Lots of Hopf Algebras, I Jeanne Duflot January 19, 1996 Abstract The purpose of this paper is to give definitions of some new Hopf algeb* *ras analo- gous to the mod-p Steenrod algebras. For example, we produce a Hopf algebr* *a over the integers whose reduction mod-p is the mod-p Steenrod algebra. Just for* * fun, we "quantize" these algebras too! Further study of the examples of this paper* * will be done in part II. 1 Deformations of Hopf Algebras Suppose that k is a commutative ring with 1, and that (A; ; ; ffl; ; S) is a Ho* *pf algebra over k. Here, is the unit, ffl the counit, the multiplication, the comultipl* *ication and S the antipode of A. An n - parameter deformation of A is a topological Hopf a* *lgebra (A"h; ; "h; ffl; "h; S"h) over the power series ring k[["h]] (where "h= (h1; : * *:;:hn) is a set of n commuting indeterminates) such that a) A"his isomorphic to A[["h]] as a topological k[["h]]-module. b) "h mod "h, "h mod "h. Two deformations A"hand B"hof the same Hopf algebra A are equivalent if ther* *e is an isomorphism f"h: A"h! B"hof Hopf algebras over k[["h]] which is the identity mo* *d "h. (We are using the notations and definitions of [2].) We are assuming that the unit and counit of A"hare obtained by simply extend* *ing those of A, so that there will be no difference in the notations for these maps, whet* *her considered on A or on A"h. Note that "h: A[["h]] ! A[["h]] ^k[["h]]A[["h]] and that "h: A[["h]] ^k[["h]]A[["h]] ! A[["h]]; the completion indicated by " ^" is the "h-adic completion. We will also make t* *he natural identification of k[["h]]-modules A[["h]] ^k[["h]]A[["h]] ~=(A k A)[["h]] 1 without further comment. The trivial n-parameter deformation of A is the deformation where "h, "hand * *S"hare obtained by extending , and S to A[["h]]. If B is any (topological) Hopf algebra (over a commutative (topological) rin* *g), with comultiplication B and antipode SB, define the set of grouplike elements of B a* *s: G(B) = {b 2 B - {0} | B(b) = b b}: We know that (see, e.g., [6]) a) G(B) is a monoid with respect to the multiplication in B, with identity th* *e unit 1 of B; in particular, for all b 2 B, bSB(b) = SB(b)b = ffl(b)1: b) For every b 2 B, we have (SB(b)) = SB(b) SB(b); so SB(b) 2 G(B) if SB(b) does not equal zero. c)If b 2 G(B), and ffl(b) = 1, then SB(b) is the inverse of b in the monoid * *G(B). The proof of the following lemma is immediate. Lemma 1.0.1 Let B be a (topological) Hopf algebra over a commutative (topolog* *ical) ring R, with antipode SB and comultiplication B. We denote the multiplication in B b* *y juxta- position. i)If a; b 2 G(B), then B(a - b) = (a - b) a + b (a - b): ii)If a; b 2 G(B) are such that ffl(a) = ffl(b) = 1, then SB(a - b) = SB(a)(b* * - a)SB(b): iii)Suppose that e; "e; g and "gare grouplike elements of B. Let z and "zin B* * satisfy B(z) = z g + g z and B("z) = "z "g+ "g "z: Then B(ze - "z"e) = (ze - "z"e) ge + ge (ze - "z"e) + "z"e (ge - "g"e) + (g* *e - "g"e) "z"e: iv)Suppose that z; "z; e; "e; g and "gare as in iii), and that furthermore, f* *fl(z) = ffl("z) = 0, and ffl(e) = ffl("e) = ffl(g) = ffl("g) = 1. Then SB(ze - "z"e) = -SB(ge)(ze - "z"e)SB(ge) - SB("z"e)(ge - "g"e)SB(ge) - SB(ge)("g"e- ge)S* *B("ge)"z"eSB(ge): 2 Let t be an indeterminate. If B is any (topological) Hopf algebra over aPco* *mmutative ring, with coalgebra map B,Pthen a curve over b0 in B is an element fl(t) = 1* *i=0biti of B[[t]] such that B(bn) = nj=0bj bn-j for every n. A curve over 1 is also call* *ed a divided power sequence in B. The set of curves in B is naturally a monoid under ordina* *ry power series multiplication. Notice that if A0 is the trivial 1-parameter deformation of A, then the mono* *id of curves in A is naturally isomorphic with the monoid of grouplike elements in A0. The i* *somorphism is obtained by setting the "curve parameter" t equal to the "deformation parame* *ter" h. Similarly, one may define multiparameter curves over b0, where B is any Hopf* * algebra over a commutative ring, and b0 2 B. Given one grouplike element of a deformation of A, we may form many others a* *s follows. Lemma 1.0.2 Suppose t = h is an indeterminate (different from the indetermina* *tes h1; : :;:hn, and commuting with these). Let p("h) 2 k[["h]] be a power series with no consta* *nt term. Then i)The function evp : A"h[[t]] ! A"hdefined by evp((t)) = (p("h)) is a k[["h]* *]-algebra homomorphism. ii)If A"h[[t]] is regarded as being the trivial one-parameter deformation of * *the Hopf algebra A"h, then evp is a homomorphism of topological Hopf algebras over k[["h]]. Proof: i) follows from the fact that A"his (isomorphic to), as a topological* * k[["h]]-module, A[["h]]. Thus it makes sense to speak of the "h-adic topology on A"h, and furth* *ermore, we have that A"his complete with respect to this topology. Since k[["h]] is also a cent* *ral subalgebra of A"h, we may evaluate power series over A"hat elements of the ideal (h1; : :;:hn* *) of k[["h]], and this is an algebra homomorphism. Similarly, ii) follows from the definition of the trivial deformation, and t* *he fact that we are tensoring over k[["h]] (at least), and the completeness of everything in si* *ght with respect to the "h-adic topology. A special case of the lemma above is Corollary 1.0.3 Let A"h;0be the trivial 1-parameter deformation of A"h. Then if* * p("h) is as above, evp : G(A"h;0) ! G(A"h) is a homomorphism of monoids. Suppose that a; b 2 G(A"h). Write a - b as a power series in "h: X a - b = x"ff(a; b)"h"ff; "ff where x"ff(a; b) 2 A for every "ff. Define I(a; b) to be the two-sided ideal of* * A generated by the x"ff(a; b). Theorem 1.0.4 Let a; b 2 G(A0), where A0 is the trivial n-parameter deformati* *on of A; suppose also that ffl(a) = ffl(b) = 1. Then I(a; b) is a Hopf ideal in A. 3 Proof: P P Let a = "ffa"ff"h"ffandPb = "ffb"ff"h"ff, where a"ffand b"ffare in A for* * every "ff. Writing a - b = "ffx"ff(a; b)"h"ff, the equation "h(a - b) = (a - b) a + b (a - b) becomes X X X X X (x"ff(a; b))"h"ff= ( x"ff(a; b)"h"ff) a"ff"h"ff+ b"ff"h"ff ( * * x"ff(a; b)"h"ff) = "ff "ff f"f "ff * *"ff X X ( x"fl(a; b) a"fi+ b"fl x"fi(a; b))"h"ff; "ff"fl+"fi="ff this implies that, for every "fl, X (x"ff(a; b)) = x"fl(a; b) a"fi+ b"fl x"fi(a; b); "fl+"fi="ff which implies that I(a; b) is a co-ideal in A. Now, let S be the antipode for A (and by extension for A0). Writing out equa* *tion ii) of the lemma above, we see also that I(a; b) is invariant under S: X X X * * X S(x"ff(a; b))h"ff= S(a - b) = S(a)(b - a)S(b) = -( S(a"ff)h"ff x"ff(a; b)* *h"ff S(b"ff)h"ff) "ff "ff "ff * * "ff X X = -( ( S(a"fi)x"ffi(a; b)S(b"fl))h"ff): "ff"fi+"ffi+"fl="ff Comparing like coefficients, we get the desired result. More generally, suppose that a and b are as above, but we have a factorizati* *on a = ge and b = "g"ein G(A"h), such that ffl(g) = ffl(e) = ffl("g) = ffl("e) = 1. Let z* *; "zbe elements of A"h such that ffl(z) = ffl("z) = 0, and "h(z) = z g + g z; "h("z) = "z "g+ "g "z: Write ze - "z"eas a power series in "h: X ze - "z"e= z"ff(z; e; "z; "e)"h"ff; "ff where z"ff(z; e; "z; "e) 2 A for every "ff. Let I(g; "g; e; "e; z; "z) be t* *he two-sided ideal of A generated by the x"ff(a; b) and the z"ff(z; e; "z; "e), for every "ff. Theorem 1.0.5 Suppose that A"h= A0 is the trivial deformation of A; and a; b;* * g; e; z; "g; "e and "zare as above. Then I(g; "g; e; "e; z; "z) is a Hopf ideal of A. Proof: One does calculations like those of the previous theorem (admittedly * *a bit more complicated to write down) based directly on properties iii) and iv) of Lemma 1* *.0.1; noting that ge = a and "g"e= b. 4 2 Examples Let U be the free associative algebra over Z on a countably infinite set X = {X* *1; X2; : :;:Xn; : :}:; for convenience, set X0 = 1. Then U is a Hopf algebra over Z with respect to th* *e diagonal defined by Xn (Xn) = Xi Xn-i: i=0 The counit ffl is the map taking a polynomial in the noncommuting variables Xit* *o its constant term. The values of the antipode S for U may be inductively computed using the * *formulas nX Xn S(Xi)Xn-i= XiS(Xn-i) = 0; i=0 i=0 which hold for all n > 0. For example, S(X1) = -X1. Notice that since U is coco* *mmutative, S2 = id, so that S is invertible and is equal to its own inverse. Let U0 = U[[h1; h2]] be the trivial two-parameter deformation ofPU. Let ff; * *fi; ffi; fl be any four power series in Z[[h1; h2]] with no constant term. Let f(t) = 1i=0Xiti; * *f(t) is clearly a curve in U0. Thus, f(ff); f(fi); f(ffi) and f(fl) are in G(U0); this implies th* *at a = f(ff)f(fi) and b = f(ffi)f(fl) are also in G(U0). By the theorems in the previous section,* * the coefficients of the element f(ff)f(fi) - f(ffi)f(fl) in U0 generate a Hopf ideal J(ff; fi; ffi; fl; f) in U, so that U=J(ff; fi; ffi* *; fl; f) is a Hopf algebra over the integers. We have deliberately not assigned a grading to the elements Xi so far, but o* *ne may do this, as follows: choose an integer K, not equal to 0, and assign deg(Xi) = Ki.* * Then U becomes a graded Hopf algebra over the integers. We only point out that the mul* *tiplication on U U that we want to use is NOT the graded multiplication (a b)(c d) = (-1)deg(b)deg(c)(ac bd) but the standard multiplication (a b)(c d) = (ac bd): (If the reader does not like this, he or she may use only even degrees for the * *Xi's.) In any case, if one grades U, then we should impose a further condition on t* *he elements ff; fi; ffi; fl above; namely that they are all homogenous polynomials in h1; h* *2 of the same de- gree. In this case, the generators that define J(ff; fi; ffi; fl; f) are homog* *eneous elements of U, and if J0(ff; fi; ffi; fl; f) is the homogeneous two-sided ideal generated b* *y these homoge- neous elements, we see that in fact J0(ff; fi; ffi; fl; f) is a graded Hopf ide* *al in U, and that U=J0(ff; fi; ffi; fl; f) is a graded Hopf algebra over the integers. Example 2.0.6 Let ff = h1; fi = h2; ffi = h2; fl = h1: Then U=J(ff; fi; ffi; fl; f) = Uc = Z [X1; : :;:Xn; : :]:, thePpolynomial ring * *in the commuting indeterminates X1; : :;:Xn; : :,:with coalgebra map (Xn) = ni=0Xi Xn-i: 5 Example 2.0.7 Assign deg(Xi) = i, and let ff = h21+ h1h2; fi = h22; ffi = h22+ h1h2; fl = h21: Then we set U=J0(ff; fi; ffi; fl; f) = A2(Z ), the integral version of the mod-* *2 Steenrod algebra. According to Bullett-Macdonald [1], the reduction mod-2 of this algebra over* * the integers is the mod-2 Steenrod algebra A2. This algebra was also studied in [7]. Example 2.0.8 Assign deg(Xi) = 2i(p - 1), where p is an odd prime. Let ff = hp1+ hp-11h2 + : :+:h1hp-12; fi = hp2; ffi = hp2+ h1hp-12+ : :+:hp-1* *1h2; fl = hp1: Then we set U=J0(ff; fi; ffi; fl; f) = Aevp(Z ), the integral version of the ev* *en part of the mod-p Steenrod algebra. Again, according to Bullett-Macdonald [1], the reduction mod-p of this algebra * *is the mod-p Steenrod algebra modulo the Bockstein. Example 2.0.9 Fix a positive integer N. Assign deg(Xi) = 2iO(N), where O is so* *me fixed integer valued function. Let ff = hN1+ hN-11h2 + : :+:h1hN-12; fi = hN2; ffi = hN2+ h1hN-12+ : :+:hN-11h2* *; fl = hN1: Then, again, U=J0(ff; fi; ffi; fl; f) is a Hopf algebra over the integers, whic* *h we call AevN(Z ). In the next example, we take quotients of an extension of U. Let @ be a sym* *bol dif- ferent from, and not commuting with, any of the symbols X1; X2; : :.: Let U@ be* * the free associative algebra over the integers on the symbols @; X1; : :;:Xn; : :m:odulo* * the two-sided ideal generated by @2. Grade U@ by setting deg(Xi) = 2iO(N) as in the previous * *example and deg(@) = 1: Now, on U@ U@ we use the graded multiplication! We may then ma* *ke U@ into a Hopf algebra by setting (@) = @ 1 + 1 @. Let U@;0be the trivial two-pa* *rameter deformation of U@ with parameters h1; h2 as above. Suppose ff; fi; ffi; fl are * *as in the previous example; let g = f(ff); e = f(fi); "g= f(ffi); "e= f(fl), set a = ge and b = "g* *"e: Finally, set z = h2(@g - g@) and "z= h1(@"g- "g@). Naturally we consider all of these as ele* *ments of U@;0. One may verify: i)Using lemma 1.0.2, g; e; "g; "e; z; "zall satisfy the properties of the el* *ements with the same names in lemma 1.0.1. (Even with the graded multiplication: this is b* *ecause the coefficients of all the grouplike elements are of even degree.) ii)Because the coefficients of all the grouplike elements have even degree, t* *he conclusions of lemma 1.0.1 remain true as well, thus allowing us to use theorem 1.0.5 * *to conclude that the ideal in theorem 1.0.5 is in this particular case a Hopf ideal. This allows us to exhibit 6 Example 2.0.10 Let AN (Z ) be the quotient of the Hopf algebra U@ (with grade* *d multipli- cation on U@ U@) by the two-sided ideal generated by the coefficients of the t* *wo power series f(hN1+ hN-11h2 + : :+:h1hN-12)f(hN2) - f(hN2+ h1hN-12+ : :+:hN-11h2)f(hN1) and h2(@f(hN1+ hN-11h2 + : :+:h1hN-12) - f(hN1+ hN-11h2 + : :+:h1hN-12)@)f(hN* *2) -h1(@f(hN2+ h1hN-12+ : :+:hN-11h2) - f(hN2+ h1hN-12+ : :+:hN-11h2)@)f(hN1): Then AN (Z ) is a Hopf algebra. If N = p is an odd prime, then the mod-p reduction of AN (Z ) is the mod-p S* *teenrod algebra, by the work of Bullett-Macdonald cited above. If k is any commutative ring with 1, then U(k) = U Z k (the free associative* * algebra on the set {X1; : :;:Xn; : :}:over k) is a Hopf algebra over k, with the same d* *iagonal map as above, and we may define Hopf algebras AevN(k) following the procedure in th* *e above examples. Example 2.0.11 Suppose that k is a commutative ring. Suppose that q is an ele* *ment of k. Assign degree Xi= i in U(k). Let ff = qh1; fi = h2; ffi = h2; fl = h1: P n Then U(k)=J0(ff; fi; ffi; fl; f) is a Hopf algebra with coalgebra map (Xn) = * *i=0Xi Xn-i: Example 2.0.12 Suppose that k is a commutative ring, and that q is an element* * of k. Assign deg(Xi) = i in U(k), and let ff = qh21+ h1h2; fi = h22; ffi = h22+ h1h2; fl = h21: P* * n Then U(k)=J0(ff; fi; ffi; fl; f) is a Hopf algebra (with coalgebra map (Xn) = * * i=0Xi Xn-i). Well, as the reader can see, we could go on and on in this way. Which of the* *se examples are "interesting"? To this author, they are all intrinsically interesting in th* *eir own ways, and it is not so clear (right now, to this author) which examples should be of more* * general interest. However, these generalizations of the Steenrod algebra, and the fact that they * *seem to have global (not just infinitesimal) deformations (as coalgebras) might have some ap* *plications in algebraic topology. We may view, for example, the classical Adams spectral seq* *uence as saying that infinitesimal deformations of the Steenrod algebra have something t* *o do with homotopy groups of spheres. Is there a direct connection between global deforma* *tions of the Steenrod algebra and homotopy theory? Also, we have the two problems of "duality" and "representations". By the pr* *oblem of "duality", we mean "What should the correct definition of the dual Hopf algebra* * be for these examples?" Since we are working over the integers (in general), and, as we shal* *l see in the 7 discussion of example 2.0.7 later in this section, "most" of the "interesting s* *tuff" in many of the examples is torsion, there is no one correct definition of "the" dual. In p* *articular, we are definitely not interested in working entirely within the category of, say, grad* *ed k-modules which are finitely generated and free over k in each degree. However, initially* *, it should be of interest to study (graded) HomZ (-; Q=Z ). By the problem of "representations",* * we mean of course "What are some interesting modules (or comodules) for these Hopf alge* *bras?" The Steenrod algebras, for example, may be defined in terms of their natural repres* *entations on mod-p cohomology rings. At present we don't have any general answers to this q* *uestion, although of course any module over the mod-2 Steenrod algebra is also a module * *for A2(Z ) (for example) because of the natural surjection of A2(Z ) onto the mod-2 Steenr* *od algebra. Also, as we shall see, there is a natural representation of A2(Z ) on the divid* *ed power algebra over the integers. We return to give just a few more somewhat different types of examples than * *the above, and finally, at the end of this section, we give an initial discussion of examp* *le 2.0.7. The special thing about U and the curve f(t) in the above lies in the fact t* *hat U represents "curves over 1", f(t) being the prototypical curve over 1. P More generally, in fact, if A is any Hopf algebra over k, and OE(t) = 1i=0* *aiti is any curve in A, then we get a Hopf ideal J(ff; fi; ffi; fl; OE) in A, exactly as ab* *ove: by taking the coefficients of the element OE(ff)OE(fi) - OE(ffi)OE(fl) in A[[h1; h2]] (regarded as the trivial deformation of A) as generators. The p* *ower series ff; fi; fl; and fl are, as before, in k[[h1; h2]] and have no constant term. Here's another specific example. We use the definitions and theorems of [5]* *. Let p be a primePnumber, and let U(p)= U(Z (p)). Grade U(p)by setting degree Xi = 2i(p * *- 1). If i = 1i=0iiti is any curve over 1 in U(p), such that degree ii = 2i(p - 1) for* * every i, there exists a unique homomorphism of graded Hopf algebras ^i: U(p)! U(p) such that ^i(Xi) = ii for every i. Thus the image of ^iis a graded Hopf subalge* *bra of U(p) which we shall call U(p);i. Notice that i is also a curve in U(p);i. AccordingPto [5], there exists a p-pure curve in the Hopf algebra U(p): thi* *s is a curve E(t) = 1 + 1i=1Eitiover 1 in U(p)such that a) degree(Ei) = 2i(p - 1), for every i, b) U(p);Eis the subalgebra of U(p)generated by the elements E1 = X1; Ep; Ep2;* * : :;:Epn; : ::: Now we get a lot more Hopf algebras! Example 2.0.13 Let A = U(p);E, where E is a p-pure curve as above. Let the po* *lynomials ff; fi; ffi; fl be as in example 2.0.7 above. Then the coefficients of E(ff)E(fi) - E(ffi)E(fl) generate a Hopf ideal of U(p);E, and we have a new Hopf algebra over Z (p)by ta* *king the quotient. 8 Notice that the Hopf ideals in all the examples above are "quadratic" ideals* * in some sense. We may get "higher-order" ideals by replacing the element OE(ff)OE(fi) - OE(ffi)OE(fl) by elements OE1(ff1)OE2(ff2) : :O:Em (ffm ) - "OE1(ffi1)O"E2(ffi2) : :":O* *Em(ffim ): Here, the OEi's and the "OEi's must be curves, and the ffi's and ffii's power s* *eries over k with no constant term. One may build even more general Hopf ideals as well. The rest of this section is devoted to a discussion of example 2.0.7. Discus* *sions of other examples will be deferred to part II of this paper [4]. The "generating function" for the graded Hopf ideal J0 in U (by this we mean* * that the coefficients of the power series generate the ideal in question) is the power s* *eries in U0 X1 X1 X1 X1 ( Xi(h21+ h1h2)i)( Xi(h22)i) - ( Xi(h22+ h1h2)i)( Xi(h21)i): i=0 i=0 i=0 i=0 Inspecting the above generating function, we see that the coefficient of hn1* *hm2is zero if n + m is odd, whereas if n + m is even, the coefficient of hn1hm2is anm - bnm, * *where [n=2]X n - k! anm = Xn-kX(m-n+2k)=2; k=max{0;(n-m)=2} k and [m=2]X m - k ! bnm = Xm-k X(n-m+2k)=2: k=max{0;(m-n)=2} k The degree of anm and bnm is (n + m)=2. Using these equations we prove a series of lemmas. For convenience, we deno* *te the generator Xiof U, and its image in A2(Z ) by the same symbol Xi. Lemma 2.0.14 For every i 0, X1Xi= (i + 1)Xi+1in A2(Z ). Proof: If i 1, notice that a1;2i+1= X1Xi, and b1;2i+1= (i + 1)Xi+1. Corollary 2.0.15 For every i 0, Xi1= i!Xi in A2(Z ). Proof: Use the previous lemma, and induction on i. Let C = [A2(Z ); A2(Z )] be the commutator ideal of A2(Z ). This is a homoge* *nous two- sided ideal in A2(Z ). Lemma 2.0.16 For every i 0 and j 0, in A2(Z ) we have ! i + j XiXj Xi+jmod C: j 9 ! i + j Proof: We prove by induction on i: For all j 0, XiXj Xi+jmod C: j For i = 1, this is implied by the above lemma. Now, consider!XnXj, where n > 1, and we know that for all i < n, and for all* * b 0, i + b XiXb Xi+bmod C. Let r = 2j + n and s = n; then j = (r - s)=2. Since b brs- ars= 0 in A2(Z ), we see that ! [(2j+n)=2] ! [n=2] ! n + j X n - k + 2j X n - k XnXj = Xn+j+ X2j+n-kXk-j- Xn-kXk+j: j k=j+1 k k=1 k Now, note that if j + 1 k [(2j + n)=2] and a = n - 2(k - j), then 0 a < n* * and ! ! 2j + n - k a + k X2j+n-k= Xa+k XaXk mod C k k by induction. Thus, [(2j+n)=2]Xn - k + 2j! [(2j+n)=2]X [n=2]X X2j+n-kXk-j XaXkXk-j= Xn-2rXj+rXr mod C: k=j+1 k k=j+1 r=1 On the other hand, [n=2]Xn - k! [n=2]X Xn-kXk+j Xn-2kXkXk+jmod C k=1 k k=1 by induction. Putting this together, we have ! [n=2] ! n + j X n + j XnXj Xn+j+ Xn-2k(Xk+jXk - XkXk+j) Xn+j mod C: j k=1 j Let (Z ) be the free abelian group on a countably infinite set of generators* * T1; T2; : :;:Tn; : :.: Grade (Z ) by setting degree(Ti) = i, and put a ring structure on (Z ) by setti* *ng ! i + j TiTj = Ti+j: j In other words, (Z ) is the divided polynomial ring over the integers; it is a * *commutative ring. Lemma 2.0.17 A2(Z )=[A2(Z ); A2(Z )] is isomorphic to (Z ) as a graded Z-alge* *bra. Proof: Define a ring homomorphism : U ! (Z ) by (Xi) = Ti, for every i. We need to show that (anm - bnm) = 0, then we will have an induced homomorph* *ism : A2(Z )=[A2(Z ); A2(Z )] ! (Z ) 10 such that (Xi) = Tifor every i. We may assume that n m, and n + m is even. Computing, we have [n=2]X n - k! (n + m)=2 ! (anm - bnm) = T(n+m)=2- k=max{0;(n-m)=2} k n - k [m=2]X m - k ! (n + m)=2! T(n+m)=2: k=max{0;(m-n)=2} k m - k Setting d = (n + m)=2 and e = (n - m)=2, we have the above equal to [(d-e)=2]Xd - l! d ! d - e - l! d ! ( - )Td = 0: l=0 l + e d - l l d - e - l Since (Z ) is free as an abelian group on the set {T1; : :;:Tn : :}:,we may * *define an inverse homomorphism (of abelian groups) to by the formula -1(Ti) = Xi+ [A2(Z ); A2(Z )] in A2(Z )=[A2(Z ); A2(Z )]: By lemma 2.0.16, this function is a homomorphism of* * rings. Corollary 2.0.18 Let : U ! (Z ) be the homomorphism of lemma 2.0.17. Then ind* *uces a isomorphism of Q -algebras 1 : A2(Z ) Q ! (Z ) Q: Since (Z ) Q is isomorphic as a Q -algebra to the polynomial ring Q [T ] in on* *e variable over Q via the homorphism Ti 1 7! T i=i!, we have A2(Z ) Q ~=Q [T ] via the map Xi 1 7! T i=i!. Proof: The inverse to the map Xi 1 7! T i=i! is the ring homomorphism given* * by T 7! X1 1: by the second of the above lemmas, Xi1= i!Xi in A2(Z ) for every i 0. Corollary 2.0.19 The set of torsion elements in A2(Z ) is equal to the commutat* *or ideal [A2(Z ); A2(Z )]: 11 Proof: One has the following exact sequence of graded abelian groups (which * *are finitely generated in each degree) 0 ! [A2(Z ); A2(Z )] ! A2(Z ) ! (Z ) ! 0 given by the above lemmas. Thus A2(Z )=[A2(Z ); A2(Z )] is torsion free. Tensoring this exact sequence with the rational numbers we get 0 ! [A2(Z ); A2(Z )] Q ! A2(Z ) Q ! (Z ) Q ! 0: Again, by the previous lemmas, the right-hand arrow here is an isomorphism, for* *cing the left-hand abelian group to be zero. Thus, [A2(Z ); A2(Z )] consists entirely of* * torsion elements. We point out that lemmas similar to the some of the above lemmas are also tr* *ue for the mod-2 Steenrod algebra [9]. Finally, we would like to point out the following interesting fact. Suppose * *that instead of the homogeneous two-sided ideal J0 of U generated by the coefficients of the* * generating function f(h21+ h1h2)f(h22) - f(h22+ h1h2)f(h21); we take instead the homogeneous ideal I0 of U generated by the "Adem relations"* * (for 0 < i < 2j) [i=2]Xj - 1 - k! XiXj = Xi+j-kXk; k=0 i - 2k and form the graded Z-algebra U=I0. (Here, we interpret the binomial!coefficie* *nts as the n integers that they in fact are, and use the convention that is zero if m* * > n.) Then m I0 is not a Hopf ideal in U, U=I0 is not a Hopf algebra with respect to the ind* *uced coalgebra map, and, in fact, U=I0 cannot be isomorphic to U=J0 as an algebra. To see this* * note that the Adem relations above imply that X21= 0 in U=I0. Thus, again by considering the possible relations among the generators* * of U=I0 in degree 1 and 2 implied by the Adem relations above, we see that in degree 1, U=* *I0 is a free abelian group generated by X1 and in degree 2, U=I0 is a free abelian group gen* *erated by X2. Thus, the elements of U=I0 U=I0 of degree 2 form a free abelian group gene* *rated by X1 X1, 1 X2 and X2 1. If the coalgebra map of U descended to give a coalgeb* *ra map for U=I0, then on the one hand, (X21) = 0; while on the other, (X21) = (1 X1 + X1 1)2 = 2(X1 X1) in U=I0. But, as noted above, X1 X1 cannot be a torsion element of U=I0 U=I0.* * This also shows that U=I0 cannot be isomorphic to U=J0 as a graded algebra, because * *X1 is a nonzero element of degree one in U=I0 such that X21= 0 in U=I0, yet no nonzero * *element of degree one in U=J0 has square equal to zero in U=J0. 12 One might say (if sense could be made of it) that A2(Z ) represents a group * *scheme over the integers whose reduction mod 2 is represented by the mod-2 Steenrod al* *gebra. (Unfortunately, A2(Z ) is neither finitely generated as an algebra over the int* *egers, nor is it commutative, nor is it torsion free; thus this description is not to be taken l* *iterally.) 3 Deformations by twisting According to Drinfel'd [3] (see also [6]), a (topological) quasi-bialgebra Q ov* *er a commutative (topological) ring R is an associative (topological) R-algebra with 1, with hom* *omorphisms of algebras : Q ! Q ^Q (the comultiplication) and ffl : Q ! R (the counit), su* *ch that: a) There exists an invertible element 2 Q ^Q ^Q (the Drinfel'd associator fo* *r Q) with (id )((a)) = (( id)((a)))-1 and (id id )()( id id)() = 234(id id)()123; for all a 2 Q. b) There exist invertible elements l; r 2 Q such that for all a 2 Q, (ffl id)((a)) = l-1al; (id ffl)((a)) = r-1ar; and (id ffl id)() = r l-1: The elements l; r in the above definition are called the unit constraints fo* *r Q. From now on, we delete the word "topological". Drinfel'd [3], [6] calls a quasi-bialgebra Q (notation as above) a quasi-Hop* *f algebra if there exist an invertible anti-automorphism S of Q and elements ff and fi ofPQ (which* * we shall call the "antipode constraints" for Q) such that if a 2 Q and (a) = a0 a00, t* *hen P 0 00 P 0 00 a) S(a )ffa = ffl(a)ff; a fiS(a ) = ffl(a)fi and P -1 P -1 -1 -1 -1 b) P i1ifiS(2i)ff3i = rP l; iS(( )1i)ff( )2ifiS(( )3i) = r l; where = i1i 2i 3iand -1 = i(-1)1i (-1)2i (-1)3i: The map S will be called the antipode for Q. Actually, Drinfel'd makes the d* *efinition of a quasi-Hopf algebra explicitly only in the case r = l = 1, but in fact the def* *inition above is implicitly contained in his work. Drinfel'd [3] comments that "In the definition of quasi-Hopf algebra, before* * the words "there exist" should probably be inserted "locally with respect to Spec(k)". [W* *e wrote R instead of k in the above.] We are not doing this, because we are primarily con* *cerned with 13 the case that k is a field or the ring of formal series over a field." Since we* * are not necessarily interested in working over fields, perhaps Drinfel'd's suggested insertion shou* *ld be made. We broaden our definition of "deformation" of a Hopf algebra A to also allow* * deformations A"hthat are perhaps only quasi-Hopf algebras. We will refer to such a deformat* *ion as a "quasi-Hopf algebra deformation". Here, we discuss various examples of quasi-Hopf algebra deformations obtaine* *d by "twist- ing" (or "skrooching" [8]). Again, we work over the integers, and restrict our attention to U as above. * *ConsiderPthe trivial one-parameterPdeformation U[[h]] of U. Now, fix any two curves E(t) = * * 1i=0Eiti and F (t) = 1i=0Fiti over 1 in the power series ring U[[t]] such that ffl(Ei)* * = ffl(Fi) = 0 for all i > 0. (If one is interested in working in a graded context, as we will be * *later in [4], one may make various homogeneity conditions on the Ei's and Fi's.) P Let FE;F = E(h) F (h) in the ring U[[h]] ^Z[[h]]U[[h]]. Let SE(t) = 1i=0S* *(Ei)ti and P 1 i SF (t) = i=0S(Fi)t ; note that E(h)SE(h) = SE(h)E(h) = F (h)SF (h) = SF (h)F * *(h) = 1; so that F-1E;F= SE(h) SF (h): Define a new homomorphism of Z-algebras E;F : U[[h]] ! U[[h]] ^Z[[h]]U[[h]] by the formula E;F(Xi) = (E(h) F (h))(Xi)(SE(h) SF (h)) = FE;F(Xi)F-1E;F: Theorem 3.0.20 E;F defines a quasi-Hopf algebra deformation of U, which we wi* *ll call UE;F. The multiplication, antipode, unit and counit of UE;F remain the same as * *in the trivial deformation (thus are obtained by extension from U). The Drinfel'd associator for UE;F is = SE(h) E(h)F (h)SE(h)SF (h) F (h).* * The elements l; r required in the definition are l = SF (h) and r = SE(h). The elem* *ents ff and fi are ff = fi = r-1l = E(h)SF (h). Proof: In fact, the proof of this theorem is a straightforward enough calcul* *ation, by now essentially standard in the quantum group setting, and usually left as an exerc* *ise to the reader (in, for example, [2], [6], [3], [8]). However we write down some detail* *s here because it interests us to do so, and also because we are not in the case r = l = 1. Also,* * the calculations do not require U to be a free algebra, but only that U is a Hopf algebra with r* *espect to , ffl, etcetera. So, in fact, we may replace U by any Hopf algebra, and E, F by a* *ny curves in that Hopf algebra. Since U is a free algebra on the Xn's, we need only check the conditions on * *each Xn. Now, we first point out that E;F() = (E(h) F (h))()(SE(h) SF (h)) for every element in U; thus X1 1X E;F( nhn) = ((E(h) F (h)) ((n)hn)(SE(h) SF (h)); n=0 n=0 14 P for every element 1n=0nhn of U[[h]]. Using this, we see that E;F(E(h)) = (E(h) F (h))(E(h) E(h))(SE(h) SF (h)) = E(h) F (h)E(h)SF (h); E;F(SE(h)) = SE(h) F (h)SE(h)SF (h); E;F(F (h)) = E(h)F (h)SE(h) F (h); E;F(SF (h)) = E(h)SF (h)SE(h) SF (h): Thus, Xn E;F(Xn) = E(h)XiSE(h) F (h)Xn-iSF (h); i=0 and so (1 E;F)(E;F(Xn)) Xn Xi = E(h)XkSE(h) E(h)F (h)Xl-kSF (h)SE(h) F (h)2Xn-lSF (h)2: l=0k=0 On the other hand, a similar computation yields (E;F 1)(E;F(Xn)) Xn Xi = E(h)2XkSE(h)2 F (h)E(h)Xl-kSE(h)SF (h) F (h)Xn-lSF (h): l=0k=0 Conjugating by above, we get ()(E;F 1)(E;F(Xn))()-1 = (1 E;F)(E;F(Xn)): Also, (from now on, we write E(h) = E; F (h) = F; SE(h) = E-1 and SF (h) = F -1) (E-1 EF E-1F -1 EF E-1 F )(E-1 F E-1F -1 EF E-1F -1 F ) = E-2 EF E-2F -1 EF 2E-1F -1 F 2 = (1E-1EF E-1F -1F )(E-1E2F E-1F -1E-1F EF E-1F -2F )(E-1EF E-1F -1F 1): Checking the condition for l and r, we have (for all n): Xn (ffl id)E;F(Xn) = ffl(EXiE-1) F Xn-iF -1= F XnF -1= l-1Xnl; i=0 similarly (id ffl)E;F(Xn) = EXnE-1 = r-1Xnr: Also, (id ffl id)(E-1 EF E-1F -1 F ) = E-1 F = r l-1: 15 For the assertion about the antipode: for all n > 0, Xn Xn EXiE-1EF -1S(F Xn-iF -1) = EXiE-1EF -1F S(Xn-i)F -1 i=0 i=0 Xn = EXiS(Xn-i)F -1= ffl(Xn)EF -1; i=0 and similarly Xn ES(Xi)E-1EF -1F Xn-iF -1) = ffl(Xn)EF -1; i=0 Finally, E-1EF -1F EF -1E-1EF -1F = EF -1= ff so that indeed S furnishes the* * an- tipode for E;F. We also point out that UE;F is "almost cocommutative" with "universal R-matr* *ix" equal to F E-1 EF -1. In fact, one can quickly check that R = F E-1 EF -1satisfies th* *e "Yang- Baxter quantum equation" [3] R12R13R23= R23R13R12 where by R12(for example) we mean F E-1 EF -1 1. Returning to consider U, we see that deformations obtained by "twisting" in * *this way havePa nice feature: If I is some two-sided Hopf ideal in U, then the two-sided* * ideal I[[h]] = { 1i=0ihi| i2 I 8 i} is a quasi-Hopf ideal in UE;F. By this we mean that a) E;F(I[[h]]) I[[h]] ^Z[[h]]UE;F + UE;F^ Z[[h]]I[[h]]; and b) S(I[[h]]) I[[h]]: This should be clear by inspecting the formulas in the above proof. Thus we * *have that UE;F=I[[h]] is a quasi-Hopf algebra; whose Drinfel'd associator, unit constrain* *ts and antipode constraints are those obtained from UE;F. The "flatness" property of deformati* *ons allows one to conclude that Theorem 3.0.21 UE;F=I[[h]] is a quasi-Hopf algebra deformation of U=I. In conclusion, we make some remarks about the graded case. Here, we add the * *require- ment that E and F satisfy the condition degree(Ei) = degree(Fi) = degree(Xi) = * *Ki for every i. Then one can calculate that the coefficients of hn in E;F(Xm ) ha* *ve degree Km + Kn; thus it is natural to assign the parameter h the degree -K. We would l* *ike to think of U[[h]] as the completion of the polynomial ring U[h] with respect to t* *he degree (in h) filtration. The ideal I of U above is replaced by a graded ideal I0 in U. In* * UE;F=I0[[h]], if is a homogeneous element of U=I0 of degree r, the coefficient of hn in E;F() h* *as degree Kn + r. It is true that this paper presents a lot of examples in a rather disconnect* *ed way, without really giving a clearly stated and well-informed reason as to why any of the ex* *amples should be interesting. This is unfortunately due to the limitations of the author at t* *his time: the 16 examples were discovered and general procedures for producing them and some of * *their global deformations were noticed, but, what's the point? The point of view of t* *he author is that at this time the discovery of these examples is interesting enough in i* *tself to merit publication. At the least, one can regard this paper as giving an extremely sho* *rt and purely algebraic proof that the mod-p Steenrod algebra is a Hopf algebra. References [1]Bullett, S.R., and Macdonald, I (1982) On the Adem Relations, Topology 21, * *329-332. [2]Chari, V., and Pressley, A. (1994) A Guide to Quantum Groups, Cambridge Uni* *versity Press, Cambridge. [3]Drinfel'd, V.G. (1990) Quasi-Hopf Algebras, Leningrad Math. Journal, 1419-5* *7. [4]Duflot, J. Lots of Hopf Algebras, II, in preparation. [5]Hazewinkel, M. (1978) Formal Groups and Applications, Academic Press, New Y* *ork. [6]Kassel, Christian (1995) Quantum Groups, Graduate Texts in Mathematics, Spr* *inger- Verlag, New York. [7]Mohammad, Hassan, Hopf Ideals in a Universal Hopf Algebra with Applications* * to Algebraic Topology, in preparation. [8]Stasheff, J. (1992) Drinfel'd's quasi-Hopf algebras and beyond, in Deformat* *ion Theory and Quantum Groups with Applications to Mathematical Physics, M. Gerstenhab* *er and J. Stasheff (editors), Contemporary Mathematics 134, pp. 297-307, American * *Mathe- matical Society, Providence, Rhode Island. [9]Steenrod, N., and Epstein, D. (1962) Cohomology Operations, Annals of Mathe* *matics Studies 50, Princeton University Press, Princeton, New Jersey. 17