Abstract
Steenrod operations are cohomology operations that are themselves natural transformations between cohomology functors. There are two distinct types of steenrod operations initially constructed by Norman Steenrod and called Steenrod squares and reduced p-th power operations usually denoted <i>Sq</i> and <i>p<sup>i</sup></i> respectively. Since their creation, it has been proved that these operations can be constructed in the cohomology of many algebraic structures, for instance in the cohomology of simplicial restricted Lie algebras, the cohomology of cocommutative Hopf algebras and the homology of infinite loop space. Later on J. P. May developped a general algebraic setting in which all the above cases can be studied. In this work we consider a cyclic group π of oder a fixed prime p and combine theπ-strongly homotopy commutative Hopf algebra structure to the May’s approach with the aim to build these natural transformations on the Hochschild cohomology groups. Moreover we give under some conditions a link of these natural transformations with the Gerstenhaber algebra structure.
Highlights
One of the main computational tool of the homotopy theory is the Steenrod algebra defined by Henri Cartan in 1955 as an algebra of stable cohomology operations for a mod p cohomology and denoted by Ap, for a given prime number p[3]
These operations are compatibles with the algebra homomorphisms commuting with the structural map θ. These operations does not satisfy in general the Cartan formula and the Adam relation. As it is well knwon from the work of Gerstenhaber[7], there is a Lie algebra structure on the supension of the Hochschild cohomology, sHH∗(A, A), of differential graded algebra (A, d) with coefficients in itself
More precisely we define the algebraic Steenrod operations on the Hochschild cohomology of a differential graded πstrongly homotopy commutative Hopf algebras (A, dA) with coefficients in itself and provide a topological illustration on which the above operations can be constructed. We show that these algebraic Steenrod operations are annihilated in some sense by the Gerstenhaber bracket
Summary
One of the main computational tool of the homotopy theory is the Steenrod algebra defined by Henri Cartan in 1955 as an algebra of stable cohomology operations for a mod p cohomology and denoted by Ap, for a given prime number p[3]. = 0, (respectively ξ) if i > k(respectively i = k) ,p=2 i > 2k(respectively i = 2k) , p is an odd prime These operations are compatibles with the algebra homomorphisms commuting with the structural map θ. These operations does not satisfy in general the Cartan formula and the Adam relation As it is well knwon from the work of Gerstenhaber[7], there is a Lie algebra structure on the supension of the Hochschild cohomology, sHH∗(A, A), of differential graded algebra (A, d) with coefficients in itself. More precisely we define the algebraic Steenrod operations on the Hochschild cohomology of a differential graded πstrongly homotopy commutative Hopf algebras (A, dA) with coefficients in itself and provide a topological illustration on which the above operations can be constructed.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
