LIMITS OF ALGEBRAS WITH SHIFTING AND A RELATIONSHIP BETWEEN THE MOD TWO STEENROD AND DYER-LASHOF ALGEBRAS DAVID J. PENGELLEY AND FRANK WILLIAMS Abstract. We provide a construction, refined from an inverse limit, that produces the mod 2 Steenrod and Dyer-Lashof algebras from each other. In fact, the construction relates various subal- gebras and quotients of the universal Steenrod algebra of opera- tions for H1 -ring spectra. We also describe how the construction transforms the axiomatic properties of homogeneous pre-Koszul algebras and Poincare-Birkhoff-Witt algebras. December 9, 1999 1. Introduction An intriguing mathematical problem is to discover connections be- tween the mod 2 Steenrod algebra A and the Dyer-Lashof algebra R. In [PW ] we treated certain aspects of this question via K, the Kudo- Araki-May algebra of lower operations. We found a vector space bi- jection from R to K that related their algebra structures in a sheared fashion. We also discovered the relation of K to A; which required an inverse limit construction to achieve a sheared algebra bijection. How- ever, we did not find a construction that related A and R directly with each other. In this paper we shall achieve this goal by describing a sim- ple construction that produces an algebra from a certain inverse limit. This algebra A(1) is built from a homogeneous pre-Koszul algebra A that has a basis of admissibles and is also endowed with a special type of vector space endomorphism F which we call a shift map. Theorem 3.3 guarantees that A(1) is also a homogeneous pre-Koszul algebra. In Section 4 we provide several examples. In particular, the construction produces the (Lie) Steenrod algebra AL when it is applied to the al- gebraic or geometric Dyer-Lashof algebras, and produces the algebraic ____________ 1991 Mathematics Subject Classification. Primary 55S99; Secondary 55S10, 55S12, 16W30, 16W50, 57T05. Key words and phrases. Homogeneous pre-Koszul algebra, Poincare-Birkhoff- Witt algebra, Steenrod algebra, Dyer-Lashof algebra, Kudo-Araki-May algebra, algebra with shifting. 1 2 DAVID J. PENGELLEY AND FRANK WILLIAMS Dyer-Lashof algebra R(-1) when it is applied to AL. We also de- scribe how our construction applies to the subalgebras and quotient algebras of the universal Steenrod algebra of operations for H1 -ring spectra studied in [Lo ]. In Section 2 we shall recall and make relevant definitions, and in Section 3 we shall describe the construction. Finally, in Section 5 we prove that if A is additionally a Poincare-Birkhoff-Witt algebra, then so is A(1). 2. Preparation Our general setting is that of homogeneous pre-Koszul algebras, about which we briefly remind the reader and refer to [Pr , Lo , PW ] for more information. The Lie Steenrod algebra, algebraic Dyer-Lashof algebra, and the Kudo-Araki-May algebra are examples [Lo , Mill, Pr, BJ , PW ]. We recall some definitions from [Pr ]. Begin with an augmented alge- bra A over F2 with generating set {bi}i2I, I a lower-bounded segment of Z, say I = {i 2 Z | i i0; for some fixedi0}: (We define bi = 0 for i < i0:) Suppose further that A has only homogeneous 2-fold rela- tions in these generators. Then A is called a homogeneous pre-Koszul algebra, and has a multiplicative length grading. We also shall require our homogeneous pre-Koszul algebras A to have a second, topological, grading. We shall denote the topological degree of a homogeneous element x by |x|: We require that |bi| = i: We write A = A*;*, in which the first subscript denotes the length grading and the second the topological grading, e.g., bi 2 A1;i. We do not require that topological degree be additive over products, but we do require that there exist an integer c 1 such that if x 2 An1;t1and y 2 An2;t2, then |xy| = t1 + cn1t2 (cf. [PW ]). Equivalently, |bi1. .b.in| = i1 + ci2 + . .+.cn-1in: (Note that A is thus of finite type with respect to bidegree.) Let I = (i1; : :;:in) denote a multi-index and call it the label of bI = bi1. .b.in, the corresponding monomial in generators. Suppose that there is a set S of multi-indices such that B = {bI}I2S is a vector space basis for A. The monomials in B are called admissibles and the expression of any element in terms of them is called its admissible expression. In particular, the two-fold admissible expressions X i j bibj = f p q bpbq LIMITS OF ALGEBRAS WITH SHIFTING 3 i j (where the coefficients f p q 2 F2), are called the admissible rela- tions. We next describe what is meant by a shift map F on a homogeneous pre-Koszul algebra A. Suppose a is a positive integer. Then a linear transformation F : A ! A is a shift map provided that F (1) = 1, F carries generators to generators or to zero, and if z1 and z2 are any monomials with z2 2 An; then n F (z1z2) = F a (z1)F (z2): Thus F preserves monomials and the length grading. Note that if a = 1; then F is simply an algebra map. In [PW ] we gave details on how shift maps can be constructed. 3. Main Construction Our construction begins with a homogeneous pre-Koszul algebra A with a basis of admissibles for length grading two or less, as described above. Further, let F : A ! A be a shift map on A given on the generators by F (bi) = bi-1 (F (bi0) = 0). We require that F carry admissibles to admissibles or to zero. Note that the constants a of the shift map and c of the topological grading interact with the length grading via F : An;t! An;t-(an-1+an-2c+...+acn-2+cn-1): We define A = blim(. . .F!A F! A); - the bigraded inverse limit. Since F preserves the length grading, A inherits a length grading l from A. We shall denote elements of A by x = {x(k)}k1 . Let x 2 An, and suppose |x(k)| = tk. Then (an-1 + an-2c + . .+.acn-2 + cn-1)k - tk is clearly independent of k, and we define it to be the topological degree of x, thus producing a second grading on A . We now define a multiplication on A by {x(k)}{y(k)} = {x(kal(y))y(k)}; where y is homogeneous of length l(y): It is straight-forward to check that this multiplication is well-defined, has identity {: :;:1; : :;:1}, and satisfies the appropriate conditions to make A an algebra. Note that A is an algebra with length and topological gradings behaving as in A, with the same constant c for the topological degree. 4 DAVID J. PENGELLEY AND FRANK WILLIAMS We next define b1-i 2 A to be determined by b1-i (k) = bk-i. (Note that |b1-i | = i.) For our purposes in this paper, A is too large, so we will use only a lower bounded subset of its generators to create A(1). Define A(1)1= {b1-i | i 1 - i0} to be the algebra generators of A(1). We now turn to defining the 2-fold relations in A(1) between these generators. We must adjust for the fact that the obvious relations in A involving these elements also involve other elements of A that we do not wish to allow in A(1). For b1-i b1-j we begin by noting that if X ka - i k - j bka-ibk-j = f ka - p k - q bka-pbk-q is an admissible relation in A, then F (bka-ibk-j) = b(k-1)a-ib(k-1)-j X (k - 1)a - i (k - 1) - j = f (k - 1)a - p (k - 1) - q b(k-1)a-pb(k-1)-q must be an admissible relation in A, since F preserves admissibility by hypothesis. (Here we note that if (k - 1)a - p or (k - 1) - q (k - 1)a - i (k - 1) - j is less than i0; then f (k - 1)a - p (k - 1) - q is undefined, since b(k-1)a-pb(k-1)-q= 0.) Hence ka - i k - j (k - 1)a - i (k - 1) - j f ka - p k - q = f (k - 1)a - p (k - 1) - q ; whenever the right hand side is defined. Thus, if we define 1 - i 1 - j ka - i k - j f 1 - p 1 - q = f ka - p k - q for k large, we have X 1 - i 1 - j b1-i b1-j = f 1 - p 1 - q b1-p b1-q in A : We note that this formal sum might be infinite and that b1-p or b1-q might not be in A(1)1. LIMITS OF ALGEBRAS WITH SHIFTING 5 Definition 3.1. Let A(1) be the free algebra generated by A(1)1subject to the relations X 1 - i 1 - j b1-i b1-j = f 1 - p 1 - q b1-p b1-q : Remark 3.2. Of course, the summation in this definition is taken only over products of elements of A(1)1: Since the topological degree of (b1-i b1-j )(k) (i.e., the topological degree of bka-ibk-j) is k(a + c) - (i + cj), then in all the summations above we have p + cq = i + cj for all p and q. For b1-p and b1-q in A(1)1; we have p; q 1 - i0: Hence the defining relations for A(1) are indeed finite sums. Moreover, it is clear from the above that A(1) has a topological grading similar to that of A , and that |b1-i1 . .b.1-in| = i1 + ci2 + . .+.cn-1in: We obtain Theorem 3.3. If A is a homogeneous pre-Koszul algebra with a basis of admissibles and a shift map, satisfying the requirements for con- structing A(1), then A(1) is a homogeneous pre-Koszul algebra. We consider now the question of an admissible basis for A(1). Definition 3.4. A monomial m in A(1) is called admissible provided that m(k) is admissible in A for some k. Lemma 3.5. A monomial m in A(1) is admissible if and only if m(k) is admissible in A for all sufficiently large k. Proof. The formulas for F ensure that it induces an epimorphism from the set of monomial strings to the set of monomial strings with zero adjoined, and that its only lack of injectivity consists of monomials sent to zero. So we need only show that F carries any inadmissible monomial to zero or another inadmissible. Suppose, on the contrary, F carried an inadmissible b to an admissible. Then applying F to the admissible expression for b would result in a sum of terms, each admissible or zero, and each different from the admissible F (b) their sum must equal, contradicting the fact that the admissibles are a basis for A. | Remark 3.6. The defining relations for A(1) provide an admissible expression for every monomial of length two, and clearly the two-fold admissibles form a basis for A(1) in length two. However, at this stage we do not have enough information to determine the existence of a basis of admissibles for all of A(1): We shall consider this question in Section 5. 6 DAVID J. PENGELLEY AND FRANK WILLIAMS 4. Examples and Applications We now give several well-known examples of homogeneous pre-Koszul algebras A with bases of admissibles and shift maps, and identify A(1) in each case as another well-known algebra. Throughout this paper, binomial coefficients will have the classical interpretation that they equal zero if either their numerator or denom- inator is negative. Our first example comes from [PW]. The algebra K is the F2-algebra generated by elements Di 2 K1;*(i = 0; 1; : :):, subject to the (Adem) relations X k - 1 - j DiDj = Di+2j-2kDk; (i > j): k 2k - i - j The bidegree of elements in K is defined inductively by the requirement that the multiplication be a map Km;i Kn;j! Km+n;i+2m j. (So the integer c is 2 in this case.) The admissible basis consists of monomials with nondecreasing indices and the shift map ff: K ! K is given on the generators by ff(Di) = Di-1 (D-1 = 0). In this case the integer a = 1, i.e. ffis actually an algebra endomorphism of K: We also recall the following algebra, constructed in [PW ], which may be thought of as a stabilized version of K [BPW ]. The algebra K(1) is described by generators D1-i 2 K(1) 1;i, for i > 0, with relations for i < j X j - p - 1 D1-i D1-j = D1-(i+2j-2p)D1-p : i + j - 2p We can now state our first result, which was proved in [PW ]. Theorem 4.1. If we take A = K and F = ff, then we obtain A(1) ' K(1) : For our next results we recall more definitions. We begin with the Dyer-Lashof algebra [CLM , Mill]. The algebraic Dyer-Lashof algebra R(-1) is generated by Qi 2 R(-1)1;ifor i 0, subject to Adem relations X t - j - 1 QiQj = Qi+j-tQt fori > 2j: i_2ti-j-1 2t - i We call a nonnegative sequence I = (i1; i2; : :;:in); and the corre- sponding monomial QI = Qi1. .Q.in2 R(1)n;i1+...+in(so c = 1), admissible provided it 2it+1 for each t, and recall that the admissible monomials form a basis for R(-1). LIMITS OF ALGEBRAS WITH SHIFTING 7 The excess of an admissible I and of QI is defined to be i1 - (i2 + . .+.in), and we recall that an admissible vanishes on all k-dimensional homology classes precisely if it has excess less than k [CLM , p. 17]. Following [CLM , Mill], we consider the ideal B(0) in R(-1) spanned by all admissibles of excess less than 0, and form the bigraded algebra R = R(-1)=B(0), called the geometric Dyer-Lashof algebra, with basis the admissibles of excess at least 0 [CLM , p. 17]. While obviously R(-1) is a homogeneous pre-Koszul algebra, R is not. It was proved in [PW , Theorem 5.11] that the map ffi: R(-1)n;i! R(-1)n;i-(2n-1) given by n-1 i -2 in-1 ffi(Qi1. .Q.in-1Qin) = Qi1-2 . .Q.n-1 Q on admissibles is a shift map. (By convention, we set Qj = 0 for negative j.) In this case a = 2: Note that ffireduces excess by one and hence passes to the quotient, producing a shift map on R as well. More generally, we extend Lomonaco [Lo ] and define, for each m 2 Z, a homogeneous pre-Koszul algebra L0m, with generators Qi 2 (L0m)1;i, i -m (Qi is called x-i in [Lo ]), subject to Adem relations given by the same formula as for R(-1), where t; i + j - t -m: Note that these Adem relations do convert inadmissibles to admissibles for any m. The admissibles as defined above form a basis, since the Adem relations always decrease the lexicographic ordering on monomials, and L0m is of finite type in each bidegree. As examples, L00= R(-1) and L01= op , where is the algebra described in [BG ]. Shift maps on the L0m are given by the above formulas for ffi[PW , Theorem 5.11], in which Qj is taken to be 0 for j < -m: Our next examples are versions of the Steenrod algebra. In the spirit of [Lo ], we define K0m, m 2 Z, to be the homogeneous pre-Koszul algebra given by generators Sqi 2 (K0m)1;i, i m (Sqj is called xj in [Lo ]), subject to the Adem relations X j - 1 - s SqiSqj = Sqi+j-sSqs for i < 2j ; s i - 2s where s; i + j - s m: (Again in this case, c = 1.) The reader may check that we have the admissible basis consisting of those Sqi1. .S.qin for which it 2it+1. Here we may again apply Theorem 5.11 of [PW ] to construct a shift map ffl: (K0m)n;i! (K0m)n;i-(2n-1)given by n-1 i -2 in-1 ffl(Sqi1. .S.qin-1Sqin) = Sqi1-2 . .S.q n-1 Sq ; where Sqj = 0 for j < m. Again, a = 2. Note that K00is the "algebraic" Steenrod algebra A of [Br ], and K01is the Lie Steenrod algebra AL (the 8 DAVID J. PENGELLEY AND FRANK WILLIAMS associated graded algebra to the usual (topological) Steenrod algebra with its length filtration.) With the definitions in hand, we move to our next results. Theorem 4.2. If we take A = L0m and F = ffi; then we obtain A(1) ' K0m+1: Proof. In this example, the lower bound i0 = -m; so (L0m)(1)1has a basis consisting of the strings Q1-i given by Q1-i (k) = Qk-i; i 1 + m. We have (Q1-i Q1-j )(k) = Q1-i (2k)Q1-j (k) = Q2k-iQk-j: Recalling the meaning of admissible from Definition 3.4, we see that the two-fold admissibles in (L0m)(1) are the Q1-i Q1-j for which i 2j: If i < 2j, we have, for k sufficiently large, X t - (k - j) - 1 (Q1-i Q1-j )(k) = Q2k-iQk-j = Q3k-i-j-tQt; t 2t - (2k - i) where t; 3k - i - j - t -m: Setting s = k - t, this sum becomes X j - s - 1 X j - s - 1 Q2k-(i+j-s)Qk-s = (Q1-(i+j-s)Q1-s )(k); s i - 2s s i - 2s in which the translated restrictions on s recede as k increases. So, eliminating the disallowed terms as in our general construction above, i.e., keeping the terms where s; i + j - s 1 + m; we obtain X j - s - 1 Q1-i Q1-j = Q1-(i+j-s)Q1-s : s i - 2s Now we can create an algebra isomorphism between K0m+1and (L0m)(1) by matching Sqi to Q1-i for i m + 1: | Thus, as promised in the introduction, our construction produces the Lie Steenrod algebra when applied to the Dyer-Lashof algebra: Corollary 4.3. If we take A = R(-1) and F = ffi; then we obtain A(1) ' AL: Although R is not a homogeneous pre-Koszul algebra, we can still apply our construction to it and obtain Corollary 4.4. If we take A = R and F = ffi; then we again obtain A(1) ' AL: LIMITS OF ALGEBRAS WITH SHIFTING 9 Proof. There is the obvious quotient map from the inverse limit system for R(-1) to the one for R. Recalling that ffireduces excess, we obtain an isomorphism R(1) ' R(-1)(1) ' AL. | Theorem 4.5. If we take A = K0m and F = ffl; then we obtain A(1) ' L0m-1: Proof. The proof parallels that of Theorem 4.2. In this case, i0 = m; so (K0m)(1)1has a basis consisting of the strings Sq1-i given by Sq1-i (k) = Sqk-i; i 1 - m. We have (Sq1-i Sq1-j )(k) = Sq1-i (2k)Sq1-j (k) = Sq2k-iSqk-j: In this case the admissibles are the Sq1-i Sq1-j for which i 2j: So if i > 2j, we have X (k - j) - s - 1 (Sq1-i Sq1-j )(k) = Sq2k-iSqk-j = Sq3k-i-j-sSqs; s (2k - i) - 2s where s; 3k - i - j - s m. Setting t = k - s, this sum becomes X t - j - 1 X t - j - 1 Sq2k-(i+j-t)Sqk-t = (Sq1-(i+j-t)Sq1-t )(k); t 2t - i t 2t - i in which, again, the restrictions on t recede as k increases. Thus our relations for (K0m)(1) are X t - j - 1 Sq1-i Sq1-j = Sq1-(i+j-t)Sq1-t ; t 2t - i in which, as required, we keep only those terms where t; i+j-t 1-m. Hence if we let Sq1-i correspond to Qi for i 1 - m, we obtain the theorem as in the proof of 4.2. | Thus, as also promised in the introduction, our construction pro- duces the Dyer-Lashof algebra when applied to the Lie Steenrod alge- bra: Corollary 4.6. If we take A = AL and F = ffl; then we obtain A(1) ' R(-1): Remark 4.7. The principal results of [Lo ] are that taking cohomology transforms L0m into K0-m+1 and vice-versa, for m 1: Thus the con- fluence of our inverse_limitoconstructionpand taking cohomology occurs only between L0 = and K1 = AL. 10 DAVID J. PENGELLEY AND FRANK WILLIAMS 5. Poincare-Birkhoff-Witt algebras Finally, we return to the question of an admissible basis for A(1). Suppose that A is a homogeneous pre-Koszul algebra with a basis of ad- missibles, as defined in Section 2. We note that the set of multi-indices of fixed length can be ordered lexicographically (from either the right or the left) based on the order of I: Under these circumstances, A is called a Poincare-Birkhoff-Witt algebra (P-B-W algebra for short) [Pr ] provided that (1) if I1 and I2 2 S, then their concatenation I1I2 either is in S or else the label of each monomial appearing in the admissible expression of bI1I2is strictly greater than I1I2; and (2) if I 2 S; then each initial and terminal segment of I is also in S. Theorem 5.1. If A is a P-B-W algebra with a shift map F satisfying the requirements for constructing A(1), then A(1) is a P-B-W algebra, with a basis consisting of the admissible monomials. Proof. First we show that the admissibles span. We define an order on the multi-indices of monomials in A(1): Let 1 - I denote the multi- index (1-i1; : :;:1-in): We define 1-I to be greater than 1-J if and only if I is less than J in the ordering for A. It now follows that if one of the defining relations is applied to any monomial b1-I , the terms that appear have multi-indices that are strictly greater than 1 - I. Since A is finite-dimensional in each bidegree, the process of applying relations must terminate at some point, i.e., when all the summands are admissibles. Moreover, it follows from Lemma 3.5 that the admissibles are independent in A(1), and thus they constitute a basis. Finally, it is now straightforward to verify that the P-B-W conditions (1) and (2) follow for A(1) from those for A. | References [BJ] T.P. Bisson, A. Joyal, Q-rings and the homology of the symmetric groups, Contemporary Mathematics 202 (1997), 235-286. [BPW] T.P. Bisson, D.J. Pengelley, F. Williams, Stabilizing the lower operatio* *ns for mod 2 cohomology, in Homotopy Invariant Algebraic Structures: A Con- ference in Honor of J. Michael Boardman (ed. J-P Meyer et al), pp. 39-47, Contemporary Mathematics 239, American Mathematical Society, 1999. 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New Mexico State University, Las Cruces, NM 88003 E-mail address: davidp@nmsu.edu E-mail address: frank@nmsu.edu