We construct a spectral sequence that computes the generalized homology E∗( Q Xα) of a product of spectra. The E2-term of this spectral sequence consists of the right derived functors of product in the category of E∗E-comodules, and the spectral sequence always converges when E is the Johnson-Wilson theory E(n) and the Xα are Ln-local. We are able to prove some results about the E2-term of this...

Let F be a number field with ring of integers OF , and let S be a finite set of places of F . Assume that S contains the set S∞ of archimedean places of F , and write Sf for the set of finite places contained in S. Let OS (or O, when there is no danger of confusion) denote the ring of Sf -integers of F . Write F for an algebraic closure of F . Let Y be any scheme over Spec(O). Suppose thatG is ...

If A is a bialgebra over a field k, a left-right Yetter-Drinfel’d module over A is a k-linear space M which is a left A-module, a right A-comodule and such that a certain compatibility condition between these two structures holds. YetterDrinfel’d modules were introduced by D. Yetter in [18] under the name of “crossed bimodules” (they are called “quantum Yang-Baxter modules” in [5]; the present ...

Let G be an affine algebraic group over an algebraically closed field k and let H be a closed subgroup of G. If V is a rational H-module (a comodule for the coordinate ring of H) there is a now well-known notion of an induced module VI G for G, defined as the space Morph/~(G, V) of all H-equivariant morphisms from G to a finite dimensional subspace of V, with obvious G-action. The question aris...

Let M be a k-vector space and R ∈ End k(M ⊗M). In [10] we introduced and studied what we called the Hopf equation: R12R23 = R23R13R12. By means of a FRT type theorem, we have proven that the category HM H of H-Hopf modules is deeply involved in solving this equation. In the present paper, we continue to study the Hopf equation from another perspective: having in mind the quantum Yang-Baxter equ...

Building on Quillen's rational homotopy theory, we obtain algebraic models for the theory of parametrised spectra. For any simply-connected space $X$ there is a dg Lie algebra $\Lambda_X$ and (coassociative cocommutative) coalgebra $C_X$ that model type. In this article, prove type an $X$-parametrised spectrum completely encoded by $\Lambda_X$-representation also $C_X$-comodule. The corresponde...

The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ is considered. This was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works (nonstandard) positive part type $U_q^{IT,+}$ its second realization (current algebra) $U_q^{T,+}$ are studied. A for given in terms a K-operator satisfying Freidel-Maillet equation condition on de...

The final goal of this paper is to introduce certain finite dimensional Hopf algebras associated with restricted Frobenius Lie algebras over a field of characteristic p > 0. The antipodes of these Hopf algebras have order either 2p or 2, and in the minimal dimension p there exists just one Hopf algebra in this class which coincides with an example due to Radford [35] of a Hopf algebra with a no...

We replace trees by multi-indices as an index set of the abstract model space to tackle quasi-linear singular stochastic partial differential equations. show that this approach is consistent with postulates regularity structures when it comes structure group, which arises from a Hopf algebra and comodule. Our approach, where dual naturally embeds into formal power series algebra, allows interpr...

Contramodules are module-like algebraic structures endowed with infinite summation (or, occasionally, integration) operations satisfying natural axioms. Introduced originally by Eilenberg and Moore in 1965 the case of coalgebras over commutative rings, contramodules experience a small renaissance now after being all but forgotten for three decades between 1970-2000. Here we present review vario...