This paper is an introduction to the series of papers [26, 28, 29, 30], in which we develop a combinatorial theory of certain important operads and their actions. The operads we consider are A∞ operads, E∞ operads, the little n-cubes operad and the framed little disks operad. Sections 2, 6 and 9, which can be read independently, are an introduction to the theory of operads. The reader is also r...

We prove a recognition principle for motivic infinite P1-loop spaces over perfect field. This is achieved by developing theory of framed spaces, which analogue the E-infinity-spaces. A space equipped with transfers along finite syntomic morphisms trivialized cotangent complex in K-theory. Our main result that grouplike are equivalent to full subcategory spectra generated under colimits suspensi...

We prove finite generation of the cohomology ring of any finite dimensional pointed Hopf algebra, having abelian group of grouplike elements, under some mild restrictions on the group order. The proof uses the recent classification by Andruskiewitsch and Schneider of such Hopf algebras. Examples include all of Lusztig’s small quantum groups, whose cohomology was first computed explicitly by Gin...

We study the various term operations on the set of skew primitive elements of Hopf algebras, generated by skew primitive semi-invariants of an Abelian group of grouplike elements. All 1-linear binary operations are described and trilinear and quadrilinear operations are given a detailed treatment. Necessary and sufficient conditions for the existence of multilinear operations are specified in t...

We show that for a discrete semigroup S there exists a uniquely determined complete Boolean algebra B(S) the algebra of clopen subsets of M(S). M(S) is the phase space of the universal minimal dynamical system for S and it is an extremally disconnected compact Hausdorff space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these alg...

Let i : A → R be a ring morphism, and χ : R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1 R) = 1 A. If R is a Frobenius A-ring, then we can define a trace map tr : A → A R. If there exists an element of trace 1 in A, then A is right FBN if and only if A R is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring. We recover resul...

We perform a renormalization-grouplike numerical analysis of geographically embedded complex networks on a two-dimensional square lattice. At each step of the coarse-graining procedure, the four vertices on each 2x2 square box are merged to a single vertex, resulting in a coarse-grained system of smaller size. Repetition of the process leads to the observation that the coarse-graining procedure...

Let A be a commutative comodule algebra over a commutative bialgebra H . The group of invertible relative Hopf modules maps to the Picard group of A, and the kernel is described as a quotient group of the group of invertible grouplike elements of the coring A⊗H , or as a Harrison cohomology group. Our methods are based on elementary K-theory. The Hilbert 90 Theorem follows as a corollary. The p...