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1. Introduction Journal of Prime Research in Mathematics Vol. 2(2006), 101-112 ON REALIZATIONS OF THE STEENROD ALGEBRAS ALEXEI LEBEDEV, DIMITRY LEITES* Abstract. The Steenrod algebra can not be realized as an enveloping of any Lie superalgebra. We list several problems that suggest a need to mod- ify the definition of the enveloping algebra, for example, to get rid of cer- tain strange deformations which we qualify as an artifact of the inadequate definition of the enveloping algebra in positive characteristic. P. Deligne appended our paper with his comments, hints and open problems. Keywords: Lie algebras, Lie superalgebras, Steenrod algebras Subjclass: 55N 1. Introduction Hereafter p is the characteristic of the ground field K. 1.1. Motivations. In the mid-1970s, Bukhshtaber and Shokurov [BS] inter- preted the Landweber-Novikov algebra as the universal enveloping algebra of the Lie algebra of the vector fields on the line with coordinate t, and with the d coefficients of dt vanishing at the origin together with their first derivative. Therefore, when (at about the same time) P. Deligne told one of us (DL) that Grothendieck told him what sounded (to DL) like a similar interpretation of the Steenrod algebra, it did not alert the listener, although one should be very careful when p > 0. From that time till recently DL remembered Deligne’s information in the following form “The Steenrod algebra A(2) is isomorphic to the enveloping algebra U(g) of a subsuperalgebra g of the Lie superalgebra of contact vec- (1) tor fields on the 1j1-dimensional superline, whose generating functions vanish at the origin together with their first derivative” MPIMiS, Inselstr. 22, DE-04103 Leipzig, Germany, E-mail: [email protected], [email protected] *We are thankful to the International Max Planck Research School and MPIMiS- Leipzig, for financial support and most creative environment and to P. Deligne for help and encouragement. 101 102 Alexei Lebedev, Dimitry Leites but the precise statement was never published and with time DL forgot what at that time he thought he understood from Deligne what Grothendieck meant under g. (For a precise statement in Deligne’s own words, quite distinct from (1), see x4.) Somewhat later, Bukhshtaber [Bu] published a paper whose title claims to interpret the Steenrod algebras A(p) for p > 2 in terms similar to (1), namely as (isomorphic to) “the enveloping algebra of the supergroup of p-adic diffeomorphisms (2) of the line”. The body of the paper [Bu] clarifies its cryptic title (it is deciphered as meant to be “the enveloping algebra of a subsuperalgebra of the Lie superalgebra of vector fields on the 1j1-dimensional superline in characteristic p > 2”), but nowhere actually states that the Steenrod algebra A(p) is identified with the U(g) for any g. Instead, A(p) is realized by differential operators but no description of the totality of these operators in more “tangible” terms, e.g., like the graphic (but wrong, as we will see) descriptions (1), (2) is offered; this is an open (but perhaps unreasonable) problem, cf. [Wp] with xx4. Our initial intention was to explicitly describe the subsuperalgebra g of the Lie superalgebra of vector fields on the 1j1-dimensional superline for which A(p) ' U(g) as we remembered (1); we also wanted to decipher (2). How- ever, having started, we have realized that we do not understand even what U(g) is if p > 0. More precisely, it is well-known ([S]) that there are two versions of the enveloping algebras (a “usual” one and a restricted one), but it seems to us that there are many more versions. An open problem is to give the appropriate definition of U(g) (this is definitely possible, at least, for classical Lie superalgebras with Cartan matrix) and related notions, such as representations and (co)homology of g. So we begin with a discussion of the notion of U(g), and next pass to realizations of the Steenrod algebras. We conclude that, under conventional definitions ([S]), there is no Lie superalgebra g such that A(p) ' U(g). This result is not appealing: we hoped to clarify known realizations, not make a negative statement that A(p) is NOT something. There are, however, realizations of A(p) by differential operators ([Bu, Wd]). These realizations, although accepted, still look somewhat mysterious to us. In his comments at the end of this note, Deligne suggests a positive characterization of A(p). 1.2. Notations. Let T (V ) be the tensor algebra of the superspace V , let S(V ) and Λ(V ) be the symmetric and exterior algebra of the space V , respec- tively. For a set x = (x1; : : : ; xn) of indeterminates that span V , we write T [x] or S[x] or Λ[x], respectively. Let Z+ be the set of nonnegative integers. On realizations of the Steenrod algebras 103 As an abstract algebra, the Steenrod algebra A(p) is defined, for any prime p, as follows (deg ¯ = 1): (¡ ¢ T [P i j deg P i = 2i(p ¡ 1) for i 2 Z ] ­ Λ[¯] =I(p) for p > 2 ¡ ¢ + A(p) = i i T [P j deg P = i for i 2 Z+] =I(2) for p = 2; (3) where the ideal of relations I(p) for p > 2 is generated by the Adem relations [a=p] ¡ ¢ a b P a+i (p¡1)(b¡i)¡1 a+b¡i i P P = (¡1) a¡pi P P for a < pb; i=0 [a=pP] ¡ ¢ a b a+i (p¡1)(b¡i) a+b¡i i (4) P ¯P = (¡1) a¡pi ¯P P ¡ i=0 [(a¡1)=p] ¡ ¢ P a+i (p¡1)(b¡i)¡1 a+b¡i i (¡1) a¡pi¡1 P ¯P for a · pb; i=0 whereas I(2) is generated by [a=P2] ¡ ¢ a b a+i b¡i¡1 a+b¡i i (5) P P = (¡1) a¡2i P P for a < 2b: i=0 Remark. For p = 2, the P i are usually denoted Sqi. 1.3. Lie superalgebras for p = 2. Observe that, for p 6= 2, for any Lie superalgebra g and any odd x, we have [x; x] = 2x2: (6) In other words, there is a squaring operation 1 x2 = [x; x] (7) 2 and to define the bracket of odd elements is the same as to define the squaring, since 2 2 2 [x; y] = (x + y) ¡ x ¡ y for any x; y 2 g1¯. (8) A Lie superalgebra for p = 2 is a superspace g such that g0¯ is a Lie algebra, g1¯ is an g0¯-module (made into the two-sided one by symmetry) and on g1¯ a squaring (roughly speaking, the halved bracket) is defined 2 2 2 2 x 7! x such that (ax) = a x for any x 2 g1¯ and a 2 K; and the map (x; y) 7! (x + y)2 ¡ x2 ¡ y2 is bilinear (9) 2 2 and g0¯-invariant, i.e., [x; y ] = (ady) (x) for any x 2 g0¯ and y 2 g1¯. Then the bracket (i.e., product in g) of odd elements is defined to be [x; y] := (x + y)2 ¡ x2 ¡ y2: (10) The Jacobi identity for three odd elements is replaced by the following relation: 2 [x; x ] = 0 for any x 2 g1¯: (11) 104 Alexei Lebedev, Dimitry Leites This completes the definition unless the ground field is Z=2 in which case we have to add the condition 2 2 [x; y ] = (ady) (x) for any x 2 g and y 2 g1¯ (12) which makes (11) and the last line in (9) redundant and replaces them over any field. The restricted Lie superalgebras are naturally defined. 1.4. Divided powers. For p > 0, there are many analogs of the polynomial algebra. These analogs break into the two types: the infinite dimensional ones and finite dimensional ones. The divided power algebra in indeterminates x1; : : : ; xm is the algebra of polynomials in these indeterminates, so, as space, it is (r1) (rm) O(m) = Spanfx1 : : : xm j r1; : : : ; rm ¸ 0g with the following multiplication: µ ¶ Ym r + s (x(r1) : : : x(rm)) ¢ (x(s1) : : : x(sm)) = i i x(ri+si): 1 m 1 m r i i=1 i For a shearing parameter N = (N1;:::;Nm), set (r1) (rm) Ni O(m; N) (or K[u; N]) := Spanfx1 : : : xm j 0 · ri < p ; i = 1; : : : ; mg; 1 where p = 1. If Ni < 1 for all i, then dim O(m; N) < 1. Observe that only the conventional polynomial algebra and the one with N = (1;:::; 1) are generated by the indeterminates that enter its definition. (pki ) For any other value of N, we have to add xi for every ki such that 1 < ki < Ni. If an indeterminate x is odd, then the corresponding shearing parameter is equal to 1 for p = 2; for p > 2, we postulate x2 = 0 and anticommutativity of odd elements: 2. The enveloping algebras of Lie algebras for p > 0 It looks strange that the following problem was never discussed in the liter- ature. For p > 0, it seems natural — in view of the Poincar´e-Birkhof-Witt theorem — to have as many types of universal enveloping algebras, as there are analogs of symmetric algebras or algebras of divided powers. Of the variety of such hypothetical definitions of enveloping algebras (the usual one and the ones labelled by various values of the shearing parameter N), only two are being considered: the usual U(g) and the one corresponding to N = (1;:::; 1). We hope that there exist U(g; N) — analogs of U(g), such that grU(g; N) ' O(dim g; N).
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