In this paper, we analyze the structure possessed by the homotopy groups of a simplicial abelian Hopf algebra over the field F2. Specifically, we review the higher-order structure that the homotopy groups of a simplicial commutative algebra and simplicial cocommutative coalgebra possess. We then demonstrate how these structures interact under the added conditions present in a Hopf algebra. Intr...

After a brief review of some basic notions concerning 1–differentiable cohomology, named here d̃-cohomology, we introduce a Lichnerowicz d̃– cohomology in a classical way. Next, following the classical study of coeffective cohomology, a special attention is paid to the study of some problems concerning coeffective cohomology in the graded algebra of 1– differentiable forms. Also, the case of an a...

An extension of bornological algebras with a bounded linear section gives rise to a six term exact sequence of the associated periodic and entire cyclic cohomology groups. 1. Introduction A convex vector bornology on a vector space is a collection of subsets satisfying some conditions 8]. A typical example is the collection of bounded subsets of a locally convex vector space. A bornological alg...

A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or Hochschild) cohomology of G with coefficients in a k[G]-comodule M . This is naturally isomorphic to the cohomology of the dual cocommutative Hopf algebra k[G] with coefficients in the k[G]-module ...

We prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups. This result was previously known only for general linear groups. We give several applications to the study of classical algebraic groups, such as a cohomological stabilization property, the injectivity of external cup products, and the existence of Hopf algebra s...

We study the mod p cohomology of the classifying space of the projective unitary group PU(p). We first prove that conjectures due to J.F. Adams and Kono and Yagita (1993) about the structure of the mod p cohomology of the classifying space of connected compact Lie groups hold in the case of PU(p). Finally, we prove that the classifying space of the projective unitary group PU(p) is determined b...

In the paper we describe class of all solvable extensions an infinite-dimensional filiform Leibniz algebra. The algebra is taken as a maximal pro-nilpotent ideal residually It proven that second cohomology group extension trivial.

We suggest a new combinatorial construction for the cohomology ring of the ag manifold. The degree 2 commutation relations satissed by the divided diierence operators corresponding to positive roots deene a quadratic associative algebra. In this algebra, the formal analogues of Dunkl operators generate a commuta-tive subring, which is shown to be canonically isomorphic to the cohomology of the ...

It is well known that the validity of the so called LenardMagri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson coho...

The geometric action of the Steenrod algebra on many mod 2 cohomology spectral sequences is complemented by the action of a completely different algebra.