Homotopy groups of a connective ring spectrum R form an Ngraded algebra π∗R which is commutative if R is commutative. We describe a secondary algebra π∗,∗R which enriches the structure of the algebra π∗R in a new unexpected way. The algebra π∗,∗R encodes secondary homotopy operations in π∗R, such as Toda brackets, and the first Postnikov invariant of R as a ring spectrum. Moreover, π∗,∗R repres...

The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second group is the set of formal diffeomorphisms with the group law being composition of series. The motivation to introduce these Hopf algebras comes ...

We introduce the notion of pseudo-commutative monad together with that of pseudoclosed 2-category, the leading example being given by the 2-monad on Cat whose 2-category of algebras is the 2-category of small symmetric monoidal categories. We prove that for any pseudo-commutative 2-monad on Cat, its 2-category of algebras is pseudo-closed. We also introduce supplementary definitions and results...

Let B be a strictly real commutative real Banach algebra with the carrier space Φ B. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism ρ : A → B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then ρ(A) is contained in the radical of B. 1. Introduction and results. Ring homomorphisms are mappings between two ...

Invariant structures on homogeneous manifolds are of fundamental importance in differential geometry. Recall that an affinor structure F (i.e., a tensor field F of type (1,1)) on a homogeneous manifold G/H is called invariant (with respect to G) if for any g ∈ G we have dτ(g)◦F = F ◦dτ(g), where τ(g)(xH)= (gx)H . An important place among homogeneous manifolds is occupied by homogeneousΦ-spaces ...

Let Λ be a commutative ring, A an augmented differential graded algebra over Λ (briefly, DGA-algebra) and X be a relatively free resolution of Λ over A. The standard bar resolution of Λ over A, denoted by B(A), provides an example of a resolution of this kind. The comparison theorem gives inductive formulae f :B(A) → X and g:X → B(A) termed comparison maps. In case that fg = 1X and A is connect...

An ideal I in the free associative algebra k〈X1, . . . ,Xn〉 over a field k is shown to have a finite Gröbner basis if the algebra defined by I is commutative; in characteristic 0 and generic coordinates the Gröbner basis may even be constructed by lifting a commutative Gröbner basis and adding commutators.

For any commutative algebra R the shuffle product on the tensor module T (R) can be deformed to a new product. It is called the quasi-shuffle algebra, or stuffle algebra, and denoted T q (R). We show that if R is the polynomial algebra, then T q (R) is free for some algebraic structure called Commutative TriDendriform (CTD-algebras). This result is part of a structure theorem for CTD-bialgebras...

Let $f : A rightarrow B$ be a ring homomorphism and $J$ an ideal of $B$. In this paper, we give a necessary and sufficient condition for the amalgamated algebra along an ideal $Abowtie^fJ$ to be $J$-Noetherian. Then we give a characterization for pseudo-irreducible ideals of $Abowtie^fJ$, in special cases.