The objects of study in this paper are Hopf algebras H which finitely generated over an algebraically closed field and extensions a commutative algebra A by finite dimensional H?:=H/A+H, where A+ is the augmentation ideal A. Basic structural homological properties recalled classes examples listed. structure theorem proved when (?) H? semisimple cosemisimple, showing that case noncommutativity a...

We define a topological space for the complexes of fixed dimension vector over the derived category of a finite dimensional algebra with finite global dimension. It has a local structure of affine variety under the action of the algebraic group. We consider the constructible functions which reflect the stratifications of the orbit spaces. By using the convolution rule, we obtain the geometric r...

abstract in this paper, we study the generalized order- jacobsthal sequences modulo for and the generalized order-k jacobsthal-padovan sequence modulo for . also, we define the generalized order-k jacobsthal orbit of a k-generator group for and the generalized order-k jacobsthal-padovan orbit a k-generator group for . furthermore, we obtain the lengths of the periods of the generalized order-3 ...

Given a Henselian and Japanese discrete valuation ring $A$ flat projective $A$-scheme $X$, we follow the approach of Biswas-dos Santos to introduce full subcategory coherent modules on $X$ which is then shown be Tannakian. We prove that, under normality generic fibre, associated affine group pro-finite in strong sense (so that its functions Mittag-Leffler $A$-module) it classifies finite torsor...

We consider a Deligne–Mumford stack $X$ which is the quotient of an affine scheme $\operatorname {Spec}A$ by action finite group $G$ and show that Balmer spectrum tensor triangulated category perfect complexes on homeomorphic to space homogeneous prime ideals in cohomology ring $H^*(G,A)$ .

Dynamical systems on an infinite translation surface with the lattice property are studied. The geodesic flow on this surface is found to be recurrent in all but countably many rational directions. Hyperbolic elements of the affine automorphism group are found to be nonrecurrent, and a precise formula regarding their action on cylinders is proven. A deformation of the surface in the space of tr...

For fixed g and T we show the finiteness of the set of affine equivalence classes of flat surfaces of genus g whose Veech group contains a cusp of hyperbolic co-area less than T . We obtain new restrictions on Veech groups: we show that any nonelementary Fuchsian group can appear only finitely many times in a fixed stratum, that any non-elementary Veech group is of finite index in its normalize...

Recall that an algebraic group G is called (linearly) reductive if any its rational (i.e., algebraic) representation is completely reducible. The finite groups, the group GLn and the products GLn1 × . . .GLnk are reductive. Below G denotes a reductive algebraic group and X is an affine algebraic variety equipped with an (algebraic) action of G. Results explained below in this section can be fou...