Let G be an affine group scheme of finite type over a global field F . (We do not assume G to be reductive or smooth or connected.) Let AF denote the locally compact adele ring of F , S be a finite non-empty set of places of F that contains the archimedean places, and AF the factor ring of adeles with vanishing component along S; for FS = ∏ v∈S Fv, we have AF = FS ×AF . Recall that if X is a fi...

We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types. Under the specialization of q and coefficients to 1, these bases are generic bases of finite and affine cluster algebras.

Let G be a connected linear algebraic group over an algebraically closed field of characteristic zero. Then the Brauer group of G is shown to be C X (Q/Z)<n) where C is finite and n = d(d l)/2, with d the Z-Ta.nk of the character group of G. In particular, a linear torus of dimension d has Brauer group (Q/Z)(n) with n as above. In [6], B. Iversen calculated the Brauer group of a connected, char...

‎let $g$ be a group and $a=aut(g)$ be the group of automorphisms of‎ ‎$g$‎. ‎then the element $[g,alpha]=g^{-1}alpha(g)$ is an‎ ‎autocommutator of $gin g$ and $alphain a$‎. ‎also‎, ‎the‎ autocommutator subgroup of g is defined to be‎ ‎$k(g)=langle[g,alpha]|gin g‎, ‎alphain arangle$‎, ‎which is a‎ ‎characteristic subgroup of $g$ containing the derived subgroup‎ ‎$g'$ of $g$‎. ‎a group is defined...

in this paper we find systems of subgroups of a finite‎ ‎group‎, ‎which $bbb p$-subnormality guarantees supersolvability‎ ‎of the whole group‎.

It is known that there is a only one invariant metric, and only one family of invariant affine connections on the the space of probability measures on finite sample spaces. Under certain conditions this may also true for finite measures on finite sample spaces. We extends these results to finite measures on arbitrary sample spaces. It is shown that the generalization of these structures are uni...

1 Affine Morphisms Definition 1. Let f : X −→ Y be a morphism of schemes. Then we say f is an affine morphism or that X is affine over Y , if there is a nonempty open cover {Vα}α∈Λ of Y by open affine subsets Vα such that for every α, fVα is also affine. If X is empty (in particular if Y is empty) then f is affine. Any morphism of affine schemes is affine. Any isomorphism is affine, and the aff...