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18
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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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
Module Homalgebras
"... Abstract. We study a twisted version of module algebras called module Homalgebras. It is shown that module algebras deform into module Homalgebras via endomorphisms. As an example, we construct certain qdeformations of the usual sl(2)action on the affine plane. 1. ..."
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Abstract. We study a twisted version of module algebras called module Homalgebras. It is shown that module algebras deform into module Homalgebras via endomorphisms. As an example, we construct certain qdeformations of the usual sl(2)action on the affine plane. 1.
The Combinatorics Of Steenrod Operations On The Cohomology Of Grassmannians
 Adv. Math
, 1997
"... The study of the action of the Steenrod algebra on the mod p cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the ..."
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The study of the action of the Steenrod algebra on the mod p cohomology of spaces has many applications to the topological structure of those spaces. In this paper we present combinatorial formulas for the action of Steenrod operations on the cohomology of Grassmannians, both in the Borel and the Schubert picture. We consider integral lifts of Steenrod operations, which lie in a certain Hopf algebra of differential operators. The latter has been considered recently as a realization of the LandweberNovikov algebra in complex cobordism theory; it also has connections with the action of the Virasoro algebra on the boson Fock space. Our formulas for Steenrod operations are based on combinatorial methods which have not been used before in this area, namely Hammond operators and the combinatorics of Schur functions. We also discuss several applications of our formulas to the geometry of Grassmannians.
Cohomology and deformation of modulealgebras
 Ohio State University Newark, 1179 University Drive
"... Abstract. An algebraic deformation theory of modulealgebras over a bialgebra is constructed. The cases of modulecoalgebras, comodulealgebras, and comodulecoalgebras are also considered. 1. ..."
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Abstract. An algebraic deformation theory of modulealgebras over a bialgebra is constructed. The cases of modulecoalgebras, comodulealgebras, and comodulecoalgebras are also considered. 1.
Deformation of algebras over the LandweberNovikov algebra
 Department of Mathematics, The Ohio State University Newark, 1179 University Drive
"... Abstract. An algebraic deformation theory of algebras over the LandweberNovikov algebra is obtained. 1. ..."
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Abstract. An algebraic deformation theory of algebras over the LandweberNovikov algebra is obtained. 1.
EXPONENTIAL BAKERCAMPBELLHAUSDORFF FORMULA AND COMPRESSED KASHIWARAVERGNE CONJECTURE
, 2006
"... Abstract. The classical BakerCampbellHausdorff formula gives a recursive way to compute the Hausdorff series H = log(e X e Y) for noncommuting X, Y. Formally H lives in a completion ˆ L of the free Lie algebra L generated by X, Y. We prove that there are F, G ∈ [ ˆ L, ˆ L] such that H = e F Xe ..."
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Abstract. The classical BakerCampbellHausdorff formula gives a recursive way to compute the Hausdorff series H = log(e X e Y) for noncommuting X, Y. Formally H lives in a completion ˆ L of the free Lie algebra L generated by X, Y. We prove that there are F, G ∈ [ ˆ L, ˆ L] such that H = e F Xe −F + e G Y e −G. We describe explicitly all symmetric solutions to the KashiwaraVergne conjecture in Lie algebras L, where commutators of commutators vanish, i.e. [ [L, L], [L, L] ] = 0. 1.1. Elementary summary. 1.
Chern numbers of Chern submanifolds
, 2002
"... We present a solution of the generalized Hirzebruch problem on the relations between the Chern numbers of a stably almost complex manifold and the Chern numbers of its virtual Chern submanifolds. ..."
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We present a solution of the generalized Hirzebruch problem on the relations between the Chern numbers of a stably almost complex manifold and the Chern numbers of its virtual Chern submanifolds.