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Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
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Cited by 30 (1 self)
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
Differential Operators and the Steenrod Algebra
, 1995
"... This article presents an elementary treatment of the Steenrod algebra from an algebraic point of view in terms of differential operators acting on polynomials. The exposition concentrates on the Steenrod algebra A over the field F 2 of two elements although the approach works for the odd prime field ..."
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Cited by 9 (3 self)
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This article presents an elementary treatment of the Steenrod algebra from an algebraic point of view in terms of differential operators acting on polynomials. The exposition concentrates on the Steenrod algebra A over the field F 2 of two elements although the approach works for the odd prime fields. From a
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"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may nd of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of ..."
Abstract
 Add to MetaCart
This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may nd of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below.
A NOTE ON THE UNSTABILITY CONDITIONS OF THE STEENROD SQUARES ON THE POLYNOMIAL ALGEBRA
, 2007
"... Abstract. We extend some results involved the action of the Steenrod operations on monomials and get some corollaries on the hit problem. Then, by multiplying some special matrices, we obtain an efficient tool to compute the action of these operations. 1. Preliminaries In 1947, Steenrod [21] introdu ..."
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Abstract. We extend some results involved the action of the Steenrod operations on monomials and get some corollaries on the hit problem. Then, by multiplying some special matrices, we obtain an efficient tool to compute the action of these operations. 1. Preliminaries In 1947, Steenrod [21] introduced the Steenrod squares Sqk in terms of cocycles in simplicial cochain complex by modifying the AlexanderĈechWhitney formula for the cup product construction. Serre [16] showed that they generate all stable operations in cohomology over F2 under composition. For an overview on algebraic topology we cite [4]. Cartan [2] discovered a formula for working out a Steenrod square on a product of cohomology classes f, g. Theorem 1.1 (Cartan formula). Sqk(fg) = 0≤r≤k Sq r(f)Sqk−r(g). Adem [1] and Serre [16] established a faithful representation of A by its action on the cohomology of a test space consisting of infinite real projective spaces whose cohomology is the polynomial algebraP(n) = F2[x1, x2,..., xn] =⊕ d≥0P d(n), viewed as a graded module over the Steenrod algebra A at prime 2. The grading is by the homogeneous polynomials Pd(n) of degree d in the variables x1, x2,..., xn of grading 1. We cite to [5] and [11] in cohomology operations and to [11] and [22] in the Steenrod algebra. The Steenrod algebra A is defined to be the graded algebra over the field F2, generated by the Steenrod squares Sqk, in grading k ≥ 0, subject to the Adem relations [7, 24]. From a topological point of view, the Steenrod algebra is the algebra of stable cohomology operations for ordinary cohomology H ∗ over F2. For present purpose we only need to know that the Steenrod algebra acts by composition of linear operators on P(n) and the action of the Steenrod squares