Results 1  10
of
12
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 ..."
Abstract

Cited by 30 (1 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
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
Flag Manifolds and the Landweber–Novikov Algebra
, 1998
"... We investigate geometrical interpretations of various structure maps associated with the Landweber–Novikov algebra S ∗ and its integral dual S∗. In particular, we study the coproduct and antipode in S∗, together with the left and right actions of S ∗ on S ∗ which underly the construction of the quan ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
We investigate geometrical interpretations of various structure maps associated with the Landweber–Novikov algebra S ∗ and its integral dual S∗. In particular, we study the coproduct and antipode in S∗, together with the left and right actions of S ∗ on S ∗ which underly the construction of the quantum (or Drinfeld) double D(S ∗). We set our realizations in the context of double complex cobordism, utilizing certain manifolds of bounded flags which generalize complex projective space and may be canonically expressed as toric varieties. We discuss their cell structure by analogy with the classical Schubert decomposition, and detail the implications for Poincaré duality with respect to double cobordism theory; these lead directly to our main results for the Landweber–Novikov algebra.
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. ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
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.
ALGEBRA OF FORMAL VECTOR FIELDS ON THE LINE AND BUCHSTABER’S CONJECTURE
, 2008
"... Let L1 denotes the Lie algebra of formal vector fields on the line which vanish at the origin together with their first derivatives. L1 is a nilpotent ”positive part ” of the Witt (Virasoro) algebra. Buchstaber and Shokurov have shown that the universal enveloping algebra U(L1) is isomorphic to th ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Let L1 denotes the Lie algebra of formal vector fields on the line which vanish at the origin together with their first derivatives. L1 is a nilpotent ”positive part ” of the Witt (Virasoro) algebra. Buchstaber and Shokurov have shown that the universal enveloping algebra U(L1) is isomorphic to the tensor product S ⊗ R, where S is the LandweberNovikov algebra in complex cobordism theory. Goncharova calculated the cohomology H ∗ (L1) = H ∗ (U(L1)), in particular it follows from her theorem that H ∗ (L1) has trivial multiplicative structure. Buchstaber conjectured that H ∗ (L1) is generated with respect to nontrivial Massey products by H 1 (L1). Feigin, Fuchs and Retakh found representation of H ∗ (L1) by trivial Massey products. Later Artelnykh found nontrivial Massey products for a part of H ∗ (L1). In the present article we prove that H ∗ (L1) is generated with respect to nontrivial Massey products by two elements from H 1 (L1).
MSp LOCALIZED AWAY FROM 2 AND ODD FORMAL GROUP LAWS
"... Abstract. We investigate the relationship between complex and symplectic cobordism localized away from the prime 2 and show that these theories are related much as a real Lie group is related to its complexification. This suggests that ideas from the theory of symmetric spaces might be used to illum ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We investigate the relationship between complex and symplectic cobordism localized away from the prime 2 and show that these theories are related much as a real Lie group is related to its complexification. This suggests that ideas from the theory of symmetric spaces might be used to illuminate these subjects. In particular, we give an explicit equivalence of ring spectra MSp[1/2] ∧ Sp/U+ ≃ MU[1/2] and deduce that MU[1/2] is a wedge of copies of MSp[1/2]. We discuss the implications for the structure of the stable operation algebra MSp[1/2] ∗ MSp[1/2] and the dual cooperation algebra MSp[1/2]∗MSp[1/2]. Finally we describe some realted Witt vector algebra and apply our results to the study of formal involutions on the category of formal group laws over a Z[1/2]algebra.
Double Cobordism, Flag Manifolds And Quantum Doubles
, 1996
"... Drinfeld's construction of quantum doubles is one of several recent advances in the theory of Hopf algebras (and their actions on rings) which may be attractively presented within the framework of complex cobordism; these developments were pioneered by S P Novikov and the first author. He ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Drinfeld's construction of quantum doubles is one of several recent advances in the theory of Hopf algebras (and their actions on rings) which may be attractively presented within the framework of complex cobordism; these developments were pioneered by S P Novikov and the first author. Here we extend their programme by discussing the geometric and homotopy theoretical interpretations of the quantum double of the LandweberNovikov algebra, as represented by a subalgebra of operations in double complex cobordism. We base our study on certain families of bounded flag manifolds with double complex structure, originally introduced into cobordism theory by the second author. We give background information on double complex cobordism, and discuss the cell structure of the flag manifolds by analogy with the classic Schubert decomposition, allowing us to describe their complex oriented cohomological properties (already implicit in the Schubert calculus of Bressler and Evens). This yields...
Operations And Quantum Doubles In Complex Oriented Cohomology Theory
, 1999
"... We survey recent developments introducing quantum algebraic methods into the study of cohomology operations in complex oriented cohomology theory. In particular, we discuss geometrical and homotopy theoretical aspects of the quantum double of the LandweberNovikov algebra, as represented by a subalg ..."
Abstract
 Add to MetaCart
We survey recent developments introducing quantum algebraic methods into the study of cohomology operations in complex oriented cohomology theory. In particular, we discuss geometrical and homotopy theoretical aspects of the quantum double of the LandweberNovikov algebra, as represented by a subalgebra of operations in double complex cobordism. We work in the context of Boardman's eightfold way, which offers an important framework for clarifying the relationship between quantum doubles and the standard machinery of Hopf algebroids of homology cooperations. These considerations give rise to novel structures in double cohomology theory, and we explore the twist operation and extensions of the quantum antipode by way of example.
Algebraic Aspects of the Theory of Product Structures in Complex Cobordism
, 2008
"... We address the general classification problem of all stable associative product structures in the complex cobordism theory. We show how to reduce this problem to the algebraic one in terms of the Hopf algebra S (the LandweberNovikov algebra) acting on its dual Hopf algebra S ∗ with a distinguished ..."
Abstract
 Add to MetaCart
(Show Context)
We address the general classification problem of all stable associative product structures in the complex cobordism theory. We show how to reduce this problem to the algebraic one in terms of the Hopf algebra S (the LandweberNovikov algebra) acting on its dual Hopf algebra S ∗ with a distinguished “topologically integral ” part Λ that coincids with the coefficient ring of the complex cobordism. We describe the formal group and its logarithm in terms of representations of S. We introduce onedimensional representations of a Hopf algebra. We give series of examples of such representations motivated by wellknown topological and algebraic results. We define and study the divided difference operators on an integral domain. We discuss certain important examples of such operators arising from analysis, representation theory, and noncommutative algebra. We give a special attention to the division operators by a noninvertible element of a ring. We give new general constructions of associative product structures (not necessarily commutative) using the divided difference operators. As application, we describe new classes of associative products in the complex cobordism theory.