algebra R: The multiplication P Q is a cartesian product P Q of posets P and Q: Let x, u P and y, v Q. Then (x, y) (u, v) if and only if x u and y v. The unit element in R is the poset with one element ˆ0 = ˆ1. The comultiplication is (P) = ˆ0 z ˆ1 [ˆ0, z] [z, ˆ1], were [x, y] is the subposet {z P x z y}. The counit ε is The antipode χ is χ(p) = k 0 {c k } ε(p) = 1, if ˆ0 = ˆ1; 0, else ( 1) k [x 0, x 1 ] [x 1, x 2 ]... [x k 1, x k ] where c k = (ˆ0 = x 0 < x 1 < < x k = ˆ1) 7

9 Example. The simplest Boolean algebra B 1 = {ˆ0, ˆ1} is the face lattice of the point pt (B 1 ) = 1 B 1 + B 1 1 χ(b 1 ) = B 1 There is a natural linear mapping L : P R, deg L = 2, that sends a polytope P to its face lattice L(P). The mapping L is injective, but it is not a ring homomorphism, since it doesn t preserve a unit: the unit of P is a point pt and L(pt) = B 1, while the unit of R is the one-element set {ˆ0 = ˆ1}. Remark. The face lattice L(P Q) of P Q contains the empty face, which can be considered as, but evidently has no faces of the form F or G, where F and G are non-empty faces of P and Q respectively. 8

10 Let P n be an n-dimensional polytope {x R n : a i, x + 1 0, i = 1,..., m}. We assume that the system of inequalities is irredundant, that is the absence of any one of them changes the set P n. In this case the polytope P has exactly m facets F i = P n {x R n : a i, x + 1 = 0}, i = 1,..., m. Definition. A dual (or polar) polytope is defined as (P n ) = {y R n : y, x + 1 0, x P n }. It is easy to see that (P ) = P, thus the correspondence P P defines a linear involution of the ring P. For any simple polytope P its polar polytope P is simplicial and vice versa. There is a bijection F F between i-faces of P and (n 1 i)-faces of P such that F G G F. Let F be a face of P. Define a face polytope P/F as (F ). It is easy to see that L(P/F) = [F, P]. 9

11 Hopf -algebra of combinatorial polytopes Let n 1 = conv(e 1,..., e n ) be the regular simplex, where e i R n is the endpoint of the i-th standard basis vector. Definition. Let P k R k+1, and Q l R l+1 be polytopes lying in k and l. A join P Q is defined as conv(p {0} {0} Q) R k+1 R l+1. It is easy to see that the polytope P Q lies in the regular simplex k+l+1 and dim P Q = k + l + 1. From this construction is evident that the join is an associative operation. Example. For the simplices k and l we have k l = k+l+1. Proposition. The face lattice L(P Q) is the direct product of the face lattices L(P) and L(Q). Proposition. For two combinatorial polytopes P and Q (P Q) = P Q. A join of polytopes defines a bilinear operation, deg = 2, on the group P. It is associative and commutative, so (P, ) is a commutative associative ring without a unit. 10

12 Let us add to P a formal unit e, deg e = 2, corresponding to the empty set. Thus P = P = P. Let us denote the ring (P Ze, ) by RP. Then RP is a commutative associative ring with the unit e. It is a bigraded ring with graduations ( ) 2(n + 1), 2(m n 1) where n = dim P and m is the number of facets of P. Theorem. 1. RP is a ring of polynomials generated by all join-indecomposable polytopes. 2. RP is a Hopf algebra with comultiplication (P) = F P F P/F. 3. Involution gives a ring homomorphism : RP RP such that = τ where τ (x y) = y x. Corollary. The correspondence P L(P) identifies RP with a graded Hopf subalgebra in R. 11

13 Leibnitz Hopf algebras Definition. A Leibnitz-Hopf algebra over A is an associative Hopf algebra H over the ring A with a fixed sequence of a countable number of multiplicative generators H i, i = 1, 2,..., satisfying the comultiplication formula Set Φ(t)= H n = i 0 i+j=n H i H j, H 0 = 1. H i t i H [[t]]. Then Φ(t)=Φ(t) Φ(t). Definition. A composition ω of a number m is an ordered set ω = (j 1,..., j k ), j i 1, such that m = j j k. Set ω = m, and l(ω) = k. The antipode in H satisfies the relation χ(φ(t))φ(t) = 1. Therefore for n 1 χ(z n ) = ω: ω =n ( 1) l(ω) Z ω where Z ω = Z j1... Z jk for the composition ω = (j 1,..., j k ). 12

14 Universal Leibnitz Hopf algebras Definition. Let Z = Z Z 1, Z 2,... be the free Leibnitz-Hopf algebra over the integers Z in an infinite number of generators Z i. Set deg Z i = 2i. Proposition. Z is the universal Leibnitz-Hopf algebra, i.e. for any Leibnitz-Hopf algebra H the correspondence Z i H i defines a Hopf algebra homomorphism Z H. Definition. Let S = Z /[Z, Z ] be the universal commutative Leibnitz-Hopf algebra. Theorem. S is a self-dual graded Hopf algebra isomorphic to the graded Hopf algebra of symmetric functions Sym = Z[σ 1, σ 2,... ] where σ i is the i-th elementary symmetric function. Definition. Let U be the universal Hopf algebra in the category of Leibnitz-Hopf algebras H with an antipode χ(h i ) = ( 1) i H i. Proposition. U = Z /J U, where J U is the two-sided Hopf ideal generated by the polynomials for all n 2. n k=0 ( 1) k Z n k Z k 13

15 Operator Hopf algebra Definition. For k 0 let us define face operators d k on P that send a polytope P n to the sum of all it s (n k)-dimensional faces. In particular, d n+1 P n =, d k = 0, k 1. Proposition. Set d = d 1. For any two polytopes P and Q d(p Q) = (dp) Q + P (dq), d(p Q) = (dp) Q + P (dq). Therefore (P s, ), (P, ) and RP are differential rings. Example. pt P = CP is a cone. We have d(cp) = d(pt P) = (d pt) P + pt (dp) = = P + C(dP) = P + C(dP). Thus, [d, C] = id in the ring RP. 14

16 Proposition. d k (P Q) = d k (P Q) = i+j=k i+j=k (d i P) (d j Q), (d i P) (d j Q). By R we will denote the rings (P s, ), (P, ) or RP. Consider Φ(t) = k 0 d k t k. Proposition. We have Φ( t)φ(t) = 1 on the ring R. Definition. Let D(R) Hom Z (R, R) be the ring of operators on the ring R generated by the faces-operators d k, k 0. Theorem. 1. The correspondence Z k d k defines a Hopf module structure on the ring R over the Hopf algebra Z. 2. This correspondence induces the isomorphism of the universal Hopf algebra U and the operator Hopf algebra D = D(R) where R = (P, ) or RP. 3. The Hopf algebra D(P s ) is isomorphic to the divided power algebra, i.e. d k d l = ( ) k+l k dk+l on the ring P s. 15

17 Part II Abstract Quasisymmetric functions arise naturally in diverse areas of mathematics such as combinatorics, noncommutative geometry, algebraic topology, Hecke algebras and quantum groups. We construct a homomorphism from the ring of convex polytopes to the ring of quasisymmetric functions over integers. Two polytopes have the same image if and only if their flag-vectors coincide. We show that the image over the rational numbers of this homomorphism is a free commutative polynomial algebra and describe this image over the integers in terms of functional equations. 16

18 Contents Ring of quasi-symmetric functions Hopf algebra of quasi-symmetric functions Ring homomorphisms Hopf comodule structure Derivation of quasi-symmetric functions Dual operator Hopf algebra References 17

19 Ring of quasi-symmetric functions Definition. Let t 1,..., t m, be a set of variables deg t i = 2i. For a composition ω = (j 1,..., j k ) consider a quasi-symmetric monomial M ω = Set deg M ω = 2 ω. Example. l 1 < <l k t j 1 l1... t j k M (1) = t 1 + t 2 + t 3, lk. M (1,2) = t 1 t t 1t t 2t 2 3, M (2,1) = t 2 1 t 2 + t 2 1 t 3 + t 2 2 t 3, M (2,2) = t 2 1 t2 2 + t2 1 t2 3 + t2 2 t2 3. Proposition. A polynomial g Z[t 1,..., t m ] is a finite linear combination of quasi-symmetric monomials if and only if g(0, t 1, t 2,..., t m 1 ) = g(t 1, 0, t 2,..., t m 1 ) =... = = g(t 1,..., t m 1, 0) The same is true in the case of an infinite number of variables and a formal power series g(t 1, t 2,... ) having bounded degree. 18

20 Hopf algebra of quasi-symmetric functions Corollary. Finite integer combinations of quasi-symmetric monomials form a ring. It is a ring of quasi-symmetric functions QSym(m) in the case of m variables, and QSym in the case of a infinite number of variables. For any two monomials M ω and M ω their product in the ring of formal series Z[[t 1, t 2,... ]] is equal to M ω M ω = ω Ω +Ω =ω M ω, where for ω = (j 1,..., j l ), ω = (j 1,..., j l ), ω = (j 1,..., j k ), Ω, Ω are the k-tuples such that Ω = (0,..., j 1,..., 0,..., j l..., 0), Ω = (0,..., j 1,..., 0,..., j l..., 0). This multiplication rule of compositions is called the overlapping shuffle multiplication. 19

21 For example, M (1) M (1, 1) = = i<j i t i j<k t 2 i t j + i<j t j t k = t i tj t i t j t k = i<j<k = M (2, 1) + M (1, 2) + 3M (1, 1, 1). The diagonal mapping : QSym QSym QSym M (j1,..., j k ) = k i=0 M (j1,..., j i ) M (j i+1,..., j k ) defines on QSym a graded Hopf algebra structure. For a composition ω = (j 1,..., j k ) set ω = (j k,..., j 1 ). The correspondence M ω M ω gives a ring homomorphism : QSym QSym such that = τ where τ (x y) = y x. 20

22 Let us order the set of all compositions lexicographically. A proper tail of a composition (j 1,..., j k ) is any composition (j i,..., j k ) with 1 < i k. (The empty composition and the one-symbol composition have no proper tails.) A composition (or a word) is Lyndon if all its proper tails are larger than the composition itself. For example, the compositions (1, 1, 2), (1, 2, 1, 2, 2), (1, 3, 1, 5) are Lyndon and the compositions (1, 1, 1, 1), (1, 2, 1, 2), (2, 1) are not Lyndon. The set of Lyndon compositions is denoted by Lyn. 21

23 We need the following known results about the Hopf algebra QSym: Theorem. 1. As a graded Hopf algebra QSym is dual to the universal Leibnitz Hopf algebra Z, where M ω, Z ω = δ ω, ω 2. The epimorphism Z S = Z /[Z, Z ] induces the embedding of the graded dual Hopf algebras Sym = S Z = QSym : σ i M ( 1, } 1, {{..., 1 } ). i 3. As an algebra QSym is a free polynomial algebra over the integers. 4. The ring QSym Q is a free polynomial algebra with generators M ω, ω Lyn. 22

24 Ring homomorphisms Definition. The correspondence P Φ(t 1 )(P) = k 0(d k P)t1 k defines a ring homomorphism Φ 1 : R R[t 1 ]. The correspondence P Φ(t 2 )Φ(t 1 )(P) = Φ(t 2 ) k 0 = P + (d k P)(t1 k + tk 2 ) + k>0 m 2 (d k P)t k 1 ω =m ω=(k,l) = d l (d k P)t k 1 tl 2 defines a ring homomorphism Φ 2 : R R QSym(2) where Φ(t 2 )t 1 = t 1. Iterating this procedure we obtain the ring homomorphisms Φ n : R R QSym(n) for each n such that Φ n+1 (P)(t 1,..., t n, 0) = Φ n (P)(t 1,..., t n ). 23

25 Example. Φ 3 (P)(t 1, t 2, t 3 ) = P + k>0(d k P)(t k 1 + tk 2 + tk 3 )+ + m 2 ω =m ω=(k,l) + m 3 d l (d k P)(t k 1 tl 2 + tk 1 tl 3 + tk 2 tl 3 ) = ω =m ω=(j 1,j 2,j 3 ) d j3 ( dj2 (d j1 P) ) t j 1 1 t j 2 2 t j 3 3. Thus we obtain the ring homomorphism Φ = lim Φ n : R R QSym. Remark. Let us mention that Φ (P n ) is defined by Φ n (P n ). 24

26 Definition. Let P n be an n-dimensional polytope and S = {a 1,..., a k } {0, 1,..., n 1}. Set l(s) = k. A flag number f S is the number of increasing sequences of faces F a 1 F a 2 F a k, dim F a i = a i. For S = {i} the number f S = f i is just the number of i-dimensional faces. Set f = 1 The collection {f S } of all flag numbers is called a flag f -vector of the polytope P n. Now composing Φ m with a ring homomorphism ϕ: R A, where A is a ring, we obtain ring homomorphisms F ϕ,m : R A QSym(m), m = 0, 1,...,. 25

27 Definition. The α-character ξ α : P Z[α] ( ξ α (P n ) = α n, n 0 ) defines the generalized f -polynomial F P : P QSym[α] : F P (P n ) = k 0 l(s)=k where ω(s) = (n a k, a k a k 1,..., a 2 a 1 ). The α-character ε α : RP Z[α] ( εα ( ) = 1, ε α (P n ) = α n+1, n 0 ) defines the generalized f -polynomial F RP : RP QSym[α] : F RP (P n ) = k 0 l(s)=k where ω(s) = (ω(s), a 1 + 1). f S α a 1M ω(s) f S ( M ω(s) + α a 1+1 M ω(s) ) The counit ε of the Hopf algebra RP defines the homomorphism F = F such that F is the Ehrenborg transformation. F(P n ) = n k=0 l(ω)=k f S M (a1 +1,ω (S)). It follows from the definition that F RP α=0 = F. 26

28 Each of these three homomorphisms encodes the whole information about the flag f -vector of the polytope P. Remark. It can be shown that F is a Hopf algebra homomorphism RP QSym, and F(P ) = F (P). Proposition. F RP (P n ) = F (P n ) + αf P (P n ). Corollary. F P (P Q) = = F P (P) F (Q) + F (P)F P (Q) + αf P (P)F P (Q) Corollary. F P (CP) = F (P) + (α + σ 1 )F P (P) 27

29 Theorem. Let g be a homogeneous polynomial of degree 2n in QSym(m)[α] and m n. Then g Im F P, m if and only if: g(α, t 1, t 1, t 3,..., t m ) = g(α, 0, 0, t 3,..., t m ), g(α, t 1, t 2, t 2,..., t m ) = g(α, t 1, 0, 0,..., t m ),... g(α, t 1,..., t m 1, t m 1 ) = g(α, t 1,..., 0, 0), and (1) g( α, t 1, t 2,..., t m 1, α) = g(α, t 1, t 2,..., t m 1, 0). Let h be a homogeneous polynomial of degree 2n in QSym(m) and m n. Then h Im F RP, m if and only if h satisfies: the equations of type (1) and the equation h( α, t 1, t 2,..., t m 1, α) = h(0, t 1, t 2,..., t m 1, 0). Let g be a homogeneous polynomial of degree 2n in QSym(m)[α] and m n. Then g Im F m if and only if g satisfies the equations of type (1). In all the cases the equations are equivalent to the Bayer Billera relations. 28

30 Hopf comodule structure Proposition. The mapping P P RP : P P = F P F P/F induces on the ring P a Hopf comodule structure over the Hopf algebra RP. Corollary. There is a ring homomorphism l α : P RP[α] : l α P = α dim F P/F. F P Example. Using the x-character ε x : RP Z[x], we obtain the homogeneous f -polynomial ε x l α P n = F P α dim F x n dim F = F P, 1 (P n ). Proposition. For any polytope P P we have F (l α P) = F P (P) Proposition. There is the following commutative diagram of ring homomorphisms P P P R F P QSym[α] F P F QSym[α] QSym 29

31 Derivation of quasi-symmetric functions Definition. Let us define θ : QSym QSym by the formula (θf )(t 1, t 2,... ) = t f (t, t 1, t 2,... ) t=0 For the monomial M ω, ω = (j 1,..., j k ) we have θm (j1,..., j k ) = M (j2,..., j k ), if j 1 = 1; 0, else. Proposition. θ is a derivation such that F P (dp) = θf P (P). Let us denote the monomial M ω, corresponding to the Lyndon composition ω by X ω Then θ = + X 2 + X 3 + X X 1 X 12 X 13 X 112 The image of the subring P s of simple polytopes under the mapping F P belongs to Z[α, σ 1 ] = Z[α, X 1 ] QSym[α]. Corollary. Let P P s then F P (dp) = X 1 F P (P). 30

32 Dual operator Hopf algebra Let us define T k : Z[t 1, t 2,... ] Z[t, t 1, t 2,... ] T k g(t 1, t 2,... ) = g(t 1,..., t k 1, t, t, t k, t k+1,... ) The epimorphism Z U D defines the injection of the graded dual Hopf algebra D into QSym. Theorem. 1. Let g QSym. Then g D if and only if T k g = g for all k D Q is a free polynomial algebra with dimension of the (2n)-th graded component equal to the Fibonacci number c n 1 ( c0 = c 1 = 1, c n+1 = c n + c n 1, n 1 ). 3. F (RP) = D. 4. The ring F P (P) Q is a free polynomial algebra with rank of (2n)-th graded component equal to c n. 31

33 Part III Topological realization of Hopf algebras In 2008 A. Baker and B. Richter showed that the ring of quasi-symmetric functions has a very nice topological interpretation. Consider CW -complexes X and their homology H (X) and cohomology H (X) with integer coefficients. Let us assume that homology groups H (X) have no torsion. The diagonal map X X X defines in H (X) the structure of a graded coalgebra with the comultiplication : H (X) H (X) H (X) and the dual structure of a graded algebra in H (X) with the multiplication : H (X) H (X) H (X). 32

34 In the case when X is an H-space with the multiplication µ: X X X we obtain the Pontryagin product µ : H (X) H (X) H (X) in the coalgebra H (X) and the corresponding structure of a graded Hopf algebra on H (X). The cohomology ring H (X) obtains the structure of a graded dual Hopf algebra with the diagonal mapping µ : H (X) H (X) H (X). For any space Y the loop space X = ΩY is an H-space. A continuous mapping f : Y 1 Y 2 induces a mapping of H-spaces Ωf : X 1 X 2, where X i = ΩY i. Thus for any space Y such that X = ΩY has no torsion in integral homology we obtain the Hopf algebra H (X). This correspondence is functorial, that is any continuous mapping f : Y 1 Y 2 induces a Hopf algebra homomorphism f : H (X 1 ) H (X 2 ). 33

35 By the Bott-Samelson theorem H (ΩΣX) is the free associative algebra T( H (X)) generated by H (X). This construction is functorial, that is a continuous mapping f : X 1 X 2 induces a ring homomorphism of the corresponding tensor algebras arising from the mapping f : H (X 1 ) H (X 2 ). Denote elements of H (ΩΣX) by (a 1... a n ), where a i H (X). Since the diagonal mapping ΣX ΣX ΣX gives the H-mapping : ΩΣX ΩΣX ΩΣX, it follows from the Eilenberg-Zilber theorem that (a 1... a n ) = ( a 1... a n ) where (a 1 b 1 a 2 b 2 ) = (a 1 a 2 ) (b 1 b 2 ). 34

36 There is a nice combinatorial model for any topological space of the form ΩΣX with X connected, namely the James construction JX on X. After one suspension this gives rise to a splitting ΣΩΣX ΣJX n 1 ΣX (n), where X (n) denotes the n-fold smash power of X. Example. There exists a homotopy equivalence ΣΩΣS 2 Σ(S 2 ) (n) Σ ( S 2n) n 1 n 1 and so there exists an H-mapping ΩΣ(ΩΣS 2 ) ΩΣ ( that is a homotopy equivalence. S 2n) n 1 35

37 Using these classical topological results let us describe topological realizations of the Hopf algebras from our talk. We have the following Hopf algebras: I. 1. H (CP ) is a divided power algebra Z[v 1, v 2,... ]/I, where the ideal I is generated by the relations v n v m ( ) n+m n vn+m. It is the Hopf algebra with the comultiplication v n = n k=0 v k v n k ; 2. H (CP ) = Z[u] a polynomial ring. It is the Hopf algebra with the comultiplication u = 1 u + u 1; 36

38 II. 1. H (ΩΣCP ) T( H (CP )) = Z v 1, v 2,... with v i being non-commuting variables of degree 2i. Thus there is an isomorphism of Hopf algebras H (ΩΣCP ) Z under which v n corresponds to Z n. Here Z is the universal Leibnitz Hopf algebra. The coproduct on H (ΩΣCP ) induced by the diagonal in ΩΣCP is compatible with the one in Z : v n = i+j=n v i v j So if we set deg Z i = 2i, then there is an isomorphism of graded Hopf algebras. This gives a geometric interpretation for the antipode χ in Z : in H (ΩΣCP ) it arises from the time-inversion of loops. 37

39 II. 2. H (ΩΣCP ) is the graded dual Hopf algebra to H (ΩΣCP ). Theorem. Let us double the graduation, that is let us set deg Z i = 2i, deg t j = 2j. Then there is an isomorphism of graded Hopf algebras: H (ΩΣCP ) Z = Z Z 1, Z 2,..., H (ΩΣCP ) M = QSym[t 1, t 2,... ]. III. H (BU) H (BU) Z[σ 1, σ 2,... ] C. It is a self-dual Hopf algebra of symmetric functions. In the cohomology σ i are represented by Chern classes. IV. 1. H (ΩΣS 2 ) = H (ΩS 3 ) = Z[v] a polynomial ring with deg v = 2. It is the Hopf algebra with the comultiplication v = 1 v + v H (ΩΣS 2 ) = Z[v n ]/I is a divided power algebra. Thus H (ΩΣS 2 ) H (CP ). 38

40 V. H (ΩΣ(ΩΣS 2 )) Z w 1, w 2,.... It is a free associative Hopf algebra with the comultiplication w n = n ( n k k=0 ) wk w n k. VI. H ( ΩΣ ( n 1 S 2n)) Z ξ 1, ξ 2,.... It is a free associative algebra and has the structure of a graded Hopf algebra with the comultiplication ξ n = 1 ξ n + ξ n 1. ( ( Therefore, H ΩΣ S 2n)) gives a topological n 1 realization of the universal Lie-Hopf algebra U. 39

41 The homotopy equivalence a: ΩΣ ( S 2n) ΩΣ(ΩΣS 2 ) n 1 induces an isomorphism of graded Hopf algebras a : Z ξ 1, ξ 2,... Z w 1, w 2,..., and its algebraic form is determined by the conditions: a ξ n = (a a )( ξ n ). For example, a ξ 1 = w 1, a ξ 2 = w 2 w 1 w 1, a ξ 3 = w 3 3w 2 w 1 + 2w 1 w 1 w 1. Thus using topological results we have obtained that two Hopf algebra structures on the free associative algebra with the comultiplications (V) and (VI) are isomorphic over Z. This result is interesting from the topological point of view, since the elements (w n a ξ n ) for n 2 are obstructions to the desuspension of the homotopy equivalence Σ(ΩΣS 2 ) Σ ( S 2n). n 1 40

42 We have the commutative diagram: Here: ΩΣ(ΩΣS 2 ) ΩΣk ΩΣCP j ΩΣS 2 k CP i is the inclusion CP = BU(1) BU; i BU det CP j arises from the universal property of ΩΣCP as a free H-space; the mapping CP ΩΣCP is induced by the identity mapping ΣCP ΣCP ; the mapping k: ΩΣS 2 CP corresponds to the generator of H 2 (ΩΣS 2 ) = Z; det is the mapping of the classifying spaces BU BU(1) induced by the mappings det : U(n) U(1), n = 1, 2,

43 As we have mentioned, H (BU) as a Hopf algebra is isomorphic to the algebra of symmetric functions (in the generators Chern classes) and is self-dual H (BU) H (BU) Sym[t 1, t 2,... ] = Z[σ 1, σ 2,... ]. Since CP gives rise to the algebra generators in H (BU), j is an epimorphism on homology, and j is a monomorphism on cohomology. j corresponds to the factorization of the non-commutative polynomial algebra Z by the commutation relations Z i Z j Z j Z i and sends Z i to σ i ; j corresponds to the inclusion Sym[t 1, t 2,... ] QSym[t 1, t 2,... ] and sends σ i to M ωi, where ω i = (1, 1,..., 1); } {{ } i k sends u to v 1 and defines an embedding of the polynomial ring Z[u] into the divided power algebra Z[v n ]/I ; the composition det j maps the algebra Z to the divided power algebra H (CP ). This corresponds to the mapping Z i d k Ps of Z to D(P s ). 42

44 The mapping λ: CP CP : (z 1 : z 2 :...) ( z 1 : z 2 :...) induces the homomorphism H (CP ) H (CP ) : v n ( 1) n v n and gives the geometric interpretation the homomorphism Z Z : Z n ( 1) n Z n, n = 1, 2,... Consider a mapping γ : ΩΣCP ΩΣCP induced by the time-inversion of loops. Corollary. D H (ΩΣCP )/J where J is a two-sided Hopf ideal generated by the elements ((ΩΣλ) (v n ) γ (v n )), n = 1, 2,

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