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 ...

A constructive procedure to obtain superintegrable deformations of the classical Smorodinsky–Winternitz Hamiltonian by using quantum deformations of its underlying Poisson sl(2) coalgebra symmetry is introduced. Through this example, the general connection between coalgebra symmetry and quasi-maximal superintegrability is analysed. The notion of comodule algebra symmetry is also shown to be app...

We have shown that the n–th Morava K –theory K∗(X) for a CW–spectrum X with action of Morava stabilizer group Gn can be recovered from the system of some height–(n + 1) cohomology groups E∗(Z) with Gn+1 –action indexed by finite subspectra Z . In this note we reformulate and extend the above result. We construct a symmetric monoidal functor F from the category of E∨ ∗ (E)–precomodules to the ca...

Isomorphisms are constructed between generalized Schur algebras in different degrees. The construction covers both the classical case (of general linear groups over infinite fields of arbitrary characteristic) and the quantized case (in type A, for any non-zero value of the quantum parameter q). The construction does not depend on the characteristic of the underlying field or the choice of q 6=...

We give a detailed comparison between the notion of a weak Hopf algebra (also called a quantum groupoid by Nikshych and Văınerman), and that of a ×R-bialgebra due to Takeuchi (and also called a bialgebroid or quantum (semi)groupoid by Lu and Xu). A weak bialgebra is the same thing as a ×R-bialgebra in which R is Frobenius-separable. We extend the comparison to cover module and comodule theory, ...

We apply group cohomological methods to calculate the cohomol-ogy of K(n) BP as a K(n) K(n)-comodule, recovering recent results of Hovey and Sadofsky. As applications we determine the Chromatic Spectral Sequence for BP based on Johnson and Wilson's E(n), showing the relationship to some generalizations of the classical Hattori-Stong Theorem and determine the change of Hopf algebroid spectral se...

Γ(m + 1) = BP∗(BP )/ (t1, . . . , tm) ∼= BP∗[tm+1, tm+2, . . . ]. In particular Γ(1) = BP∗(BP ) by definition. To get the structure of ExtΓ(m+1)(BP∗, BP∗), we will use the chromatic method introduced in [3]. Denote an ideal (p, v1, · · · vn−1) of BP∗ by In, and a comodule v−1 n+sBP∗/(p, v1, · · · , vn−1, v∞ n , · · · , v∞ n+s−1). by M s n. Then we can consider the chromatic spectral sequence co...