Commutators in the Steenrod algebra
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1 Commutators in the Steenrod algebra J. H. Palmieri and J. J. Zhang University of Washington Vancouver, 5 October 2008 J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
2 Let A be the mod 2 Steenrod algebra. The elements Sq 2i, i 0, form a minimal set of generators for A. Theorem (Wall) A minimal set of relations for A consists of two families: (Sq 2i ) 2 =... for i 0, [Sq 2i, Sq 2i+j ] =... for j 2. Want to find a standard form for elements of A: try sums of products Sq 2i1 Sq 2i2... Sq 2i n with i 1 < i 2 < < i n. A problem with this: no way to rewrite Sq 2i+1 Sq 2i : see the j 2 restriction in Wall s theorem. This is the only problem, though, and you can fix it. J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
3 Define elements s? as follows: s 2 i = Sq 2i, s 2 i,2 i+1 = [Sq2i, Sq 2i+1 ], s 2 i,...,2 i+j = [s 2 i,...,2 i+j 1, Sq2i+j ]. Example s 12 = [Sq 1, Sq 2 ] = Sq 3 + Sq 2 Sq 1 = Sq(0, 1) s 24 = [Sq 2, Sq 4 ] = Sq 4 Sq 2 + Sq 5 Sq 1 + Sq 6 = Sq(0, 2) s 48 = [Sq 4, Sq 8 ] = Sq 8 Sq 4 + Sq 10 Sq 2 + Sq 11 Sq 1 + Sq 12 = Sq(0, 4) + Sq(3, 3) s 124 = [s 12, Sq 4 ] = Sq 4 Sq 2 Sq 1 + Sq 5 Sq 2 + Sq 6 Sq 1 + Sq 7 = Sq(0, 0, 1) Call these elements iterated commutators. J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
4 Now we can achieve our standard form, using computations like this: Sq 2i+1 Sq 2i = Sq 2i Sq 2i+1 +[Sq 2i, Sq 2i+1 ]. That is, we can use iterated commutators to rewrite elements of A in a standard form, for example like this: Sq 2 Sq 1 Sq 8 = s 2 s 1 s 8 = s 1 s 2 s 8 + s 12 s 8. J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
5 Theorem Choose a linear ordering on the set {s 2 i,...,2 i+j commutators. Then the set of products : i 0, j 0} of iterated s 2 i 1,...,2 i 1 +j 1... s 2 in,...,2 i n+jn, where s 2 i 1,...,2 i 1 +j 1 < < s 2 in,...,2 i n+jn, forms a basis for A. These bases are called commutator bases. J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
6 Example For example, the sets of commutators whose products lie in dimension 7 are {s 1, s 2, s 4 }, {s 1, s 24 }, {s 12, s 4 }, and {s 124 }. To form a commutator basis in this dimension, we choose an ordering on the set {s 1, s 2, s 4, s 12, s 24, s 124 }, and then multiply distinct elements in increasing order. Here are two different commutator bases: (s 1 s 2 s 4, s 1 s 24, s 12 s 4, s 124 ), (s 1 s 4 s 2, s 1 s 24, s 4 s 12, s 124 ). All together, there are 24 different commutator bases in this dimension, one for each of the possible permutations of the factors from the above sets of factors. Some iterated commutators commute with each other, so e.g., in dimension 6 the factors {s 1, s 2, s 12 } combine to give only two different products, s 1 s 2 s 12 and s 2 s 1 s 12. J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
7 The commutator bases restrict to give bases for the sub-hopf algebras A(n). Example To obtain a basis for A(2), first order the iterated commutators in A(2), e.g.: {s 1, s 2, s 4, s 12, s 24, s 124 }. Then take products of iterated commutators, in increasing order; this will yield the 64 basis elements. E.g., in dimension 10: (s 1 s 2 s 124, s 12 s 124, s 4 s 24, s 1 s 12 s 24, s 1 s 2 s 4 s 12 ) They don t restrict well to arbitrary sub-hopf algebras of A: there is a sub-hopf algebra B with dim B = 4 but which contains only 3 commutator basis elements. J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
8 Sketch of proof. Filter A by powers of the augmentation ideal (the May filtration). The associated graded algebra gr A is a restricted enveloping algebra on the restricted Lie algebra generated by certain classes Pt s, s 0 and t 1, with trivial restriction. (Here Pt s is the image in gr A of an element Pt s in A.) The Poincaré-Birkhoff-Witt theorem says if you choose an ordering on the Pt s s and then form all monomials P s 1 t 1... P sn t n where P s 1 t 1 < < P sn t n, you get a basis for gr A. J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
9 To prove our theorem, then: show that s 2 i,...,2 i+j is congruent to Pi j+1 modulo the May filtration (a straightforward computation) Example s 2 i = Sq(2 i ) = P1 i s 1,2 = Sq(0, 1) = P2 0 s 1,2,...,2 j = Sq(0,..., 0, 1) = Pj+1 0 s 2,4 = Sq(0, 2) = P2 1 s 4,8 = Sq(0, 4) + Sq(3, 3) = P2 2 + Sq(3, 3) observe that any basis for gr A lifts to one for A. (One also obtains Monks Pt s -bases from this.) J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
10 Application: computations in A(2) The iterated commutators in A(2): {s 1, s 12, s 2, s 4, s 24, s 124 }. There are 21 relations among them: the restriction: s 2 1 = 0, s2 12 = 0, s2 24 = 0, s2 124 = 0, s2 2 = s 1s 12, and s 2 4 = s 2s 24 defining relations: s 12 = [s 1, s 2 ], s 24 = [s 2, s 4 ], and s 124 = [s 12, s 4 ] other nonzero commutators: [s 1, s 4 ] = s 12 s 2, [s 1, s 24 ] = s 124, [s 2, s 24 ] = s 1 s 124, [s 4, s 24 ] = s 1 s 2 s s 12 s 124. all other commutators are zero. J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
11 Application, continued Summary of commutators and relations: {s 1, s 12, s 2, s 4, s 24, s 124 } s 2 2 = s 1s 12, s 2 4 = s 2s 24, [s 1, s 4 ] = s 12 s 2, [s 1, s 24 ] = s 124, [s 2, s 24 ] = s 1 s 124, [s 4, s 24 ] = s 1 s 2 s s 12 s 124. Example (s 2 s 24 )(s 1 ) = s 2 s s 2 s 1 s 24 (using [s 1, s 24 ]) = s 2 s s 1 s 2 s 24 + s 12 s 24 (using [s 1, s 2 ]) (Things get more complicated with A(3): there are 55 relations instead of 21.) J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
12 The odd prime case. Fix an odd prime p and let A be the mod p Steenrod algebra. Define elements s p i,...,p i+j of A as above, replacing Sq2n with P pn. More precisely, s p i = P pi and s p i,...,p i+j = [Ppi+j, s p i,...,p i+j 1]. Theorem Choose a linear ordering on the set of iterated commutators. Then the set of products Q i1... Q im s e 1 p i 1,...,p i 1 +j 1... sen p i n,...,p i n+jn, where i 1 < < i m, s p i 1,...,p i 1 +j 1 < < s p in,...,p i n+jn, and 1 e k p 1 for each k, forms a basis for A. The proof is the same as when p = 2. J. H. Palmieri and J. J. Zhang (Washington) Commutators in the Steenrod algebra Vancouver, 5 October / 12
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